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Mechanisms of cavity growth in creep
Scripta
METALLURGICA
V o l . 17, pp. Printed in
MECHANISMS
17-22, 1983 the U.S.A.
Pergamon
Press
VIEWPOINT
SET
Ltd.
No.
5
OF CAVITY GROWTH IN CREEP*
l-Wei Chen t Metals and Ceramics Division Oak Ridge National Laboratory Oak Ridge, Tennessee 37830 (Received
September
9,
1982)
Introduction The growth of intergranular cavities under creep conditions is of considerable technological interest. It is well-documented in many engineering and model materials and is the subject of numerous investigations. However, the phenomenon is a complex one, which is to be expected on several accounts. First, kinetic and mechanical processes at elevated temperature are many, e.g. lattice diffusion, grain-boundary diffusion and surface diffusion of the former category and dislocation creep, diffusional creep and graln-boundary sliding of the latter. Second, the size distribution of cavities, being a function of time, varies from one grainboundary to the other due to the heterogeneous and continuous nucleation of new cavities. Third, the orientation and the surroundings of each grain-boundary is different, giving rise to a broad spectrum of growth conditions of different mechanical descriptions. These considerations, when taken into account in toto, result in an almost infinite number of cases which are too n~nerous to analyze deterministically. For a mechanistic understanding, certain idealizations have to be made. This paper attempts to give an up-to-date account of such understanding, with the necessary idealization, and to point out the deficiencies in the simplified picture in each case. As an outline, we pose the following three problems in the order of increasing complexity. The simplest case, on which a large amount of work has been done to date, essentially pertains to cavitation on the transverse grain-boundary in a bicrystal under a normal stress. Since many mechanisms are possible at elevated temperature, even this simplest case is not trivial and was solved in a satisfactory way only recently. The second idealized case deals with cavitation on transverse boundaries in a polycrystal. Currently, only the limiting case in which all transverse boundaries cavitate has been treated. While an alternative solution of the above problem is given in the paper, attention must be drawn to the more general problem which have been hitherto overlooked. The third case deals with inclined boundaries when the additional component of grain-boundary sliding sometimes causes "anomalous" effects. Since in reality cavitation also occurs on such boundaries, the problem warrants attention as well. These three problems are treated briefly in the following. Cavity Growth in a Bicrystal Under Normal Loading (Fig. i) The early model of cavity growth, of Hull and Rimmer (i), considered an array of spherical cavities which enlarge by absorbing vacancies from the adjoining grain-boundaries. The reverse
*Research sponsored by the Division of Materials Sciences, U.S. Department of Energy, under contract W-7405-eng-26 with the Union Carbide Corporation. ~Permanent address: Department of Nuclear Engineering and Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139.
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flow of vacancies plates out a layer of atoms onto the grain-boundary, causing elongation normal to the plane (Fig. la). Obviously the process strongly favors those grain-boundaries transverse to the applied tensile stress and, indeed, in this important respect the model captures one of the most significant characteristics of cavity growth, i.e. micrographically most cavities are found on transverse boundaries. The morphology of cavities, especially those revealed by direct observations of the post-creep intergranular fracture surfaces (2), of pure metals and engineering materials alike, exhibits many features which are consistent with the diffusive mechanism. The above model, which, strictly speaking is only applicable to a bicrystal under normal tension, has been extended to treat the following general problem. An array of identical cavities of radius a at a regular Imlf spacing b, under a normal stress c~ which causes a uniform distant creep rate ~ , grows at ~ while maintaining a shape to be determined by the combined action of surface and grain-boundary diffusion. The analytic form of solution of Chen and Argon (3) is shown in Fig. 2. Here h(@) is the ratio of the volume of an equilibrium-shaped cavity to a sphere of the stone radius, and is only a function of the dihedral angle ~, while the rest of the symbols are the ones commonly used in the field. In the following, we describe how the problem was solved. Three important developments of previous investigators contribute to the final formulation of the above result. The Hull and Rimmer solution, later revised by Speight and Beere (4), is for the case of equilibrium cavities (Fig. la) between ri$id grains. The corresponding solution for the case of non-equilibrium cavities (Fig. ib) between rigid grains was due to Chuang et al. (5). In both cases, cavities have to communicate diffusionally as they grow, to lay a unifo~l matter deposition onto the grain-boundary, but the crack-like cavities grow faster and take over when the radius or tile dimensionless constant ~ (which is the ratio of grain boundary conductance to the surface conductance, essentially) becomes large. If the restriction of rigid grains is relaxed by matrix creep, the growth rate is enhanced as the required diffusion distance shortens (Fig. ic), according to Beere and Speight (6). Noting the analogy between fluid flow and the present problem, i.e. diffusionally expanding cavities in a flowing creep field, Chen and Argon (3) adopted the boundary layer approach which