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Introduction to the viewpoint set on creep cavitation
Scripta
METALLURGICA
V o l . 17, pp. Printed in
1 - 4, 1 9 8 3 the U.S.A.
Pergamon P r e s s Ltd. All rights reserved
VIEWPOINT
INTRODUCTION
TO THE VIEWPOINT
SET
No.
5
SET ON CREEP CAVITATION
W. D. Nix Department of Materials Science and Engineering Stanford University, Stanford, California 94305 (Received
November
15,
1982)
It is well known that creep cavitation occurs by the nucleation, growth and coalescence of intergranular cavities. These cavities form microcracks which propagate through the material to cause failure. Recent research on creep cavitation has greatly improved our understanding of these processes, but there is still considerable debate about the relative importance of the various mechanisms involved. The viewpoints expressed here have been brought together to report on progress in this field and to stimulate further thought and research on these topics. As an introduction to this Viewpoint processes involved in creep cavitation. Nucleation
Set we will review briefly some of the physical
and Early Growth of Cavities
It is generally necessary to nucleate cavities to initiate fracture. Some materials may contain pre-existing voids left over from manufacturing processes or. as discussed by Shewmon, et al. (i), produced during previous phase transformations. In these cases, cavity nucleation may not be required. However, many metals appear to be fully dense; these metals fail by cavitation only after nucleation has occurred. It is usually assumed that cavity nucleation involves the clustering of vacancies on interfaces subjected to high tensile stresses (2,3). The analysis of this problem indicates that extremely high normal stresses are required to nucleate a cavity; typically stresses of the order of E/100 are needed. From this it is clear that the applied stress alone is insufficient to nucleate cavities and that very large stress concentrations are required. Cavity nucleation could start without the clustering of vacancies by thermally activated bond breaking, but this, too, requires very large tensile stresses. Large stress concentrations can be produced at particles in sliding boundaries, at the intersections of slip bands and grain boundary particles and, perhaps, at grain boundary triple junctions. However, the ease with which cavities can be produced by these processes has been greatly over estimated in the past. Early models envisioned such processes as grain boundary sliding with elastic accommodation at the particles in the boundary. Recent studies by Argon et al. (3) show that the stress concentrations are greatly reduced by both power law creep (plasticity) and diffusional processes. When proper account is taken of these relaxation processes, the conditions for cavity nucleation are not easily achieved. This explains why large numbers of cavities are not formed at the beginning of a creep test when the load is applied. It is now well documented that cavities form continuously throughout the creep process. Attention has been drawn to this fact by Argon (4) who concluded that boundary sliding leading to cavity nucleation must be, like creep itself, a stochastic process with localized sliding displacements occurring irregularly throughout the creep process. Cavity nucleation occurs only when a large boundary segment containing a small number of particles slides rapidly. Such events are presumably related to the stochastic hardening and recovery processes that occur within the matrix. Dyson (5) has argued that the density of grain boundary particles in engineering materials is so large that cavity nucleation will not occur by boundary sliding. Rather, he suggests that cavities nucleate when slip bands intersect with grain boundary particles. However this also requires very high normal stresses on the particle/matrix interface, even if nucleation does not involve vacancy clustering. This, too, is a stochastic process as it relates directly to the stochastic processes of slip creep. Whether
cavities
form by localized
grain boundary sliding or by localized slip,
1 0036-9748/83/010001-04503.00/0 Copyright (c) 1 9 8 3 P e r g a m o n Press
Ltd.
the cavity
2
INTRODUCTION
TO VIEWPOINT
SET
Vol.
17,
No.
