- Email: [email protected]
On the strain-induced growth of neighboring voids
Pergamon
ScriptaMetallurgicaet Materialia, Vol. 31, No. 4, pp. 419-422, 1994 Copyright©1994ElsevierScienceLtd Printed in the USA. All rights reserved 0956-716X/94 $6.00 + 00
ON THE STRAIN-INDUCED G R O W T H OF N E I G H B O R I N G VOIDS L. E. Forero+ and D. A. Koss Department of Materials Science and Engineering, Penn State University, University Park, PA +Currently: Universidad Industrial De Santander, Bucaramanga, Columbia
(Received February 22, 1994) (Revised April 19, 1994) Introduction The tensile fracture of most structural alloys is usually limited by accumulation of damage in the form of void nucleation, growth, and linking. From the standpoint of predicting the fracture strains of components undergoing microvoid ductile fracture, an accurate description of the strain-induced growth and linking behavior of voids is essential. This includes an ability to describe not only the growth of isolated voids but also any void interaction effects. The growth of isolated, initially spherical voids has been modeled using theoretical analyses and computational techniques for both non-linearviscous and strain-hardeningmaterials (1-3). Recent experimental results indicate that the initial analysis of Rice and Tracey (1) remains an acceptable basis for predicting the effect of plastic strain and stress triaxiality on the growth rates of isolated cavities (4-8). In order to predict macroscopic behavior such as the ductility of a tensile specimen, void interaction effects describing the growth and linking of neighboring voids must also be taken into account (9-17). Attempts to incorporate void interaction effects have been performed analytically and computationally by modeling cavities as either cylindrical holes or spheres within specimens (10,11,15-17). Cylindrical holes are predicted to interact strongly within specimens even in uniaxial tension (11). However, computational modeling of spherical cavities located one cavity diameter apart predicts a strong void interaction only at highly triaxial states of stress; no significantinteraction is predicted for uniaxial tension [15]. The pro'pose of this communication is to report preliminary experimental results which describe the straininduced growth of a pair of spherical cavities, spaced about one cavity diameter apart, and embedded in material (titanium) subjected to far-field uniaxial tension. Ultrasonic imaging is used to monitor cavity dimensions during the incremental straining of the specimens, and data are presented for both longitudinal and transverse cavity growth for extensional strains up to ~ 0.5. As will be evident, the data indicate a significant degree of void interaction even in this uniaxial tension case. Experimen~l Procedure The specimens used in this study contain two nearly spherical cavities centrally located in a tensile specimen with a square cross section to optimize ultrasonic imaging; see Figure 1. Using a technique similar to one originally used by Tittmann et al. [18], we rely on the excellent diffusion bonding capability of Ti to fabricate the specimens. The initial step of the procedure consisted of drilling two - 0.9 mm close-end holes, about 0.9 mm deep and located about 1 mm apart, into one end of a 19 mm diameter round bar of Ti which had been ground flat and polished. The end face of the cylinder containing the holes was diffusion bonded to the blank end of a companion cylinder at 695°C at a pressure of 30 MPa for 4h in an argon gas atmosphere. After diffusion bonding, the specimen were subjected to an anneal at 705°C for 23 h. The diffusion bonded joint was free of porosity and was characterized by the same grain size as the adjoining material. There was no evidence that the bond line acted as an imperfection to bias either deformation or specimen fracture. The initial shape of the internal cavil(s) was dose to spherical; if a and b represent the longitudinal and transverse diameters of the "spheres," thenb = 1.0 :t: 0.1. The specimens were machined with a square cross section of 10 m m x 10 mm and a gauge length of 20 ram, as also shown in Figure 1. In all cases the flat surfaces of the specimen were at least 3.5 cavity diameters from a cavity surface. For specimens containing two cavities, two of the specimen faces were oriented parallel to the axis between the cavities to facilitate accurate monitoring of cavity dimension using ultrasonics. Ullrasonie Cscan imaging was employed to monitor the cavity dimensions using a Sonix system with a 30 MHz focused transducer. Image analysis was used to determine the major and minor axes of the cavities. Using similarly shaped Ti specimens but with blind-end cylindrical holes, this ultrasonic system predicted the hole diameters as well as the spacing between hole ends with an accuracy of +.05 mm of the measured values. Thus, we estimate 419
420
GROWTHVOIDS
Vol. 31, No. 4