permits the approximate but straightforward solution of adequate accuracy by prescribing a boundary-layer thickness which may be determined by a reasonable dimensional argument. In the present case, the above thickness A~ which has the physical meaning of the diffusion distance, is found by compatibility which states that at the division line of the inner diffusion zone and the outer creep zone (Fig. Ic), the strain rate due to either mechanism must be comparable. With this method, the entire range of the solution, for equilibrium and crack-like cavities alike, in a creeping field of various strength, is mapped out in Fig. 2. The L-shaped branch on the left is for equilibrium cavities [which is in excellent agreement with the FEM numerical result (7)] while the V-shaped branches for crack-like cavities. In either case, creep flow becomes important as a/A increases. In fact, ignoring the V-shaped branches, we can verify that the growth rate of equilibrium-shaped cavities at large a/A eventually approaches that for a plastically expanding void in a creeping matrix (8). Meanwhile, at small creep rates, A approaches b and the growth rate is superceded by the Hull and Rimmer solution (4) given as the dotted lines on the left in Fig. 2. Thus the above solution incorporates all the essential variations of the general problem, including cavity shape, diffusion kinetics, and creep modifications. Two experimental observations of a very general nature support the above picture. First, for a typical material constant ~ = i0, a minimum in growth rate is expected near a ~ A. Since nucleation is continuous and often proceeds at a roughly constant rate, cavities will spend more time passing this size range where their number per unit size range increases. Measurements of cavity size distributions indeed confirm such a peak at the appropriate size range, and their profile suggests that some accelerated growth takes place for cavities of larger sizes (2). Furthermore, the same phase of slow growth will in turn contribute a ma~or portion to the time to fracture as well as creep ductility. The estimation of Chen and Argon (3) predicts a strain to failure between 0.05 and 0.5 with a gradual increase with the creep rate which ranges from 10 -7.5 to 10-4 per second, in good accord, both qualitatively and quantitatively, with the Monkman-Grant correlation which was also established in the same creep regime. We note that other attempts to explain the Monkman-Grant correlation based on the transition from diffusion control to creep control, for equilibrium-shaped cavities alone, were unsuccessful, predicting a precipitous drop of ductility from several hundred percent to roughly one percent at creep rates ~I0 -2 per second which is not observed experimentally.
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It is pleasing to see that the results developed from the rather drastic idealization of the bicrystal model are reasonable and useful. However, at very slow creep rate, the model breaks down as we shall see in the next section. Cavity Growth on Transverse Grain Boundaries in a Polycr~stal (Fi$. 3 ) Cavitation in a polycrystal is heterogeneous among grain-boundaries. Generally, transverse grain-boundaries cavitate more. The remaining ligaments between cavitated transverse boundaries exert upon the cavitating boundaries a mechanical constraint, which may be strong if the grains are nearly rigid~ and vanishingly small if grains creep easily. This is illustrated experimentally by Dyson (9--12), for a special case in which essentially all transverse grainboundaries are pre-cavitated before the test (Fig. 3a). He found the cavity growth rate in the material controlled by the macroscopic creep rate much slower than that expected in the bicrystal model. It is interesting to note that even in such heavily damaged material, grainboundary sliding alone cannot entirely relieve the mechanical constraint. Approximate analysis due to Dyson (9--10) and Rice (13) has been advanced. The analysis demonstrates that the thickening of the boundary due to cavitation unloads over the cavitating graln-boundary and overloads in the immediate vicinity. This "load shedding" toward the sides continues until the thickening rate under the reduced load is matched by the permissible opening rate of a penny-shaped crack carrying the same reduced normal traction which is significantly lower than the applied stress since the surrounding creep rate is low. The growth rate of cavities in this limit scales with the creep rate and, if it can be assumed (as Rice did) that isolated facet fracture is synomymous with overall fracture, then the strain to fracture is predicted to be approximately b/d, where d is the diameter of the grain-boundary facet. It can be deduced from the above that the constraint is more effective in a fine grain material, for the cavitational strain scales with b/d. The above analysis involves some geometrical and mechanical approximations which are difficult to assess without a comparison with the experiment. We give below a simple treatment which reproduces the experimental results satisfactorily. First divide the polycrystal into spherical cells each of which contains one transverse (cavitated) and on average two longitudinal (uncavitated) grain-boundaries. We then assign the same strain rate ~ to each cell. Meanwhile we recognize that the cell is encased in a much stiffer solid matrix in the immediate surroundings, which creeps at a strain rate ~= -- @c, where ~c is the strain rate due to cavitation which needs to be discounted for the solid matrix. Here the cavitational strain can be measured metallographically or by densitometry. Following Eshelby, the strain concentration in a soft spherical inclusion within a much stiffer matrix, as evaluated for creeping material by Chen and Argon (14), is 5/3. Hence,
