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Low amplitude cyclic deformation behavior of single crystalline silicon
Scripta Materialia, Vol. 41, No. 1, pp. 109 –115, 1999 Elsevier Science Ltd Copyright © 1999 Acta Metallurgica Inc. Printed in the USA. All rights reserved. 1359-6462/99/$–see front matter
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LOW AMPLITUDE CYCLIC DEFORMATION BEHAVIOR OF SINGLE CRYSTALLINE SILICON Jianghong Gong* Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, People’s Republic of China
Alain Jacques and Amand George Laboratoire de Physique des Mate´riaux, Ecole des Mines de Nancy, 54052 Nancy Cedex, France (Received January 7, 1999) (Accepted in revised form March 26, 1999) Introduction Single crystalline silicon has been commonly used as a model material for detailed dislocation studies, owing to its high purity and crystalline perfection (1). A series of experiments have shown that, when subjecting pre-cracked silicon specimens to an increasing stress at higher temperatures and/or lower loading rates, a dense plastic zone can be detected easily around the crack tip, exhibiting a brittle-ductile transition (BDT) behavior (2,3). Such a brittle-ductile transition has been suggested to be a phenomenon associated with rate-dependent plasticity driven by thermally activated dislocation motion, although understanding of the fundamental process during the BDT remains incomplete (4). Undoubtedly, similar rate-controlled brittle-ductile transition may also be expected to occur under cyclic loads. In other words, a significant plastic deformation may be expected to be observed in silicon cyclically deformed at higher temperatures and/or lower strain rates. Recently, a preliminary study on the cyclic deformation behavior of single crystalline silicon was conducted by Degli-Esposti et al. (5). In their work, cylindrical samples oriented for single slip (stress axis parallel to [123]) were cyclically fatigued in tension/compression at high temperatures (1073 to 1373 K) in neutral atmosphere (70% Ar, 30% He). In conducting the fatigue test, the resolved shear-strain amplitude, ⌬⑀p, and the strain rate, ⑀˙ p, were kept constants for a given sample and varied from sample to sample. Some of the experimental results measured at 1173 K are now illustrated in Figure 1, where the abscissa is the cumulative shear-strain, ␥pc, and the ordinate is the maximum stress in each cycle, . These cyclic fatigue deformation curves are similar to those observed in FCC metals (6,7): an initial cyclic hardening is pronounced in each test, and then a cyclic saturation may be expected to occur after some more cycles. However, the intrinsic brittleness of silicon makes it difficult to observe a real saturation in a smaller number of cycles. Especially, brittle fracture may occur before the saturation achieves when a higher strain rate is used. For example, a specimen tested at a strain rate of 5 ⫻ 10⫺3/s (specimen C in Figure 1) failed after only 125 cycles (5). The widely used method to analyze the cyclic deformation behavior for a given material is to set up the cyclic stress-strain relation firstly by measuring the saturation stress as a function of the resolved * Corresponding author.
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Figure 1. Cyclic fatigue hardening curves for silicon measured at 1173K. Data from Degli-Esposti et al. (5) Specimens were cyclically fatigued with a resolved shear-strain amplitude of 5 ⫻ 10⫺3 and different strain rates: 5 ⫻ 10⫺5/s for specimen A, 5 ⫻ 10⫺4/s for B and 5 ⫻ 10⫺3/s for C.