1
generated must be large enough to grow when the stochastic event is over (4,5). Following Dyson (5) we can assume that the cavity formed must be sufficiently large that it does not shrink due to capillarity forces. For the case of a particle of diameter a in a boundary which slides a distance of 6 in a short period of time, the cavity formed would have a volume of about a26, or after surface diffusion, a radius of about (a2~) I/3. Taking the capillarity stress to be about 2"(s/r , it follows that unstable cavity growth will occur only if the sliding displacement exceeds a critical value 2Ys 3 1 6 > ~2 (--~--) where o is the applied stress. Otherwise capillarity forces would gradually cause the cavity to sinter by diffusion. This simple analysis shows that although high normal stresses are necessary for cavity nucleation, they are not sufficient. Stable cavities form only if relatively large sliding displacements occur. All of the above indicates that cavity formation and early growth is intimately linked to the creep process itself. This is most likely why the fracture time correlates so well with the minimum creep rate for so many structural materials (the Monkman-Grant relation). Cavity Growth There are two limiting kinds of cavity growth that occur under creep conditions. They have been called unconstrained and constrained cavity growth, respectively. In the case of unconstrained cavity growth, cavities are present on all of the grain boundaries in the solid and are free to grow to the point of complete failure. In the case of constrained cavity growth, cavities are present only on isolated boundaries. Here cavity growth on the eavitated boundary can proceed only if the surrounding matrix creeps, as the relative grain displacements associated with cavitation have to be accommodated by corresponding displacements in the matrix. Thus, in this case, cavity growth may be limited entirely by creep flow of the matrix. Unconstrained Cavity Growth A great amount of attention has been devoted to the subject of intergranular cavity growth, with the result that growth processes are very much better understood than the processes of nucleation. When pre-existing cavities are present, cavity growth can be the controlling process for fracture. However, as discussed by Goods and Nieh (6), when cavities are artificially introduced into metals by chemical means the fracture time is greatly reduced, suggesting that the time for cavity formation is much greater than that for growth. This, in turn, indicates that cavity growth is not usually the limiting factor in creep rupture. Nevertheless, cavity growth is an essential part of the creep cavitation process and it deserves the attention it has received. The first important treatment of cavity growth was given by Hull and Rimmer (7) who showed the volumetric growth of a grain boundary cavity should be limited by diffusion in the adjoining grain boundary. Many corrections and modifications of this analysis have been made, with the recent treatment of Speight and Beere (8) being regarded as the most complete. A key feature of this analysis is the linear stress dependence of the growth rate. Many authors have noted that if fracture were limited by Hull-Rimmer growth, the rupture time would vary inversely with the stress. There is almost no evidence to support this prediction. Fracture times invariably depend on ~-n where n is greater than one, even if care is taken to avoid nucleation by implanting pre-existing cavities. Thus, the Hull-Rimmer model fails to describe the fracture process. However, density change measurements appear to provide strong support for the Hull-Rimmer model. Miller and Langdon (9) have reviewed the literature on density measurements of cavitation and have concluded that if proper account is taken of the formation of new cavities in the course of creep, the growth rate depends linearly on the stress. Hanna and Greenwood (i0) have recently measured both density changes and fracture times for copper containing pre-existing voids. At low stresses their density change measurements show a linear stress dependence, whereas the fracture time depends inversely on the third power of the stress. Thus the experimental data appear to be contradictory. That this is not so is evident from the following arguments. Density change measurements and fracture time measurements are not equivalent. All cavities within the solid contribute to the density changes whereas only a very small fraction of the total population are responsible for fracture. Typically fracture occurs when the larger, more closely spaced cavities grow together. It is possible that the physical mechanisms which limit their growth may not be the
Vol.
17,
No.