~=o -- ~c
3
(i)
which is in numerical agreement with tile data of precavitated Nimonic 80 A giving ~ = 2.5 ~c (11--12). The cavitational strain rate can be evaluated geometrically by computing the total cavitation rate in each cell which contains one cavitating grain-boundary. We thus find
Sc
1 r 2 d (ha3) R 3 b 2 dt
(2)
where r and R are the radii of the transverse grain-boundary and the cell respectively. Finally noting, from quantitative metallography, that the linear intercept grain size L is given by [(2/~) (grain boundary area)/(unit volume)] -I, or L = 2~R3/9r 2
we obtain from above 3
(3)
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The above result compares well with Rice's analysis (13), but the more readily measurable quantity L is used instead of d. Despite the fact that the creep rate plays the key role in constrained growth, it must be recognized that physically cavitation is still diffusional. The low ductility and the high proportion of cavitatlonal strain in overall fracture, both characteristic of the Hull and Rimmer model, are common to the constrained growth in precavitated materials. In fact, if Eq. (4) is compared to the result in the Hull and Rimmer regime, on the left of the L-shaped branch in Fig. i, it is indistinguishable from the case without any constraint but with a much longer "constraint" diffusion distance, bc, approximately given by b = 2.1 exp(5~A3/6b2L)
(5)
which lengthens as the creep rate decreases. This is understandable, since the slow-down of the diffusive mechanism can be alternatively interpreted as the result of a reduced driving force (due to the constraint), an increased diffusion distance, or even a reduced dlffusivity while the basic features of diffusive growth remain unchanged. At this time, we wish to call attention to one serious deficiency in the present modelling of constrained growth. Unfortunately, Eq. (4) does not immediately lead to an estimation of time to fracture even though the cavity growth becomes constrained. This is so because overall fracture does not usually take place when the first set of cavitated boundaries separate, except in the heavily pre-damaged material (Fig. 3a). Instead, the damage will be incurred at this time on neighboring graln-boundaries and the damage zone spreads slowly (Fig. 3b). Not until a critical fraction of the material is damaged does the overall fracture occur. The critical fraction of damage, the spreading speed of the damage zone, and the initial damage are all material properties dependent on past history. Obviously the load shedding process will occur at every stage of the evolution, and the problem bears some resemblance to that of crack propagation in the context of stress concentrations and damage spreading. Since a very low ductility must be expected if constrained growth of the type Dyson studied (11--12) were the general case, the fact that many engineering alloys (such as austenitic stainless steel) do exhibit excellent ductility even at very low creep rate (~I0 -8 s-1) is a strong testimony that they must have accumulated their creep damage in such a gradual way. Hence, without a better understanding of the damage spreading process, Eq. (4) and the appreciation of constrained growth have no direct utility for predicting the failure time. Indeed, although we believe the constrained growth to be very important in engineering service at slow creep rate, it must be reminded that at stress concentrators such as notches, welds, and especially the crack front, the local creep rate may very well be sufficiently high to alleviate the constraint substantially. Clearly, all the above problems warrant more study in the future in order to furnish a better picture of the damage accumulation by cavitation. But next let us turn to the problem of grain-boundary sliding which plays, perhaps, an important role in the early stage and the last stage of cavity growth, hence a potentially crucial link in the damage process. Cavity Growth on Inclined Grain Boundaries (Fi~. 4) The obvious consequence of an inclined grain-boundary is graln-boundary sliding. Indeed, even for essentially transverse graln-boundaries, sliding may result due to compatibility. It has been proposed many times that sliding is responsible for cavity growth. This cannot be the case for cavities which are as small as a few Bm when sliding is much slower than 0. I um/s, for surface diffusion will restore any distortion due to sliding. In addition, many proposals entailed the use of the elasticity analogy which must fall at creep temperatures due to creep relaxation. A possible exception may come with coarse grain materials, with close spacing of cavities or at high strain rates. Good metallographic evidence indicating the importance of graln-boundary sliding and a clear demonstration of distorted cavities (schematically drawn in Fig. 4) were given in the study of type 304 stainless steel by Chen and Argon (2). In a recent analysis of this phenomenon, Chen (15) established a correlation between the sliding-controlled growth and the surface-controlled growth, which was found to be the case in H2O implanted Cu and Ag studied and analyzed by Goods, Nix and Nieh (16--18). The effect appears to be the result of a coupling between surface flux and the sliding-enabled matter transport, occurring on the two opposite halves of a cavity subject to heavy shearing. In this case, grain-boundary diffusion is not required to participate at all, yet the demand for surface diffusion is strong enough, due to grain-boundary sliding, to create a situation in which the matter transport is controlled by surface diffusion (5,19). In the above limit, a cavity is laterally spread out,