shear-strain amplitude (6,7). Clearly, this traditional method can not be used for the analysis of the experimental results of silicon shown in Figure 1, for the real saturation stress can not be observed easily. The objective of the present work is, therefore, to establish a new approach for analyzing the low amplitude cyclic deformation behavior in single crystalline silicon. Description of the Cyclic Hardening Curve In the literature (6,7), the measured cyclic hardening curve was usually plotted on a ⫺␥pc scale, as shown in Figure 1. The cumulative shear-strain ␥pc, the quantity used to scale the abscissa of Figure 1, is given by:
␥ pc ⫽ 2N⌬ p
(1)
where N is the cycle number and ⌬⑀p is the resolved shear-strain amplitude, i.e., the full-width of the hysteresis loop at zero stress. On the other hand, the maximum stress , the quantity used to scale the ordinate of Figure 1, can be regarded as a measure of the material resistance to cyclic deformation at a given shear-strain amplitude. We name it the cyclic deformation resistance. Thus, the cyclic hardening curve can be considered as a cyclic deformation resistance curve. Furthermore, the cyclic hardening can also be considered as a phenomenon associated with the variation of the material resistance to deformation during cyclic fatigue. It is interesting to make a comparison between the cyclic fatigue hardening curves shown in Figure 1 and the typical crack-growth-resistance curve for ceramics. The origin of the crack-growth-resistance curve for ceramics has been attributed to the crack-tip shielding effect due to the interactions between the microstructure and the crack (8,9). During crack propagation, a shielding zone forms and develops gradually behind the crack tip. The formation and development of the shielding zone provides an extra, gradually increased resistance to crack propagation. Thus, the apparent crack growth resistance, KR,
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increases continuously with crack extension, ⌬c. A plateau value of KR can be observed when the shielding zone is well developed. Similar analysis may also be conducted for the cyclic fatigue hardening curves, or the cyclic deformation resistance curves, shown in Figure 1. Clearly, the variation of cyclic deformation resistance, , with the cumulative shear-strain, ␥pc, can be considered as a result of the changes in material microstructure during cyclic fatigue, due mainly to the generation and motion of dislocation, the possible micro-mechanism for cyclic deformation. Note that many of the recent understandings about the increasing tendency in crack growth resistance for ceramics come from the analysis of the measured crack-growth-resistance curve (8,9). If we can establish a reliable relationship between the cyclic deformation resistance, R, and the cumulative shear-strain ␥pc, the cyclic hardening behavior can be understood more quantitatively, clearly, and directly. We now wish to give a quantitative description for the cyclic deformation resistance curve shown in Figure 1. This can be done easily by considering the similarity between the cyclic fatigue hardening curves in Figure 1 and the typical crack-growth-resistance curve. Ramachandran and Shetty (10) have proposed an empirical equation to describe the crack-resistance curve for ceramics:
冉 冊
K R ⫽ K ⬁ ⫺ K 0 exp ⫺
⌬c
(2)
where the parameters K⬁, K0, and are constants. Equation 2 predicts an increasing tendency in KR with the increasing crack extension, ⌬c. When crack extension tends to infinite, KR reaches its plateau value, K⬁. On the other hand, a crack threshold, KR0 ⫽ K⬁ ⫺ K0, exists below which no crack propagation occurs (⌬c ⫽ 0). According to the analysis of Ramachandran and Shetty, the parameter in Equation 2 is a measure of the crack-tip shielding effect. A similar empirical equation may also be established to describe the cyclic fatigue hardening curve, i.e., the variation of cyclic deformation resistance with the cumulative shear-strain. Note that, in general, there are two different processes, dislocation generation and dislocation motion, which should be considered for the determination of the cyclic deformation resistance, while there is only one process, crack propagation, which should be considered for crack resistance. Thus, the following empirical equation is selected to fit the experimental data shown in Figure 1:
冉 冊
⫽ ⬁ ⫺ G exp ⫺
冉 冊
␥ pc ␥ pc ⫺ M exp ⫺ G M
(3)
where parameters ⬁, G, M, G, and M are adjustable constants. The second and the third terms in the right-hand side of Equation 3 represent the effects of the dislocation generation and the dislocation motion on the cyclic deformation resistance, respectively. The solid lines in Figure 1 represent the best-fit results of the experimental data according to Equation 3 by a conventional exponential decay regression analysis. It is clear that Equation 3 gives excellent fits to the experimental data. The best-fit values of the parameters included in Equation 3 for each specimen are summarized in Table 1. There seems to be no direct criterion for judging which of the two exponential terms in the right-hand side of Equation 3 represents the effect of the dislocation generation or the dislocation motion on the cyclic deformation resistance. But an empirical method can be proposed by analyzing the general features of the generation and motion of dislocation during cyclic fatigue. As mentioned above, the variation of cyclic deformation resistance with the cumulative shear-strain is associated with the change in material microstructure. One can expect that, as a result of the gradual development of the dislocation structure during the cyclic fatigue, the contribution of dislocation motion to the cyclic deformation resistance should increase continuously until a saturation state of deformation achieves. On the other hand, it has been confirmed that a significant variation in the mean dislocation