that our C-scan technique predicts at least the initial cavity image dimensions (when the cavities are still ~1 cavity diameter apart) with an accuracy of about +.05 ram. The strain-induced growth behavior of the cavities was determined by ultrasonic imaging of the tensile specimens subjected to interrupted tensile tests. Using a constant extension rate of 2.5x10 "4 turn/s, the tests were interrupted each ~ 3 % total extensional strain to determine the cavity and intercavity ligament dimensions. This sequence was continued until specimen fracture. The presence of the cavities creates a strain gradient and diffuse necking along the gauge section of the specimen. The following procedure was used to determine the local tensile strains in the vicinity of the cavities. Transverse thickness- and width-strain profiles normal to the tensile axis were measured after each increment of plastic deformation. This permitted the determination of the local values of the maximum principal tensile strains el in the vicinity of the cavities from measuring the thickness strain (e2) data and width strain (e3) data assuming el _.- - (e2 + e3). This relationship assumes conservation of volume, which in our case is violated by the cavity growth. However, the initial volume of cavities is very small (_<0.8%) when compared to the volume of the segment of the gauge section of the specimen which contains the cavities. As a result, the incremental change in the volume of the gauge section due to cavity growth, which amounts to an additional --2% at el _- 0.5 for two cavities, is still small. Thus, the conservation-of-volumeassumption is a reasonable one and introduces only a small error (~ 0.05 el) to the el - values. The material used in this study was commercially pure Ti with a microstructure consisting of equiaxed grains with an average grain size of ~ 75 ~ n . The Ti had a yield strength of 300 MPa and a strain hardening exponent n = dino/dlne = 0.23. For our experiments, the initial cavity dimensions and intercavity ligament spacing is roughly 900 ~ n ; this means that roughly 12 grains exist in the minimum ligament dimension as well as along the length of the cavity. This condition should result in reasonably good approximation of polycrystalline plasticity within the ligament and near the cavity. Results and Discussion The swain-induced growth of a single isolated cavity in uniaxial tension is expected to undergo a shape change such that an initially spherical caviaty elongates in the tensile direction into an ellipsoidal shape. Denoting a and b as the major and minor dimensions of the cavity, the Rice-Tracey analysis (1), supported by subsequent computational analyses (2,3) as well as experiment (4-8), predict that cavity dimensions change in manner nearly linear with strain. Furthermore while the major diameter a increases with strain, the minor diameter b decreases. Thus in the absence of void interaction effects, we expect strain-inducedvoid growth in uniaxial tension such that extensional growth is coupled with transverse contraction of the voids. Experiments performed on tensile specimens containing pairs of neighboring cavities indicate behavior different from that described above for a single cavity. Figure 2 shows results from one test typical of the five performed on Ti specimens, each of which contained a pair of initially spherical cavities spaced about one cavity diameter apart. In all cases, the specimens were subjected to far-field uniaxial tensile deformation, and the cavities were oriented such that the axis between cavity centers was normal to the tensile axis. The most striking of the data in Figure 2 is that both the major and the minor axial dimensions of the cavity increase with strain. Thus there is cavity growth in not only the longitudinal direction but also in the transverse direction. In both cases, within the accuracy of these data, cavity growth appears to be nearly linear with strain for the range of strains examined. A natural consequence of the positive growth of the cavities in the lateral direction is a decrease in the intercavity spacing, d, with increasing strain. Furthermore, our preliminary data indicate that all of the lateral cavity growth occurs at the expense of the ligament. This is a consequence of the observation that the transverse distance between the ouIer surfaces of the cavities [bl + 132+ d] remains constant or decreases slightly with strain. Thus all of the lateral growth is directed into the intercavity ligament. As a result, the cavity interaction effects cause a much faster decrease of the intercavity dimension than if non-interactingcavity growth occurred. The presence of such cavity interaction effects, even in uniaxial tension, at least partly explains the difficulty of predicting fracture strains based on the growth of isolated cavities (10). We also note that in most of our five initial tests there appears to indicate an acceleration of ligament thinning at macroscopic tensile strains between 0.1 and 0.2; see Figure 2. Whether or not this is an artifact of the data or an indication of a ligament necking phenomenon remains to be seen. It also might be recognized that, due to specimen necking (19), there is a small increase in stress triaxiality, Ond~, from 0.33 initially to --0.45 at el =-0.35 (here, t~m denotes the mean stress and ~ the equivalent stress). Such small increases in ~m/~ are predicted to have only very minor effects on the growth behavior of isolated cavities (1-3). Similarly, we do not expect the observed void interaction effects in Figure 2 to be a result of the small change in stress triaxiality.