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while very little volumetric growth actually takes place. The growth rate also turns out to be identical to that given by Chuang et al. (5) for the case of surface diffusion control, although it arises from an entirely different physical origin. The other interesting case in which grain-boundary sliding and surface diffusion appear to play a major role pertains to the instability of cavity morphology. The original analysis of the phenomenon, which was first observed in = iron (20--21), concludes that a steep gradient of the tensile stress ahead of the cavity is necessary for the instability (22). This may happen when the growth rate, or the cavity size, or the intercavity spacing, becomes large. The incorporation of surface diffusion substantially stabilizes the growth (23), and the condition for instability after this modification becomes very similar to that for crack-like growth, as might be expected, for the same surface diffusion mechanism which kinetically is the fastest process is involved in both cases (3,23). The phenomenon appears to be triggered by grainboundary sliding at some stage. At any rate it seems appropriate to regard crack-like growth, unstable growth, and sliding-enabled growth as related variations which are possible ways for accelerated growth toward the final stage of creep. In addition, the role of grain-boundary sliding in the early stage of growth at the juncture of nucleation may also be quite important. It is well recognized now that the stress required for thermal nucleation of intergranular cavities is rather high. It may be brought about only by transient loading resulting from stress concentrations at grain-boundary obstacles or particles which impede sliding, while the transient condition ensues following each event of intermittent grain-boundary sliding (2,24). The cavities thus nucleated are extremely small and must sinter shut once the transient has elapsed and the stress relaxed. It is nevertheless possible that the supercritical nuclei may be aided still by the transient grain-boundary sliding, which would immediately wedge open the nuclei along the particles in an attempt to further relieve the strain energy as envisioned in the Stroh process (24,25), with an important difference that the nucleation is initiated only by the thermal nucleation process. Chen and Yoo (24) have recently examined this process and concluded that the inclusion of such an early stage growth mechanism, even with the subsequent rounding-off of the microwedge crack by surface diffusion, appears to stabilize the thermally nucleated nuclei against shrinkage when the stress is relaxed to approach the nominal stress, In this way, therefore, the alternative mode of cavity growth impelled by grain-boundary sliding serves as the link between nucleation and growth in creep cavitation. Acknowledgment The related research at MIT leading to this work was supported by the U. S. Department of Energy under contract EG-77-S-02-4461. References I. 2. 3. 4. 5. 6. 7. 8. 9. i0. II. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
D. Hull and R. E. Rimmer, Phil. Mag. 4, 673 (1959). I-W. Chen and A. S. Argon, Acta Me|all. 29, 1321 (1981). I-W. Chen and A. S. Argon, Acta Metall. 29, 1759 (1981). M. V. Speight and W. Beere, Metal Sci. 9, 190 (1975). T. J. Chuang, K. T. Kagawa, J. R. Rice, and L. B. Sills, Acta Me tall. 27, 265 (1979). W. Beere and M. V. Speight, Metal Sci. ]2, 172 (1978). A. Needleman and J. R. Rice, Acta Metall. 28, 1315 (1980). F. A. McClintock, Proc. Roy. Soc. ~ 8 5 , 58 (1965). B. F. Dyson, Metal Sci. |0, 349 (1976). B. F. Dyson, Can. Metall. Quart. 18, 31 (1979). B. F. Dyson and M. J. Rogers, Metal Sci. 8, 261 (1974). B. F. Dyson and M. J. Rogers, in Fracture 77, ed. D.M.R. Taplin, 2, 621, University of Waterloo, Canada (1977). J. R. Rice, Acta Me|all. 29, 675 (1981). I-W. Chen and A. S. Argon, Acta Metall. 27, 785 (1979). I-W. Chen, "Effect of Grain-Boundary Sliding on Diffusive Growth of Intergranular Cavities," submitted to Acta. Metall. (1982). S. H. Goods and W. D. Nix, Acta Me|all. 26, 739 (1978). T. G. Nieh and W. D. Nix, Acta Metall. 27, 1097 (1979). T. G. Nieh and W. D. Nix, Acta Me|all. 28, 557 (1980). L. Martinez and W. D. Nix, Me|all. Trans. [3A, 427 (1982). D.M.R. Taplin and A. W. Wingrove, Acta Metall. 15, 1231 (1967). R. J. Fields, T. Weerasooriya and M. F. Ashby, Me|all. Trans. |IA, 333 (I~o).
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R. J. Fields and M. F. Ashby, Phil. Mag. 33, 33 (1976). W. Beere, Phil. Mag. 38A, 691 (1978). I-W. Chen and M. H. Yoo, "Nucleation of Intergranular Cavities During Creep," to be published (1982). A. N. Stroh, Adv. Phys. 6, 418 (1957).
25.
° s ~
s
D
S
Og
°s De
De
107
Os
Ds
OS
I
I
I
0"~ 106 $ .w ,m
I
,
,.
:
105 __
~ . ~
a jib = 0.1
a/b = 0.0 I - , ~
I <
104
-1t-
FIG. 1 •
"D~o possible modes of cavity growth, with (a) quasi-equilibrlum and (b) crack-llke cavity shape, are both subject to the influence of creep, (c), which shortens the diffusion length from b to A.
O nr" (.9
103
-J
102
rr O Z
\ _
\
kT~=
/
101 (~=
4 "a'h(V/)
(DbSb~A/Y'
(4sin ~/2) 3/2 100 10-3
\Ds6sTs /
I
I
I
I
10-2
10-1
100
101
102
NORMALIZED CAVITY RADIUS (a/A) 2b
2
FIG. 2.
(a) HEAVILYCAVITATED
FIG. 3.
FIG. 4.
fb;SLIGHTLYCAVITATED
Diffusive growth is relatively fast in comparison with cavity growth by creep flow alone, i.e. the lower right limit. The growth slows down when cavity size increases but speeds up again with the transition to the crack-like mode.
Cavity growth subject to the matrix constraint in both (a) heavily cavitared and (b) slightly cavitated polycrystal.
/
Cavity under graln-boundary sliding develops a crack-llke morphology on two asymmetric sides near the tips, which requires considerable surface diffusion. (a) B E F O R E S L I D I N G
(b) A F T E R S L I D I N G
/
METALLURGICA
V o l . 17, pp. Printed in
MECHANISMS
17-22, 1983 the U.S.A.