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TABLE 1 Descriptive Parameters for the Cyclic Fatigue Hardening Curves from the Regression Analysis Test parameteres Specimen A B C
Parameters in Equation 3 ⫺1
⌬p
dp/dt (s )
⬁ (MPa)
G (MPa)
M (MPa)
G
M
5 ⫻ 10⫺3 5 ⫻ 10⫺3 5 ⫻ 10⫺3
5 ⫻ 10⫺5 5 ⫻ 10⫺4 5 ⫻ 10⫺3
35.9 53.3 71.0
5.6 10.2 8.1
16.0 22.0 24.3
0.04 0.15 0.11
1.96 4.52 1.67
density occurs generally at the early stage of the fatigue (11). For single crystalline silicon fatigued in tension-compression, due to its intrinsic brittleness, only a few slip systems can be activated and the number of dislocations in each active slip system is limited. So the contribution of the dislocation generation to the cyclic deformation resistance would tend to saturation faster than that of the dislocation motion. Denoting
冉 冊 冉 冊
V G ⫽ G exp ⫺
␥ pc G
(4a)
V M ⫽ M exp ⫺
␥ pc M
(4b)
Using the best-fit values of the parameters listed in Table I, the values of VG and VM for specimen A are calculated and plotted as functions of the cumulative shear-strain in Figure 2, respectively. It can be seen that, with the increasing cumulative shear-strain, the VG-value decreases sharply at first and then tends to invariable, while the VM-value decreases smoothly and continuously. It can be concluded immediately from Figure 2 that the VG term in Equation 3 may represent the effect of the dislocation generation on the cyclic deformation resistance, while the VM term represents the effect of the dislocation motion. The same conclusions can also be obtained for specimens B and C.
Figure 2. The variations of VG (Equation 4a) and VM (Equation 4b) with the cumulative shear-strain, ␥pc, for specimen A.
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Discussion Having established the empirical equation for describing the variation of the cyclic deformation resistance with the cumulative shear-strain, it is necessary to attribute some physical meanings to each of the parameters included in this equation. (1) Cyclic Hardening The essence of the present new approach for studying the cyclic deformation behavior is that the cyclic deformation resistance is composed of two components, i.e., resistance to dislocation generation, RG, and resistance to dislocation motion, RM:
⫽ RG ⫹ RM
(5)
The following relationships can be obtained directly from Equation 3:
冉 冊 冉 冊
RG ⫽ SG ⫺ G exp ⫺
␥ pc G
(6a)
RM ⫽ SM ⫺ M exp ⫺
␥ pc M
(6b)
where SG and SM are the saturation values of RG and RM, respectively; SG ⫹ SM ⫽ ⬁. The similarity between Equations 6 and 2 makes it easier to discuss the cyclic hardening behavior shown as Figure 1, in the light of the work of Ramachandran and Shetty (10) on the crack-growthresistance curves of ceramics. Obviously, the cyclic hardening results from the increasing tendency of the resistances to both dislocation generation and dislocation motion during cyclic fatigue. As can be predicted from Figure 2, the resistance to dislocation generation would achieve its saturation state more rapidly than that to dislocation motion. Accordingly, the cyclic hardening can be divided into three stages. At the first stage, both RG and RM increase sharply with the cumulative shear-strain and a rapid hardening is resulted. Since RG reaches its saturation value after only a few cycles while RM increases continuously, the cyclic hardening in the second stage is dominated mainly by the dislocation motion. Finally, a transition from the hardening to the saturation for the cyclic deformation occurs at the third stage as a result of the formation of a well-developed dislocation structure. Certainly, the characteristics of such a well-developed dislocation structure is dependent on the test conditions, such as temperature, loading rate, resolved shear-strain amplitude, etc. (2) Cyclic Saturation The present new approach provides a simple and convenient method to determine the states of cyclic saturation characterized by the saturation value of the cyclic deformation resistance, S, and the corresponding value of cycle number, NS. In fact, the value of parameter ⬁ listed in Table I can be treated directly as the saturation value of the cyclic deformation resistance, S. The increase saturation resistance to deformation, ⬁ or S, with the increasing strain rate, as indicated in Table I, seems to be reasonable, for it has been concluded that the average dislocation kinetics is rate-controlled for silicon cyclically fatigued in tension-compression (5) and a higher loading rate would impede the dislocation motion.