1
INTRODUCTION
TO
VIEWPOINT
SET
3
same as those that limit the growth of the smaller cavities. Also, the volumetric growth of a cavity does not uniquely determine its tip velocity. The extent of cavity growth in the plane of the grain boundary for a given volume change depends on the shape of the cavity, which may change in the course of growth. It is now well known that the shape depends mainly on the relative rates of surface and grain boundary diffusion (ii). The importance of surface diffusion in cavity growth was overlooked for a long time. It was assumed implicitly that surface diffusion is very fast and that the cavity retains its equilibrium shape as it grows. Chuang et al. (12) have shown that when the rate of surface diffusion is finite, the cavity naturally becomes crack-like as it grows. When surface diffusion is much slower than boundary diffusion the velocity of the crack tip varies as DsO3 where D s is the surface self diffusivity. This suggests that the rupture time should vary as 0 -3 . As noted by Goods and Nieh (6) this kind of stress dependence has been observed for Cu, Ag, Fe and Ni under a variety of different conditions. This strongly suggests that the growth of those cavities leading to fracture is limited primarily by surface diffusion. As noted above, this is not in conflict with the observation that the volumetric growth of the entire cavity population depends linearly on stress. Although cavity growth appears to be controlled by surface diffusion, it is hard to understand why surface diffusion should be so much slower than grain boundary diffusion. For most metals the surface diffusivity is greater than the boundary diffusivity. Chen (13) has recently suggested that crack-like cavity growth controlled by surface diffusion may be caused by grain boundary sliding. It is easy to see that boundary sliding changes the shape of the cavity and that this prompts surface diffusion to occur (14). It is less clear how this would affect the growth of the cavity since surface diffusion to restore the shape does not change the volume of the cavity. However, Chen has argued that sharpening of the cavity tip by sliding causes the tip velocity during growth to be limited by surface diffusion, even if the surface diffusivity is greater than the boundary diffusivity. The resulting cavity growth law has the same form as that given by Chuang and Rice (ii). It is now recognized that the diffusional processes described above can be affected by power law creep of the surrounding matrix. A number of authors have shown that power law creep reduces the necessary diffusion distance in the grain boundary and thus speeds up the rate of cavity growth (13, 15-19). This effect becomes important when the cavities are large and widely spaced. When such cavity microstructures are artificially produced, the coupling of power law creep with diffusive cavity growth is necessary to describe the failure process (20). Very small cavities are not typically limited by creep flow because it takes so little creep flow to maintain compatibility with a growing cavity. Power law creep processes are also of importance in the final stages of cavity growth when the cavities grow together (coalescence) (21), but this requires so little time that it is of minor importance in the overall creep fracture process. Constrained
Cavity Growth
When cavities are present on isolated boundaries their growth may be constrained by creep flow of the surrounding grains. This situation was discussed first by Dyson (22) and later treated by Rice (23) and Raj and Ghosh (24). A new treatment of this is given by Chen (13) in the present volume. As discussed above, cavitation of an isolated boundary can occur only if the relative grain displacements associated with cavitation are accommodated by creep flow in the surrounding grains. When cavitation is much faster than creep, then a redistribution of stress occurs within the solid causing the load to be shed by the boundary and supported by the surrounding material. Cavity Coalescence
and the Final Stages of Creep Fracture
The final stages of creep fracture involve the coalescence of cavities to form microcracks and the interlinkage of these microcracks to cause failure. These processes are not well understood. In most models fracture is assumed to occur as soon as the cavities start to coalesce. If the boundaries were uniformly cavitated at the beginning of the creep process, as in the case of metals containing gas bubbles, this might be a reasonably good approximation. However, when the cavities form naturally they are distributed very inhomogeneously on the boundaries, with the consequence that cavitation on some boundaries reaches the point of
4
INTRODUCTION
coalescence before it has started plicated by the redistribution of consequence of this, entire grain A coherent picture of these final
TO VIEWPOINT
SET
Vol.
17,
No.
on others. The final stages of fracture are further comstress that occurs during inhomogeneous cavitation. As a boundary facets can separate long before fracture occurs. stages of fracture has not yet emerged. Acknowledgement
This work was supported by the Division of Materials Science, Office of Basic Energy Sciences of the United States Department of Energy under Contract No. DE-AT03-79-ERI0378. References i. 2. 3. 4.