Pergamon
Press
VIEWPOINT
SET
Ltd.
No.
5
OF CAVITY GROWTH IN CREEP*
l-Wei Chen t Metals and Ceramics Division Oak Ridge National Laboratory Oak Ridge, Tennessee 37830 (Received
September
9,
1982)
Introduction The growth of intergranular cavities under creep conditions is of considerable technological interest. It is well-documented in many engineering and model materials and is the subject of numerous investigations. However, the phenomenon is a complex one, which is to be expected on several accounts. First, kinetic and mechanical processes at elevated temperature are many, e.g. lattice diffusion, grain-boundary diffusion and surface diffusion of the former category and dislocation creep, diffusional creep and graln-boundary sliding of the latter. Second, the size distribution of cavities, being a function of time, varies from one grainboundary to the other due to the heterogeneous and continuous nucleation of new cavities. Third, the orientation and the surroundings of each grain-boundary is different, giving rise to a broad spectrum of growth conditions of different mechanical descriptions. These considerations, when taken into account in toto, result in an almost infinite number of cases which are too n~nerous to analyze deterministically. For a mechanistic understanding, certain idealizations have to be made. This paper attempts to give an up-to-date account of such understanding, with the necessary idealization, and to point out the deficiencies in the simplified picture in each case. As an outline, we pose the following three problems in the order of increasing complexity. The simplest case, on which a large amount of work has been done to date, essentially pertains to cavitation on the transverse grain-boundary in a bicrystal under a normal stress. Since many mechanisms are possible at elevated temperature, even this simplest case is not trivial and was solved in a satisfactory way only recently. The second idealized case deals with cavitation on transverse boundaries in a polycrystal. Currently, only the limiting case in which all transverse boundaries cavitate has been treated. While an alternative solution of the above problem is given in the paper, attention must be drawn to the more general problem which have been hitherto overlooked. The third case deals with inclined boundaries when the additional component of grain-boundary sliding sometimes causes "anomalous" effects. Since in reality cavitation also occurs on such boundaries, the problem warrants attention as well. These three problems are treated briefly in the following. Cavity Growth in a Bicrystal Under Normal Loading (Fig. i) The early model of cavity growth, of Hull and Rimmer (i), considered an array of spherical cavities which enlarge by absorbing vacancies from the adjoining grain-boundaries. The reverse
*Research sponsored by the Division of Materials Sciences, U.S. Department of Energy, under contract W-7405-eng-26 with the Union Carbide Corporation. ~Permanent address: Department of Nuclear Engineering and Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139.
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flow of vacancies plates out a layer of atoms onto the grain-boundary, causing elongation normal to the plane (Fig. la). Obviously the process strongly favors those grain-boundaries transverse to the applied tensile stress and, indeed, in this important respect the model captures one of the most significant characteristics of cavity growth, i.e. micrographically most cavities are found on transverse boundaries. The morphology of cavities, especially those revealed by direct observations of the post-creep intergranular fracture surfaces (2), of pure metals and engineering materials alike, exhibits many features which are consistent with the diffusive mechanism. The above model, which, strictly speaking is only applicable to a bicrystal under normal tension, has been extended to treat the following general problem. An array of identical cavities of radius a at a regular Imlf spacing b, under a normal stress c~ which causes a uniform distant creep rate ~ , grows at ~ while maintaining a shape to be determined by the combined action of surface and grain-boundary diffusion. The analytic form of solution of Chen and Argon (3) is shown in Fig. 2. Here h(@) is the ratio of the volume of an equilibrium-shaped cavity to a sphere of the stone radius, and is only a function of the dihedral angle ~, while the rest of the symbols are the ones commonly used in the field. In the following, we describe how the problem was solved. Three important developments of previous investigators contribute to the final formulation of the above result. The Hull and Rimmer solution, later revised by Speight and Beere (4), is for the case of equilibrium cavities (Fig. la) between ri$id grains. The corresponding solution for the case of non-equilibrium cavities (Fig. ib) between rigid grains was due to Chuang et al. (5). In both cases, cavities have to communicate diffusionally as they grow, to lay a unifo~l matter deposition onto the grain-boundary, but the crack-like cavities grow faster and take over when the radius or tile dimensionless constant ~ (which is the ratio of grain boundary conductance to the surface conductance, essentially) becomes large. If the restriction of rigid grains is relaxed by matrix creep, the growth rate is enhanced as the required diffusion distance shortens (Fig. ic), according to Beere and Speight (6). Noting the analogy between fluid flow and the present problem, i.e. diffusionally expanding cavities in a flowing creep field, Chen and Argon (3) adopted the boundary layer approach which permits the approximate but straightforward solution of adequate accuracy by prescribing a boundary-layer thickness which may be determined by a reasonable dimensional argument. In the present case, the above thickness A~ which has the physical meaning of the diffusion distance, is found by compatibility which states that at the division line of the inner diffusion zone and the outer creep zone (Fig. Ic), the strain rate due to either