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Although the saturation value of the cyclic deformation resistance is calculated as the cyclic deformation resistance for a cycle number of infinite, a quasi-steady state of deformation can still be expected to occur after a finite numbers of cycles. Rewriting Equation 3 as:
冉 冊
⌬ ⫽ ⬁ ⫺ ⫽ G exp ⫺
冉 冊
␥ pc ␥ pc ⫹ M exp ⫺ G M
(7)
it can be seen that the absolute value of ⌬ in Equation 7 decreases with increasing ␥pc, thereby the cycle number, N. When the cycle number increases to a characteristic value, the value of ⌬ would become so small that it can be neglected compared with the value of the saturation resistance to deformation, ⬁. Thus, the measured R would remain nearly constant and a quasi-steady state of deformation would be observed. The characteristic value of the cycle number corresponding to such a quasi-steady state of deformation can be treated approximately as the value of NS. In continuation of this idea, the values of NS are calculated to be about 820 for specimen A, about 2000 for specimen B, and about 760 for C, respectively, by reasonably assuming ⌬ ⫽ 0.05 GPa based on the consideration of the measurement precision of the test system. Besides the parameters S and NS discussed above, G and M in Equation 3 are also important parameters for describing the cyclic saturation behavior. According to the analysis of Ramachandran and Shetty (10), the parameter in Equation 2 is related to the dimension of the well-developed shielding zone around the crack tip. Similarly, the parameters G and M in Equation 3 can be related (but not equal) to the cumulative shear-strain corresponding to the saturation states of the resistances to dislocation generation and dislocation motion, respectively. Because of the rate-controlled nature of the dislocation motion in silicon, a higher loading rate would result in large values of G and M. This seems to be the reason why the values of G and M for specimen B are larger than those for specimen A, respectively. One can predict that, among all the three specimens tested, the largest values of G and M would be observed in specimen C. The inconsistency of this prediction with the experimental results listed in Table I can be explained as a result of the competition between brittle fracture and plastic deformation. Generally, two different physical processes exist in silicon cyclically fatigued in tension-compression (4): one being the generation and the motion of dislocation, which may result in an increase in the material resistance to cyclic deformation, hence the maximum stress in each cycle; the other being the increase in the stress intensity at the crack tips of pre-existing defects, which results from the increase in the applied stress, i.e., the maximum stress in each cycle. At a lower level of loading rate, such as those used for specimens A and B, even the maximum applied stress resulted from the cyclic hardening, i.e., the saturation stress, may not result in a stress intensity higher than its critical value, the fracture toughness, at the crack tip. So a quasi-steady state of deformation can be achieved. At a higher level of loading rate, however, a relatively higher applied stress is needed for the generation and the motion of dislocation, thus the stress intensity at the crack tip can reach its critical value more easily. As a result, the brittle fracture may occur usually before the achievement of the quasi-steady state of cyclic deformation in this case. Since the brittle fracture of specimen C occurs after only 125 cycles, it is reasonable to conclude that a real quasi-steady state of the dislocation motion may not achieve and the best-fit values of G and M may be unreliable due to the uncompleted experimental data. (3) The Initial State of Test Specimen Setting ␥pc ⫽ 0 in Equation 6 would yield the values of the cyclic deformation resistance corresponding to the initial state for a specimen. In other words, the values of (SG ⫺ G) and (SM ⫺ M) are the intrinsic values of the material resistances to dislocation generation and dislocation motion, respec-
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tively. Note that, according to the description of Degli-Esposti et al. (5), the specimens were prestrained at 1373K for a few cycles before the deformation measurement at 1173K to introduce dislocation with a sufficient density in order to prevent specimens from brittle fracture during the subsequent tests. Such a pre-straining treatment may result in some uncertainties in the determination of the values of G and M. Furthermore, the values of SG and SM in Equation 6 can also not be determined with the present new approach. So it seems to be impossible at present to discuss the physical meanings of G and M more clearly. Acknowledgments One of the authors, J. Gong, wishes to thank the Center National de la Recherche Scientifique (CNRS, France) for the award of a CNRS-K.C.WONG post-doctoral fellowship. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
A. George, J. Phys. III. 1, 909 (1991). G. Michot and A. George, Scrip. Mater. 20, 1495 (1986). R. Behrensmier, M. Brede, and P. Haasen, Scripa Mater. 21, 1581 (1987). Y.-B. Xin and K. J. Hsia, Acta Mater. 45, 1747 (1997). J. Degli-Esposti, A. Jacques, and A. George, Mater. Sci. Eng. A234 –236, 1000 (1997). H. Mughrabi, Mater. Sci. Eng. 38, 207 (1978). Z. S. Basinski and S. J. Basinski, Progr. Mater. Sci. 36, 89 (1992). A. G. Evans, J. Am. Ceram. Soc. 73, 187 (1990). R. W. Steinbrech, in Fracture Mechanics of Ceramics, vol. 9, ed. R. C. Brade, D. P. H. Hasselman, D. Munz, M. Sakai, and V. Ya. Shevchenko, pp. 187–208, Plenum, New York (1992). N. Ramachandran and D. K. Shetty, J. Am. Ceram. Soc. 74, 2634 (1991). S. Suresh, Fatigue of Materials, Ch. 2, Cambridge University Press, London (1991).