5. 6. 7. 8. 9. I0. ii. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
P. G. Shewmon, H. Lopez and T. A. Parthasarathy, Scripta Met. 17, 39 (1983). R. Raj, Acta Met. 26, 995 (1978). A. S. Argon, I-W. Chen and C. W. Lau, in Creep-Fatigue-Environment Interactions, R.M. Pelloux and N.S. Stoloff (Eds.) AIME, New York, 1980, p. 46. A. S. Argon, in Recent Advances in Creep and Fracture of Engineering Materials and Structures, B. Wilshire and D.R.J. Owen (Eds.), Pineridge Press, Swansea U.K. (1982) p.l. B. F. Dyson, Scripta Met. 17, 31 (1983). S. H. Goods and T. G. Nieh, Scripta Met. 17, 23 (1983). D. Hull and D. E. Rimmer, Phil. Mag. 4, 673 (1959). M. V. Speight and W. Beere, Met. Sci. 9, 190 (1975). D. A. Miller and T. G. Langdon, Metall. Trans. IIA, 955 (1980). M. D. Hanna and G. W. Greenwood, Acta Met. 30, 719 (1982). T. J. Chuang and J. R. Rice, Acta Met. 21, 1625 (1973). T. J. Chuang, K. T. Kagawa, J. R. Rice and L. B. Sills, Acta Met. 27, 265 (1979). I-W. Chen, Scripta Met. 17, 17 (1983). W. D. Nix, K. S. Yu and J. S. Wang, to be published in Met. Trans. W. Beere and M. V. Speight, Met. Sci. 12, 172 (1978). G. H. Edward and M. F. Ashby, Acta Met. 27, 1505 (1979). A. Needleman and J. R. Rice, Acta Met. 28, 1315 (1980). I-W. Chen and A. S. Argon, Acta Met. 29, 1759 (1981). W. Beere, Scripta Met. 17, 13 (1983). J. S. Wang, L. Martinez and W. D. Nix, unpublished research. S. H. Goods and W. D. Nix, Acta Met. 26, 753 (1978). B. F. Dyson, Met. Sci. iO, 349 (1976). J. R. Rice, Acta Met. 29, 675 (1981). R. Raj and A. K. Ghosh, Met. Trans. 12A, 1291 (1981).
i
METALLURGICA
V o l . 17, pp. Printed in
1 - 4, 1 9 8 3 the U.S.A.
Pergamon P r e s s Ltd. All rights reserved
VIEWPOINT
INTRODUCTION
TO THE VIEWPOINT
SET
No.
5
SET ON CREEP CAVITATION
W. D. Nix Department of Materials Science and Engineering Stanford University, Stanford, California 94305 (Received
November
15,
1982)
It is well known that creep cavitation occurs by the nucleation, growth and coalescence of intergranular cavities. These cavities form microcracks which propagate through the material to cause failure. Recent research on creep cavitation has greatly improved our understanding of these processes, but there is still considerable debate about the relative importance of the various mechanisms involved. The viewpoints expressed here have been brought together to report on progress in this field and to stimulate further thought and research on these topics. As an introduction to this Viewpoint processes involved in creep cavitation. Nucleation
Set we will review briefly some of the physical
and Early Growth of Cavities
It is generally necessary to nucleate cavities to initiate fracture. Some materials may contain pre-existing voids left over from manufacturing processes or. as discussed by Shewmon, et al. (i), produced during previous phase transformations. In these cases, cavity nucleation may not be required. However, many metals appear to be fully dense; these metals fail by cavitation only after nucleation has occurred. It is usually assumed that cavity nucleation involves the clustering of vacancies on interfaces subjected to high tensile stresses (2,3). The analysis of this problem indicates that extremely high normal stresses are required to nucleate a cavity; typically stresses of the order of E/100 are needed. From this it is clear that the applied stress alone is insufficient to nucleate cavities and that very large stress concentrations are required. Cavity nucleation could start without the clustering of vacancies by thermally activated bond breaking, but this, too, requires very large tensile stresses. Large stress concentrations can be produced at particles in sliding boundaries, at the intersections of slip bands and grain boundary particles and, perhaps, at grain boundary triple junctions. However, the ease with which cavities can be produced by these processes has been greatly over estimated in the past. Early models envisioned such processes as grain boundary sliding with elastic accommodation at the particles in the boundary. Recent studies by Argon et al. (3) show that the stress concentrations are greatly reduced by both power law creep (plasticity) and diffusional processes. When proper account is taken of these relaxation processes, the conditions for cavity nucleation are not easily achieved. This