mechanism must be comparable. With this method, the entire range of the solution, for equilibrium and crack-like cavities alike, in a creeping field of various strength, is mapped out in Fig. 2. The L-shaped branch on the left is for equilibrium cavities [which is in excellent agreement with the FEM numerical result (7)] while the V-shaped branches for crack-like cavities. In either case, creep flow becomes important as a/A increases. In fact, ignoring the V-shaped branches, we can verify that the growth rate of equilibrium-shaped cavities at large a/A eventually approaches that for a plastically expanding void in a creeping matrix (8). Meanwhile, at small creep rates, A approaches b and the growth rate is superceded by the Hull and Rimmer solution (4) given as the dotted lines on the left in Fig. 2. Thus the above solution incorporates all the essential variations of the general problem, including cavity shape, diffusion kinetics, and creep modifications. Two experimental observations of a very general nature support the above picture. First, for a typical material constant ~ = i0, a minimum in growth rate is expected near a ~ A. Since nucleation is continuous and often proceeds at a roughly constant rate, cavities will spend more time passing this size range where their number per unit size range increases. Measurements of cavity size distributions indeed confirm such a peak at the appropriate size range, and their profile suggests that some accelerated growth takes place for cavities of larger sizes (2). Furthermore, the same phase of slow growth will in turn contribute a ma~or portion to the time to fracture as well as creep ductility. The estimation of Chen and Argon (3) predicts a strain to failure between 0.05 and 0.5 with a gradual increase with the creep rate which ranges from 10 -7.5 to 10-4 per second, in good accord, both qualitatively and quantitatively, with the Monkman-Grant correlation which was also established in the same creep regime. We note that other attempts to explain the Monkman-Grant correlation based on the transition from diffusion control to creep control, for equilibrium-shaped cavities alone, were unsuccessful, predicting a precipitous drop of ductility from several hundred percent to roughly one percent at creep rates ~I0 -2 per second which is not observed experimentally.
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It is pleasing to see that the results developed from the rather drastic idealization of the bicrystal model are reasonable and useful. However, at very slow creep rate, the model breaks down as we shall see in the next section. Cavity Growth on Transverse Grain Boundaries in a Polycr~stal (Fi$. 3 ) Cavitation in a polycrystal is heterogeneous among grain-boundaries. Generally, transverse grain-boundaries cavitate more. The remaining ligaments between cavitated transverse boundaries exert upon the cavitating boundaries a mechanical constraint, which may be strong if the grains are nearly rigid~ and vanishingly small if grains creep easily. This is illustrated experimentally by Dyson (9--12), for a special case in which essentially all transverse grainboundaries are pre-cavitated before the test (Fig. 3a). He found the cavity growth rate in the material controlled by the macroscopic creep rate much slower than that expected in the bicrystal model. It is interesting to note that even in such heavily damaged material, grainboundary sliding alone cannot entirely relieve the mechanical constraint. Approximate analysis due to Dyson (9--10) and Rice (13) has been advanced. The analysis demonstrates that the thickening of the boundary due to cavitation unloads over the cavitating graln-boundary and overloads in the immediate vicinity. This "load shedding" toward the sides continues until the thickening rate under the reduced load is matched by the permissible opening rate of a penny-shaped crack carrying the same reduced normal traction which is significantly lower than the applied stress since the surrounding creep rate is low. The growth rate of cavities in this limit scales with the creep rate and, if it can be assumed (as Rice did) that isolated facet fracture is synomymous with overall fracture, then the strain to fracture is predicted to be approximately b/d, where d is the diameter of the grain-boundary facet. It can be deduced from the above that the constraint is more effective in a fine grain material, for the cavitational strain scales with b/d. The above analysis involves some geometrical and mechanical approximations which are difficult to assess without a comparison with the experiment. We give below a simple treatment which reproduces the experimental results satisfactorily. First divide the polycrystal into spherical cells each of which contains one transverse (cavitated) and on average two longitudinal (uncavitated) grain-boundaries. We then assign the same strain rate ~ to each cell. Meanwhile we recognize that the cell is encased in a much stiffer solid matrix in the immediate surroundings, which creeps at a strain rate ~= -- @c, where ~c is the strain rate due to cavitation which needs to be discounted for the solid matrix. Here the cavitational strain can be measured metallographically or by densitometry. Following Eshelby, the strain concentration in a soft spherical inclusion within a much stiffer matrix, as evaluated for creeping material by Chen and Argon (14), is 5/3. Hence,
~=o -- ~c
3
(i)
which is in numerical agreement with tile data of precavitated Nimonic 80 A giving ~ = 2.5 ~c (11--12). The cavitational strain rate can be evaluated geometrically by computing the total cavitation rate in each cell which contains one cavitating grain-boundary. We thus find
Sc
1 r 2 d (ha3) R 3 b 2 dt
(2)
where r and R are the radii of the transverse grain-boundary and the cell respectively. Finally noting, from quantitative metallography, that the linear intercept grain size L is given by [(2/~) (grain boundary area)/(unit volume)] -I, or L = 2~R3/9r 2
we obtain from above 3
(3)
20
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GROWTH
Vol.