Vol. 31, No. 4
GROWTHVOIDS
421
Many investigators have previously recognized the importance of void interaction effects in determining ductile fracture (5,9-17). Experimental observations have indicated void interaction effects must become especially strong at high volume fractions of voids (6,12). Furthermore, it is also possible that clusters of voids concentrate on planes such as to accentuate void interaction effects, so as to trigger rapid void linking by flow localization effects (9,13,14,20,21). The present behavior suggests that, whle void interaction effects definitely occur between a pair of neighboring voids, there is no evidence of shear instability causing void linking. Rather fairly "stable", (if accelerated) cavity growth to impingement appears to occur in our experiments. Void interaction effects have been examined by finite element modeling (FEM) of specimens containing a regular arrays of either cylindrical or spherical cavities which are spaced one cavity diameter apart (10,15), as in our experiments. The computational results for the case of cylindrical cavities predict significant cavity interactions even in uniaxial tension (10). However, in three dimensions, void interactions are predicted to be much weaker. For example, FEM of a regular array of initially spherical cavities each spaced one cavity diameter apart predicts no cavity interactions in uniaxial tension (15). Void interactions are predicted to occur only if the stress state is highly triaxial (15). Thus, there appears to be a significant lack of agreement between our experimental results, which indicate void interactions during deformation which is close to uniaxial tension, and the previous computational predictions, which do not. Summary_ Preliminary experimental results are also presented for the strain-induced growth behavior of a pair of nearly spherical cavities which are equal in size, spaced roughly one cavity diameter apart, and oriented transverse to the tensile axis. In this case, void interaction effects are quite evident during far-field uniaxial tension. Specifically, both the longitudinal and transverse cavity dimensions increase with increasing strain in roughly a linear manner. The growth behavior in the transverse direction results in a much more rapid decrease of the intercavity ligament than would be expected if the cavities behaved as non-interacting, isolated cavities. This implies that accurate predictions of failure strains during ductile, microvoid fracture must include the acceleration of void linking due to void interaction effects. Acknowledgments The authors with to thank Dr's. A. Geltmacher, P. Matic and P. Thomason for stimulating discussions and suggestions. The support of the Office of Naval Research is gratefully acknowledged. References 1. A.R. Rice and D. M. Tracey, J. Mech. Phys. Solids 17,201 (1969). 2. B. Budiansky, J. W. Hutchinson and S. Slutsky, in Mechanics of Solids (Pergamon Press, Oxford), 1982, p. 13. 3. C.J. Worswick and R. J. Pick, J. Mech. Phys. Solids 38,601 (1990). 4. J.D. Atkinson, Ph.D. Thesis, University of Cambridge, 1973. 5. P.F. Thomason, D~¢tile Fracture of Metals (Pergamon Press, Oxford), p. 47, 1990. 6. B. Marini, F. Mudry, and A. Pineau, Eng. Frac. Mech. 22, 989 (1985). 7. G. LeRoy, J. D. Embury, G. Edward, and M. F. Ashby, Acta Metall. 29, 1509 (1981). 8. L. Forero and D. A. Koss, unpublished research, 1993. 9. R.J. Bourcier, D. A. Koss, R. E. Smelser, and O. Richmond, Acta Metall. 34, 2443 (1986). 10. D.M. Tracey, Eng. Frac. Mech. 3, 301 (1971). 11. A. Needleman, J. Appl. Mech. 39, 963 (1972). 12. L.M. Brown and J. D. Embury in Proc. 3rd Int. Conf. on Strength of Metals and Alloys, 1973, p. 14. 13. V. Tvergaard and A. Needleman, Acta Metall. 32, 157 (1984). 14. A. Needleman and V. Tvergaard, J. Mech. Phys. Solids 32, 461 (1984). 15. C.L. Horn and R. M. McMeeking, J. Appl. Mech. 56, 309 (1989). 16. N.P. O'Dowd, M. Ortiz, and C. F. Shih, unpublished research. 17. P.F. Thomason, Acta Metall. et Mater. 41, 2127 (1993). 18. B.R. Tittmann, H. Nadler, and N. E. Paton, Metall. Trans. A 7A, 320 (1976). 19. P.W. Bridgeman, Studies of Large Plastic Flow and Fracture (McGraw Hill, New York) p. 9, 1952. 20. K. Yamamoto, Int. J. Frac. 14, 347 (1978). 21. L. Sage, J. Pan and A. Needleman, Int. J. Frac. 19, 163 (1982).