Pergamon
PII S1359-6462(99)00116-5
LOW AMPLITUDE CYCLIC DEFORMATION BEHAVIOR OF SINGLE CRYSTALLINE SILICON Jianghong Gong* Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, People’s Republic of China
Alain Jacques and Amand George Laboratoire de Physique des Mate´riaux, Ecole des Mines de Nancy, 54052 Nancy Cedex, France (Received January 7, 1999) (Accepted in revised form March 26, 1999) Introduction Single crystalline silicon has been commonly used as a model material for detailed dislocation studies, owing to its high purity and crystalline perfection (1). A series of experiments have shown that, when subjecting pre-cracked silicon specimens to an increasing stress at higher temperatures and/or lower loading rates, a dense plastic zone can be detected easily around the crack tip, exhibiting a brittle-ductile transition (BDT) behavior (2,3). Such a brittle-ductile transition has been suggested to be a phenomenon associated with rate-dependent plasticity driven by thermally activated dislocation motion, although understanding of the fundamental process during the BDT remains incomplete (4). Undoubtedly, similar rate-controlled brittle-ductile transition may also be expected to occur under cyclic loads. In other words, a significant plastic deformation may be expected to be observed in silicon cyclically deformed at higher temperatures and/or lower strain rates. Recently, a preliminary study on the cyclic deformation behavior of single crystalline silicon was conducted by Degli-Esposti et al. (5). In their work, cylindrical samples oriented for single slip (stress axis parallel to [123]) were cyclically fatigued in tension/compression at high temperatures (1073 to 1373 K) in neutral atmosphere (70% Ar, 30% He). In conducting the fatigue test, the resolved shear-strain amplitude, ⌬⑀p, and the strain rate, ⑀˙ p, were kept constants for a given sample and varied from sample to sample. Some of the experimental results measured at 1173 K are now illustrated in Figure 1, where the abscissa is the cumulative shear-strain, ␥pc, and the ordinate is the maximum stress in each cycle, . These cyclic fatigue deformation curves are similar to those observed in FCC metals (6,7): an initial cyclic hardening is pronounced in each test, and then a cyclic saturation may be expected to occur after some more cycles. However, the intrinsic brittleness of silicon makes it difficult to observe a real saturation in a smaller number of cycles. Especially, brittle fracture may occur before the saturation achieves when a higher strain rate is used. For example, a specimen tested at a strain rate of 5 ⫻ 10⫺3/s (specimen C in Figure 1) failed after only 125 cycles (5). The widely used method to analyze the cyclic deformation behavior for a given material is to set up the cyclic stress-strain relation firstly by measuring the saturation stress as a function of the resolved * Corresponding author.
109
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Figure 1. Cyclic fatigue hardening curves for silicon measured at 1173K. Data from Degli-Esposti et al. (5) Specimens were cyclically fatigued with a resolved shear-strain amplitude of 5 ⫻ 10⫺3 and different strain rates: 5 ⫻ 10⫺5/s for specimen A, 5 ⫻ 10⫺4/s for B and 5 ⫻ 10⫺3/s for C.