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The above result compares well with Rice's analysis (13), but the more readily measurable quantity L is used instead of d. Despite the fact that the creep rate plays the key role in constrained growth, it must be recognized that physically cavitation is still diffusional. The low ductility and the high proportion of cavitatlonal strain in overall fracture, both characteristic of the Hull and Rimmer model, are common to the constrained growth in precavitated materials. In fact, if Eq. (4) is compared to the result in the Hull and Rimmer regime, on the left of the L-shaped branch in Fig. i, it is indistinguishable from the case without any constraint but with a much longer "constraint" diffusion distance, bc, approximately given by b = 2.1 exp(5~A3/6b2L)
(5)
which lengthens as the creep rate decreases. This is understandable, since the slow-down of the diffusive mechanism can be alternatively interpreted as the result of a reduced driving force (due to the constraint), an increased diffusion distance, or even a reduced dlffusivity while the basic features of diffusive growth remain unchanged. At this time, we wish to call attention to one serious deficiency in the present modelling of constrained growth. Unfortunately, Eq. (4) does not immediately lead to an estimation of time to fracture even though the cavity growth becomes constrained. This is so because overall fracture does not usually take place when the first set of cavitated boundaries separate, except in the heavily pre-damaged material (Fig. 3a). Instead, the damage will be incurred at this time on neighboring graln-boundaries and the damage zone spreads slowly (Fig. 3b). Not until a critical fraction of the material is damaged does the overall fracture occur. The critical fraction of damage, the spreading speed of the damage zone, and the initial damage are all material properties dependent on past history. Obviously the load shedding process will occur at every stage of the evolution, and the problem bears some resemblance to that of crack propagation in the context of stress concentrations and damage spreading. Since a very low ductility must be expected if constrained growth of the type Dyson studied (11--12) were the general case, the fact that many engineering alloys (such as austenitic stainless steel) do exhibit excellent ductility even at very low creep rate (~I0 -8 s-1) is a strong testimony that they must have accumulated their creep damage in such a gradual way. Hence, without a better understanding of the damage spreading process, Eq. (4) and the appreciation of constrained growth have no direct utility for predicting the failure time. Indeed, although we believe the constrained growth to be very important in engineering service at slow creep rate, it must be reminded that at stress concentrators such as notches, welds, and especially the crack front, the local creep rate may very well be sufficiently high to alleviate the constraint substantially. Clearly, all the above problems warrant more study in the future in order to furnish a better picture of the damage accumulation by cavitation. But next let us turn to the problem of grain-boundary sliding which plays, perhaps, an important role in the early stage and the last stage of cavity growth, hence a potentially crucial link in the damage process. Cavity Growth on Inclined Grain Boundaries (Fi~. 4) The obvious consequence of an inclined grain-boundary is graln-boundary sliding. Indeed, even for essentially transverse graln-boundaries, sliding may result due to compatibility. It has been proposed many times that sliding is responsible for cavity growth. This cannot be the case for cavities which are as small as a few Bm when sliding is much slower than 0. I um/s, for surface diffusion will restore any distortion due to sliding. In addition, many proposals entailed the use of the elasticity analogy which must fall at creep temperatures due to creep relaxation. A possible exception may come with coarse grain materials, with close spacing of cavities or at high strain rates. Good metallographic evidence indicating the importance of graln-boundary sliding and a clear demonstration of distorted cavities (schematically drawn in Fig. 4) were given in the study of type 304 stainless steel by Chen and Argon (2). In a recent analysis of this phenomenon, Chen (15) established a correlation between the sliding-controlled growth and the surface-controlled growth, which was found to be the case in H2O implanted Cu and Ag studied and analyzed by Goods, Nix and Nieh (16--18). The effect appears to be the result of a coupling between surface flux and the sliding-enabled matter transport, occurring on the two opposite halves of a cavity subject to heavy shearing. In this case, grain-boundary diffusion is not required to participate at all, yet the demand for surface diffusion is strong enough, due to grain-boundary sliding, to create a situation in which the matter transport is controlled by surface diffusion (5,19). In the above limit, a cavity is laterally spread out,
Vol.