GROWTH VOIDS
422
Vol. 31, No. 4
1
I
II d
Figure 1.
A schematic showing the diffusion bonding procedure used to make a tensile specimen containing a pair of internal cavities. The test procedure involves tests based on -3% extensional strain increments.
2.5
|
I
|
,,.
!
!
O ~"o
0
o
~
8
oo o
1.5 9 ,.O
4-
4-
+
0 (a/ao)l ~,
+ +
0.5
(a/ao)2
0 (b/bo)2 4-
+
+
+
+
D (b/bo)l + (d/do)
0 0
, 0.1
i
i
z
0.2
0.3
0.4
0.5
TENSILE S T R A I N Figure 2.
The su~dn-induced cavity growth of a specimen containing a pair of nearly spherical cavities spaced about one cavity diameter apart. The measured initial cavity values arc ao = 1.0 ram, Do = 0.9 mm, and do = 0.9 mm.
ScriptaMetallurgicaet Materialia, Vol. 31, No. 4, pp. 419-422, 1994 Copyright©1994ElsevierScienceLtd Printed in the USA. All rights reserved 0956-716X/94 $6.00 + 00
ON THE STRAIN-INDUCED G R O W T H OF N E I G H B O R I N G VOIDS L. E. Forero+ and D. A. Koss Department of Materials Science and Engineering, Penn State University, University Park, PA +Currently: Universidad Industrial De Santander, Bucaramanga, Columbia
(Received February 22, 1994) (Revised April 19, 1994) Introduction The tensile fracture of most structural alloys is usually limited by accumulation of damage in the form of void nucleation, growth, and linking. From the standpoint of predicting the fracture strains of components undergoing microvoid ductile fracture, an accurate description of the strain-induced growth and linking behavior of voids is essential. This includes an ability to describe not only the growth of isolated voids but also any void interaction effects. The growth of isolated, initially spherical voids has been modeled using theoretical analyses and computational techniques for both non-linearviscous and strain-hardeningmaterials (1-3). Recent experimental results indicate that the initial analysis of Rice and Tracey (1) remains an acceptable basis for predicting the effect of plastic strain and stress triaxiality on the growth rates of isolated cavities (4-8). In order to predict macroscopic behavior such as the ductility of a tensile specimen, void interaction effects describing the growth and linking of neighboring voids must also be taken into account (9-17). Attempts to incorporate void interaction effects have been performed analytically and computationally by modeling cavities as either cylindrical holes or spheres within specimens (10,11,15-17). Cylindrical holes are predicted to interact strongly within specimens even in uniaxial tension (11). However, computational modeling of spherical cavities located one cavity diameter apart predicts a strong void interaction only at highly triaxial states of stress; no significantinteraction is predicted for uniaxial tension [15]. The pro'pose of this communication is to report preliminary experimental results which describe the straininduced growth of a pair of spherical cavities, spaced about one cavity diameter apart, and embedded in material (titanium) subjected to far-field uniaxial tension. Ultrasonic imaging is used to monitor cavity dimensions during the incremental straining of the specimens, and data are presented for both longitudinal and transverse cavity growth for extensional strains up to ~ 0.5. As will be evident, the data indicate a significant degree of void interaction even in this uniaxial tension case. Experimen~l Procedure The specimens used in this study contain two nearly spherical cavities centrally located in a tensile specimen with a square cross section to optimize ultrasonic imaging; see Figure 1. Using a technique similar to one originally used by Tittmann et al. [18], we rely on the excellent diffusion bonding capability of Ti to fabricate the specimens. The initial step of the procedure consisted of drilling two - 0.9 mm close-end holes, about 0.9 mm deep and located about 1 mm apart, into one end of a 19 mm diameter round bar of Ti which had been ground flat and polished. The end face of the cylinder containing the holes was diffusion bonded to the blank end