explains why large numbers of cavities are not formed at the beginning of a creep test when the load is applied. It is now well documented that cavities form continuously throughout the creep process. Attention has been drawn to this fact by Argon (4) who concluded that boundary sliding leading to cavity nucleation must be, like creep itself, a stochastic process with localized sliding displacements occurring irregularly throughout the creep process. Cavity nucleation occurs only when a large boundary segment containing a small number of particles slides rapidly. Such events are presumably related to the stochastic hardening and recovery processes that occur within the matrix. Dyson (5) has argued that the density of grain boundary particles in engineering materials is so large that cavity nucleation will not occur by boundary sliding. Rather, he suggests that cavities nucleate when slip bands intersect with grain boundary particles. However this also requires very high normal stresses on the particle/matrix interface, even if nucleation does not involve vacancy clustering. This, too, is a stochastic process as it relates directly to the stochastic processes of slip creep. Whether
cavities
form by localized
grain boundary sliding or by localized slip,
1 0036-9748/83/010001-04503.00/0 Copyright (c) 1 9 8 3 P e r g a m o n Press
Ltd.
the cavity
2
INTRODUCTION
TO VIEWPOINT
SET
Vol.
17,
No.
1
generated must be large enough to grow when the stochastic event is over (4,5). Following Dyson (5) we can assume that the cavity formed must be sufficiently large that it does not shrink due to capillarity forces. For the case of a particle of diameter a in a boundary which slides a distance of 6 in a short period of time, the cavity formed would have a volume of about a26, or after surface diffusion, a radius of about (a2~) I/3. Taking the capillarity stress to be about 2"(s/r , it follows that unstable cavity growth will occur only if the sliding displacement exceeds a critical value 2Ys 3 1 6 > ~2 (--~--) where o is the applied stress. Otherwise capillarity forces would gradually cause the cavity to sinter by diffusion. This simple analysis shows that although high normal stresses are necessary for cavity nucleation, they are not sufficient. Stable cavities form only if relatively large sliding displacements occur. All of the above indicates that cavity formation and early growth is intimately linked to the creep process itself. This is most likely why the fracture time correlates so well with the minimum creep rate for so many structural materials (the Monkman-Grant relation). Cavity Growth There are two limiting kinds of cavity growth that occur under creep conditions. They have been called unconstrained and constrained cavity growth, respectively. In the case of unconstrained cavity growth, cavities are present on all of the grain boundaries in the solid and are free to grow to the point of complete failure. In the case of constrained cavity growth, cavities are present only on isolated boundaries. Here cavity growth on the eavitated boundary can proceed only if the surrounding matrix creeps, as the relative grain displacements associated with cavitation have to be accommodated by corresponding displacements in the matrix. Thus, in this case, cavity growth may be limited entirely by creep flow of the matrix. Unconstrained Cavity Growth A great amount of attention has been devoted to the subject of intergranular cavity growth, with the result that growth processes are very much better understood than the processes of nucleation. When pre-existing cavities are present, cavity growth can be the controlling process for fracture. However, as discussed by Goods and Nieh (6), when cavities are artificially introduced into metals by chemical means the fracture time is greatly reduced, suggesting that the time for cavity formation is much greater than that for growth. This, in turn, indicates that cavity growth is not usually the limiting factor in creep rupture. Nevertheless, cavity growth is an essential part of the creep cavitation process and it deserves the attention it has received. The first important treatment of cavity growth was given by Hull and Rimmer (7) who showed the volumetric growth of a grain boundary cavity should be limited by diffusion in the adjoining grain boundary. Many