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MECHANISMS
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GROWTH
21
while very little volumetric growth actually takes place. The growth rate also turns out to be identical to that given by Chuang et al. (5) for the case of surface diffusion control, although it arises from an entirely different physical origin. The other interesting case in which grain-boundary sliding and surface diffusion appear to play a major role pertains to the instability of cavity morphology. The original analysis of the phenomenon, which was first observed in = iron (20--21), concludes that a steep gradient of the tensile stress ahead of the cavity is necessary for the instability (22). This may happen when the growth rate, or the cavity size, or the intercavity spacing, becomes large. The incorporation of surface diffusion substantially stabilizes the growth (23), and the condition for instability after this modification becomes very similar to that for crack-like growth, as might be expected, for the same surface diffusion mechanism which kinetically is the fastest process is involved in both cases (3,23). The phenomenon appears to be triggered by grainboundary sliding at some stage. At any rate it seems appropriate to regard crack-like growth, unstable growth, and sliding-enabled growth as related variations which are possible ways for accelerated growth toward the final stage of creep. In addition, the role of grain-boundary sliding in the early stage of growth at the juncture of nucleation may also be quite important. It is well recognized now that the stress required for thermal nucleation of intergranular cavities is rather high. It may be brought about only by transient loading resulting from stress concentrations at grain-boundary obstacles or particles which impede sliding, while the transient condition ensues following each event of intermittent grain-boundary sliding (2,24). The cavities thus nucleated are extremely small and must sinter shut once the transient has elapsed and the stress relaxed. It is nevertheless possible that the supercritical nuclei may be aided still by the transient grain-boundary sliding, which would immediately wedge open the nuclei along the particles in an attempt to further relieve the strain energy as envisioned in the Stroh process (24,25), with an important difference that the nucleation is initiated only by the thermal nucleation process. Chen and Yoo (24) have recently examined this process and concluded that the inclusion of such an early stage growth mechanism, even with the subsequent rounding-off of the microwedge crack by surface diffusion, appears to stabilize the thermally nucleated nuclei against shrinkage when the stress is relaxed to approach the nominal stress, In this way, therefore, the alternative mode of cavity growth impelled by grain-boundary sliding serves as the link between nucleation and growth in creep cavitation. Acknowledgment The related research at MIT leading to this work was supported by the U. S. Department of Energy under contract EG-77-S-02-4461. References I. 2. 3. 4. 5. 6. 7. 8. 9. i0. II. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
D. Hull and R. E. Rimmer, Phil. Mag. 4, 673 (1959). I-W. Chen and A. S. Argon, Acta Me|all. 29, 1321 (1981). I-W. Chen and A. S. Argon, Acta Metall. 29, 1759 (1981). M. V. Speight and W. Beere, Metal Sci. 9, 190 (1975). T. J. Chuang, K. T. Kagawa, J. R. Rice, and L. B. Sills, Acta Me tall. 27, 265 (1979). W. Beere and M. V. Speight, Metal Sci. ]2, 172 (1978). A. Needleman and J. R. Rice, Acta Metall. 28, 1315 (1980). F. A. McClintock, Proc. Roy. Soc. ~ 8 5 , 58 (1965). B. F. Dyson, Metal Sci. |0, 349 (1976). B. F. Dyson, Can. Metall. Quart. 18, 31 (1979). B. F. Dyson and M. J. Rogers, Metal Sci. 8, 261 (1974). B. F. Dyson and M. J. Rogers, in Fracture 77, ed. D.M.R. Taplin, 2, 621, University of Waterloo, Canada (1977). J. R. Rice, Acta Me|all. 29, 675 (1981). I-W. Chen and A. S. Argon, Acta Metall. 27, 785 (1979). I-W. Chen, "Effect of Grain-Boundary Sliding on Diffusive Growth of Intergranular Cavities," submitted to Acta. Metall. (1982). S. H. Goods and W. D. Nix, Acta Me|all. 26, 739 (1978). T. G. Nieh and W. D. Nix, Acta Metall. 27, 1097 (1979). T. G. Nieh and W. D. Nix, Acta Me|all. 28, 557 (1980). L. Martinez and W. D. Nix, Me|all. Trans. [3A, 427 (1982). D.M.R. Taplin and A. W. Wingrove, Acta Metall. 15, 1231 (1967). R. J. Fields, T. Weerasooriya and M. F. Ashby, Me|all. Trans. |IA, 333 (I~o).
22
MECHANISMS
22. 23. 24.
OF
CAVITY
GROWTH
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No.
I
R. J. Fields and M. F. Ashby, Phil. Mag. 33, 33 (1976). W. Beere, Phil. Mag. 38A, 691 (1978). I-W. Chen and M. H. Yoo, "Nucleation of Intergranular Cavities During Creep," to be published (1982). A. N. Stroh, Adv. Phys. 6, 418 (1957).
25.
° s ~
s
D
S
Og
°s De
De
107
Os
Ds
OS
I
I
I
0"~ 106 $ .w ,m
I
,
,.
:
105 __
~ . ~
a jib = 0.1
a/b = 0.0 I - , ~
I <
104
-1t-
FIG. 1 •
"D~o possible modes of cavity growth, with (a) quasi-equilibrlum and (b) crack-llke cavity shape, are both subject to the influence of creep, (c), which shortens the diffusion length from b to A.
O nr" (.9
103
-J
102
rr O Z
\ _
\
kT~=
/
101 (~=
4 "a'h(V/)
(DbSb~A/Y'
(4sin ~/2) 3/2 100 10-3
\Ds6sTs /
I
I
I
I
10-2
10-1
100
101
102
NORMALIZED CAVITY RADIUS (a/A) 2b
2
FIG. 2.
(a) HEAVILYCAVITATED
FIG. 3.
FIG. 4.
fb;SLIGHTLYCAVITATED
Diffusive growth is relatively fast in comparison with cavity growth by creep flow alone, i.e. the lower right limit. The growth slows down when cavity size increases but speeds up again with the transition to the crack-like mode.
Cavity growth subject to the matrix constraint in both (a) heavily cavitared and (b) slightly cavitated polycrystal.
/
Cavity under graln-boundary sliding develops a crack-llke morphology on two asymmetric sides near the tips, which requires considerable surface diffusion. (a) B E F O R E S L I D I N G
(b) A F T E R S L I D I N G
/