of a companion cylinder at 695°C at a pressure of 30 MPa for 4h in an argon gas atmosphere. After diffusion bonding, the specimen were subjected to an anneal at 705°C for 23 h. The diffusion bonded joint was free of porosity and was characterized by the same grain size as the adjoining material. There was no evidence that the bond line acted as an imperfection to bias either deformation or specimen fracture. The initial shape of the internal cavil(s) was dose to spherical; if a and b represent the longitudinal and transverse diameters of the "spheres," thenb = 1.0 :t: 0.1. The specimens were machined with a square cross section of 10 m m x 10 mm and a gauge length of 20 ram, as also shown in Figure 1. In all cases the flat surfaces of the specimen were at least 3.5 cavity diameters from a cavity surface. For specimens containing two cavities, two of the specimen faces were oriented parallel to the axis between the cavities to facilitate accurate monitoring of cavity dimension using ultrasonics. Ullrasonie Cscan imaging was employed to monitor the cavity dimensions using a Sonix system with a 30 MHz focused transducer. Image analysis was used to determine the major and minor axes of the cavities. Using similarly shaped Ti specimens but with blind-end cylindrical holes, this ultrasonic system predicted the hole diameters as well as the spacing between hole ends with an accuracy of +.05 mm of the measured values. Thus, we estimate 419
420
GROWTHVOIDS
Vol. 31, No. 4
that our C-scan technique predicts at least the initial cavity image dimensions (when the cavities are still ~1 cavity diameter apart) with an accuracy of about +.05 ram. The strain-induced growth behavior of the cavities was determined by ultrasonic imaging of the tensile specimens subjected to interrupted tensile tests. Using a constant extension rate of 2.5x10 "4 turn/s, the tests were interrupted each ~ 3 % total extensional strain to determine the cavity and intercavity ligament dimensions. This sequence was continued until specimen fracture. The presence of the cavities creates a strain gradient and diffuse necking along the gauge section of the specimen. The following procedure was used to determine the local tensile strains in the vicinity of the cavities. Transverse thickness- and width-strain profiles normal to the tensile axis were measured after each increment of plastic deformation. This permitted the determination of the local values of the maximum principal tensile strains el in the vicinity of the cavities from measuring the thickness strain (e2) data and width strain (e3) data assuming el _.- - (e2 + e3). This relationship assumes conservation of volume, which in our case is violated by the cavity growth. However, the initial volume of cavities is very small (_<0.8%) when compared to the volume of the segment of the gauge section of the specimen which contains the cavities. As a result, the incremental change in the volume of the gauge section due to cavity growth, which amounts to an additional --2% at el _- 0.5 for two cavities, is still small. Thus, the conservation-of-volumeassumption is a reasonable one and introduces only a small error (~ 0.05 el) to the el - values. The material used in this study was commercially pure Ti with a microstructure consisting of equiaxed grains with an average grain size of ~ 75 ~ n . The Ti had a yield strength of 300 MPa and a strain hardening exponent n = dino/dlne = 0.23. For our experiments, the initial cavity dimensions and intercavity ligament spacing is roughly 900 ~ n ; this means that roughly 12 grains exist in the minimum ligament dimension as well as along the length of the cavity. This condition should result in reasonably good approximation of polycrystalline plasticity within the ligament and near the cavity. Results and Discussion The swain-induced growth of a single isolated cavity in uniaxial tension is expected to undergo a shape change such that an initially spherical caviaty elongates in the tensile direction into an ellipsoidal shape. Denoting