corrections and modifications of this analysis have been made, with the recent treatment of Speight and Beere (8) being regarded as the most complete. A key feature of this analysis is the linear stress dependence of the growth rate. Many authors have noted that if fracture were limited by Hull-Rimmer growth, the rupture time would vary inversely with the stress. There is almost no evidence to support this prediction. Fracture times invariably depend on ~-n where n is greater than one, even if care is taken to avoid nucleation by implanting pre-existing cavities. Thus, the Hull-Rimmer model fails to describe the fracture process. However, density change measurements appear to provide strong support for the Hull-Rimmer model. Miller and Langdon (9) have reviewed the literature on density measurements of cavitation and have concluded that if proper account is taken of the formation of new cavities in the course of creep, the growth rate depends linearly on the stress. Hanna and Greenwood (i0) have recently measured both density changes and fracture times for copper containing pre-existing voids. At low stresses their density change measurements show a linear stress dependence, whereas the fracture time depends inversely on the third power of the stress. Thus the experimental data appear to be contradictory. That this is not so is evident from the following arguments. Density change measurements and fracture time measurements are not equivalent. All cavities within the solid contribute to the density changes whereas only a very small fraction of the total population are responsible for fracture. Typically fracture occurs when the larger, more closely spaced cavities grow together. It is possible that the physical mechanisms which limit their growth may not be the
Vol.
17,
No.
1
INTRODUCTION
TO
VIEWPOINT
SET
3
same as those that limit the growth of the smaller cavities. Also, the volumetric growth of a cavity does not uniquely determine its tip velocity. The extent of cavity growth in the plane of the grain boundary for a given volume change depends on the shape of the cavity, which may change in the course of growth. It is now well known that the shape depends mainly on the relative rates of surface and grain boundary diffusion (ii). The importance of surface diffusion in cavity growth was overlooked for a long time. It was assumed implicitly that surface diffusion is very fast and that the cavity retains its equilibrium shape as it grows. Chuang et al. (12) have shown that when the rate of surface diffusion is finite, the cavity naturally becomes crack-like as it grows. When surface diffusion is much slower than boundary diffusion the velocity of the crack tip varies as DsO3 where D s is the surface self diffusivity. This suggests that the rupture time should vary as 0 -3 . As noted by Goods and Nieh (6) this kind of stress dependence has been observed for Cu, Ag, Fe and Ni under a variety of different conditions. This strongly suggests that the growth of those cavities leading to fracture is limited primarily by surface diffusion. As noted above, this is not in conflict with the observation that the volumetric growth of the entire cavity population depends linearly on stress. Although cavity growth appears to be controlled by surface diffusion, it is hard to understand why surface diffusion should be so much slower than grain boundary diffusion. For most metals the surface diffusivity is greater than the boundary diffusivity. Chen (13) has recently suggested that crack-like cavity growth controlled by surface diffusion may be caused by grain boundary sliding. It is easy to see that boundary sliding changes the shape of the cavity and that this prompts surface diffusion to occur (14). It is less clear how this would affect the growth of the cavity since surface diffusion to restore the shape does not change the volume of the cavity. However, Chen has argued that sharpening of the cavity tip by sliding causes the tip velocity during growth to be limited by surface diffusion, even if the surface diffusivity is greater than the boundary diffusivity. The resulting cavity growth law has the same form as that given by Chuang and Rice (ii). It is now recognized that the diffusional processes described above can be affected by power law creep of the surrounding matrix. A number of authors have