a and b as the major and minor dimensions of the cavity, the Rice-Tracey analysis (1), supported by subsequent computational analyses (2,3) as well as experiment (4-8), predict that cavity dimensions change in manner nearly linear with strain. Furthermore while the major diameter a increases with strain, the minor diameter b decreases. Thus in the absence of void interaction effects, we expect strain-inducedvoid growth in uniaxial tension such that extensional growth is coupled with transverse contraction of the voids. Experiments performed on tensile specimens containing pairs of neighboring cavities indicate behavior different from that described above for a single cavity. Figure 2 shows results from one test typical of the five performed on Ti specimens, each of which contained a pair of initially spherical cavities spaced about one cavity diameter apart. In all cases, the specimens were subjected to far-field uniaxial tensile deformation, and the cavities were oriented such that the axis between cavity centers was normal to the tensile axis. The most striking of the data in Figure 2 is that both the major and the minor axial dimensions of the cavity increase with strain. Thus there is cavity growth in not only the longitudinal direction but also in the transverse direction. In both cases, within the accuracy of these data, cavity growth appears to be nearly linear with strain for the range of strains examined. A natural consequence of the positive growth of the cavities in the lateral direction is a decrease in the intercavity spacing, d, with increasing strain. Furthermore, our preliminary data indicate that all of the lateral cavity growth occurs at the expense of the ligament. This is a consequence of the observation that the transverse distance between the ouIer surfaces of the cavities [bl + 132+ d] remains constant or decreases slightly with strain. Thus all of the lateral growth is directed into the intercavity ligament. As a result, the cavity interaction effects cause a much faster decrease of the intercavity dimension than if non-interactingcavity growth occurred. The presence of such cavity interaction effects, even in uniaxial tension, at least partly explains the difficulty of predicting fracture strains based on the growth of isolated cavities (10). We also note that in most of our five initial tests there appears to indicate an acceleration of ligament thinning at macroscopic tensile strains between 0.1 and 0.2; see Figure 2. Whether or not this is an artifact of the data or an indication of a ligament necking phenomenon remains to be seen. It also might be recognized that, due to specimen necking (19), there is a small increase in stress triaxiality, Ond~, from 0.33 initially to --0.45 at el =-0.35 (here, t~m denotes the mean stress and ~ the equivalent stress). Such small increases in ~m/~ are predicted to have only very minor effects on the growth behavior of isolated cavities (1-3). Similarly, we do not expect the observed void interaction effects in Figure 2 to be a result of the small change in stress triaxiality.
Vol. 31, No. 4
GROWTHVOIDS
421
Many investigators have previously recognized the importance of void interaction effects in determining ductile fracture (5,9-17). Experimental observations have indicated void interaction effects must become especially strong at high volume fractions of voids (6,12). Furthermore, it is also possible that clusters of voids concentrate on planes such as to accentuate void interaction effects, so as to trigger rapid void linking by flow localization effects (9,13,14,20,21). The present behavior suggests that, whle void interaction effects definitely occur between a pair of neighboring voids, there is no evidence of shear instability causing void linking. Rather fairly "stable", (if accelerated) cavity growth to impingement appears to occur in our experiments. Void interaction effects have been examined by finite element modeling (FEM) of specimens containing a regular arrays of either cylindrical or spherical cavities which are spaced one cavity diameter apart (10,15), as in our experiments. The computational results for the case of cylindrical cavities predict significant cavity interactions even in uniaxial tension (10). However, in three dimensions, void interactions are predicted to be much weaker. For example, FEM of a regular array of initially spherical cavities each spaced one cavity diameter apart predicts no cavity interactions in uniaxial tension (15). Void interactions are predicted to occur only if the stress state is highly triaxial (15). Thus, there appears to be a significant lack of agreement between our experimental results, which indicate void interactions during deformation which is close to uniaxial tension, and the previous computational predictions, which do not. Summary_ Preliminary experimental results are also presented for the strain-induced growth behavior of a pair of nearly spherical cavities which are equal in size, spaced roughly one cavity diameter apart, and oriented transverse to the tensile axis. In this case, void interaction effects are quite evident during far-field uniaxial tension. Specifically, both the longitudinal and transverse cavity dimensions increase with increasing strain in roughly a linear manner. The growth behavior in the transverse direction results in a much more rapid decrease of the intercavity ligament than would be expected if the cavities behaved as non-interacting, isolated cavities. This implies that accurate predictions of failure strains during ductile, microvoid fracture must include the acceleration of void linking due to void interaction effects. Acknowledgments The authors with to thank Dr's. A. Geltmacher, P. Matic and P. Thomason for stimulating discussions and suggestions. The support of the Office of Naval Research is gratefully acknowledged. References 1. A.R. Rice and D. M. Tracey, J. Mech. Phys. Solids 17,201 (1969). 2. B. Budiansky, J. W. Hutchinson and S. Slutsky, in Mechanics of Solids (Pergamon Press, Oxford), 1982, p. 13. 3. C.J. Worswick and R. J. Pick, J. Mech. Phys. Solids 38,601 (1990). 4. J.D. Atkinson, Ph.D. Thesis, University of Cambridge, 1973. 5. P.F. Thomason, D~¢tile Fracture of Metals (Pergamon Press, Oxford), p. 47, 1990. 6. B. Marini, F. Mudry, and A. Pineau, Eng. Frac. Mech. 22, 989 (1985). 7. G. LeRoy, J. D. Embury, G. Edward, and M. F. Ashby, Acta Metall. 29, 1509 (1981). 8. L. Forero and D. A. Koss, unpublished research, 1993. 9. R.J. Bourcier, D. A. Koss, R. E. Smelser, and O. Richmond, Acta Metall. 34, 2443 (1986). 10. D.M. Tracey, Eng. Frac. Mech. 3, 301 (1971). 11. A. Needleman, J. Appl. Mech. 39, 963 (1972). 12. L.M. Brown and J. D. Embury in Proc. 3rd Int. Conf. on Strength of Metals and Alloys, 1973, p. 14. 13. V. Tvergaard and A. Needleman, Acta Metall. 32, 157 (1984). 14. A. Needleman and V. Tvergaard, J. Mech. Phys. Solids 32, 461 (1984). 15. C.L. Horn and R. M. McMeeking, J. Appl. Mech. 56, 309 (1989). 16. N.P. O'Dowd, M. Ortiz, and C. F. Shih, unpublished research. 17. P.F. Thomason, Acta Metall. et Mater. 41, 2127 (1993). 18. B.R. Tittmann, H. Nadler, and N. E. Paton, Metall. Trans. A 7A, 320 (1976). 19. P.W. Bridgeman, Studies of Large Plastic Flow and Fracture (McGraw Hill, New York) p. 9, 1952. 20. K. Yamamoto, Int. J. Frac. 14, 347 (1978). 21. L. Sage, J. Pan and A. Needleman, Int. J. Frac. 19, 163 (1982).
GROWTH VOIDS
422
Vol. 31, No. 4
1
I
II d
Figure 1.
A schematic showing the diffusion bonding procedure used to make a tensile specimen containing a pair of internal cavities. The test procedure involves tests based on -3% extensional strain increments.
2.5
|
I
|
,,.
!
!
O ~"o
0
o
~
8
oo o
1.5 9 ,.O
4-
4-
+
0 (a/ao)l ~,
+ +
0.5
(a/ao)2
0 (b/bo)2 4-
+
+
+
+
D (b/bo)l + (d/do)
0 0
, 0.1
i
i
z
0.2
0.3
0.4
0.5
TENSILE S T R A I N Figure 2.
The su~dn-induced cavity growth of a specimen containing a pair of nearly spherical cavities spaced about one cavity diameter apart. The measured initial cavity values arc ao = 1.0 ram, Do = 0.9 mm, and do = 0.9 mm.