shown that power law creep reduces the necessary diffusion distance in the grain boundary and thus speeds up the rate of cavity growth (13, 15-19). This effect becomes important when the cavities are large and widely spaced. When such cavity microstructures are artificially produced, the coupling of power law creep with diffusive cavity growth is necessary to describe the failure process (20). Very small cavities are not typically limited by creep flow because it takes so little creep flow to maintain compatibility with a growing cavity. Power law creep processes are also of importance in the final stages of cavity growth when the cavities grow together (coalescence) (21), but this requires so little time that it is of minor importance in the overall creep fracture process. Constrained
Cavity Growth
When cavities are present on isolated boundaries their growth may be constrained by creep flow of the surrounding grains. This situation was discussed first by Dyson (22) and later treated by Rice (23) and Raj and Ghosh (24). A new treatment of this is given by Chen (13) in the present volume. As discussed above, cavitation of an isolated boundary can occur only if the relative grain displacements associated with cavitation are accommodated by creep flow in the surrounding grains. When cavitation is much faster than creep, then a redistribution of stress occurs within the solid causing the load to be shed by the boundary and supported by the surrounding material. Cavity Coalescence
and the Final Stages of Creep Fracture
The final stages of creep fracture involve the coalescence of cavities to form microcracks and the interlinkage of these microcracks to cause failure. These processes are not well understood. In most models fracture is assumed to occur as soon as the cavities start to coalesce. If the boundaries were uniformly cavitated at the beginning of the creep process, as in the case of metals containing gas bubbles, this might be a reasonably good approximation. However, when the cavities form naturally they are distributed very inhomogeneously on the boundaries, with the consequence that cavitation on some boundaries reaches the point of
4
INTRODUCTION
coalescence before it has started plicated by the redistribution of consequence of this, entire grain A coherent picture of these final
TO VIEWPOINT
SET
Vol.
17,
No.
on others. The final stages of fracture are further comstress that occurs during inhomogeneous cavitation. As a boundary facets can separate long before fracture occurs. stages of fracture has not yet emerged. Acknowledgement
This work was supported by the Division of Materials Science, Office of Basic Energy Sciences of the United States Department of Energy under Contract No. DE-AT03-79-ERI0378. References i. 2. 3. 4.
5. 6. 7. 8. 9. I0. ii. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
P. G. Shewmon, H. Lopez and T. A. Parthasarathy, Scripta Met. 17, 39 (1983). R. Raj, Acta Met. 26, 995 (1978). A. S. Argon, I-W. Chen and C. W. Lau, in Creep-Fatigue-Environment Interactions, R.M. Pelloux and N.S. Stoloff (Eds.) AIME, New York, 1980, p. 46. A. S. Argon, in Recent Advances in Creep and Fracture of Engineering Materials and Structures, B. Wilshire and D.R.J. Owen (Eds.), Pineridge Press, Swansea U.K. (1982) p.l. B. F. Dyson, Scripta Met. 17, 31 (1983). S. H. Goods and T. G. Nieh, Scripta Met. 17, 23 (1983). D. Hull and D. E. Rimmer, Phil. Mag. 4, 673 (1959). M. V. Speight and W. Beere, Met. Sci. 9, 190 (1975). D. A. Miller and T. G. Langdon, Metall. Trans. IIA, 955 (1980). M. D. Hanna and G. W. Greenwood, Acta Met. 30, 719 (1982). T. J. Chuang and J. R. Rice, Acta Met. 21, 1625 (1973). T. J. Chuang, K. T. Kagawa, J. R. Rice and L. B. Sills, Acta Met. 27, 265 (1979). I-W. Chen, Scripta Met. 17, 17 (1983). W. D. Nix, K. S. Yu and J. S. Wang, to be published in Met. Trans. W. Beere and M. V. Speight, Met. Sci. 12, 172 (1978). G. H. Edward and M. F. Ashby, Acta Met. 27, 1505 (1979). A. Needleman and J. R. Rice, Acta Met. 28, 1315 (1980). I-W. Chen and A. S. Argon, Acta Met. 29, 1759 (1981). W. Beere, Scripta Met. 17, 13 (1983). J. S. Wang, L. Martinez and W. D. Nix, unpublished research. S. H. Goods and W. D. Nix, Acta Met. 26, 753 (1978). B. F. Dyson, Met. Sci. iO, 349 (1976). J. R. Rice, Acta Met. 29, 675 (1981). R. Raj and A. K. Ghosh, Met. Trans. 12A, 1291 (1981).
i