shear-strain amplitude (6,7). Clearly, this traditional method can not be used for the analysis of the experimental results of silicon shown in Figure 1, for the real saturation stress can not be observed easily. The objective of the present work is, therefore, to establish a new approach for analyzing the low amplitude cyclic deformation behavior in single crystalline silicon. Description of the Cyclic Hardening Curve In the literature (6,7), the measured cyclic hardening curve was usually plotted on a ⫺␥pc scale, as shown in Figure 1. The cumulative shear-strain ␥pc, the quantity used to scale the abscissa of Figure 1, is given by:
␥ pc ⫽ 2N⌬ p
(1)
where N is the cycle number and ⌬⑀p is the resolved shear-strain amplitude, i.e., the full-width of the hysteresis loop at zero stress. On the other hand, the maximum stress , the quantity used to scale the ordinate of Figure 1, can be regarded as a measure of the material resistance to cyclic deformation at a given shear-strain amplitude. We name it the cyclic deformation resistance. Thus, the cyclic hardening curve can be considered as a cyclic deformation resistance curve. Furthermore, the cyclic hardening can also be considered as a phenomenon associated with the variation of the material resistance to deformation during cyclic fatigue. It is interesting to make a comparison between the cyclic fatigue hardening curves shown in Figure 1 and the typical crack-growth-resistance curve for ceramics. The origin of the crack-growth-resistance curve for ceramics has been attributed to the crack-tip shielding effect due to the interactions between the microstructure and the crack (8,9). During crack propagation, a shielding zone forms and develops gradually behind the crack tip. The formation and development of the shielding zone provides an extra, gradually increased resistance to crack propagation. Thus, the apparent crack growth resistance, KR,
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increases continuously with crack extension, ⌬c. A plateau value of KR can be observed when the shielding zone is well developed. Similar analysis may also be conducted for the cyclic fatigue hardening curves, or the cyclic deformation resistance curves, shown in Figure 1. Clearly, the variation of cyclic deformation resistance, , with the cumulative shear-strain, ␥pc, can be considered as a result of the changes in material microstructure during cyclic fatigue, due mainly to the generation and motion of dislocation, the possible micro-mechanism for cyclic deformation. Note that many of the recent understandings about the increasing tendency in crack growth resistance for ceramics come from the analysis of the measured crack-growth-resistance curve (8,9). If we can establish a reliable relationship between the cyclic deformation resistance, R, and the cumulative shear-strain ␥pc, the cyclic hardening behavior can be understood more quantitatively, clearly, and directly. We now wish to give a quantitative description for the cyclic deformation resistance curve shown in Figure 1. This can be done easily by considering the similarity between the cyclic fatigue hardening curves in Figure 1 and the typical crack-growth-resistance curve. Ramachandran and Shetty (10) have proposed an empirical equation to describe the crack-resistance curve for ceramics:
冉 冊
K R ⫽ K ⬁ ⫺ K 0 exp ⫺
⌬c
(2)
where the parameters K⬁, K0, and are constants. Equation 2 predicts an increasing tendency in KR with the increasing crack extension, ⌬c. When crack extension tends to infinite, KR reaches its plateau value, K⬁. On the other hand, a crack threshold, KR0 ⫽ K⬁ ⫺ K0, exists below which no crack propagation occurs (⌬c ⫽ 0). According to the analysis of Ramachandran and Shetty, the parameter in Equation 2 is a measure of the crack-tip shielding effect. A similar empirical equation may also be established to describe the cyclic fatigue hardening curve, i.e., the variation of cyclic deformation resistance with the cumulative shear-strain. Note that, in general, there are two different processes, dislocation generation and dislocation motion, which should be considered for the determination of the cyclic deformation resistance, while there is only one process, crack propagation, which should be considered for crack resistance. Thus, the following empirical equation is selected to fit the experimental data shown in Figure 1:
冉 冊
⫽ ⬁ ⫺ G exp ⫺
冉 冊
␥ pc ␥ pc ⫺ M exp ⫺ G M
(3)
where parameters ⬁, G, M, G, and M are adjustable constants. The second and the third terms in the right-hand side of Equation 3 represent the effects of the dislocation generation and the dislocation motion on the cyclic deformation resistance, respectively. The solid lines in Figure 1 represent the best-fit results of the experimental data according to Equation 3 by a conventional exponential decay regression analysis. It is clear that Equation 3 gives excellent fits to the experimental data. The best-fit values of the parameters included in Equation 3 for each specimen are summarized in Table 1. There seems to be no direct criterion for judging which of the two exponential terms in the right-hand side of Equation 3 represents the effect of the dislocation generation or the dislocation motion on the cyclic deformation resistance. But an empirical method can be proposed by analyzing the general features of the generation and motion of dislocation during cyclic fatigue. As mentioned above, the variation of cyclic deformation resistance with the cumulative shear-strain is associated with the change in material microstructure. One can expect that, as a result of the gradual development of the dislocation structure during the cyclic fatigue, the contribution of dislocation motion to the cyclic deformation resistance should increase continuously until a saturation state of deformation achieves. On the other hand, it has been confirmed that a significant variation in the mean dislocation
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TABLE 1 Descriptive Parameters for the Cyclic Fatigue Hardening Curves from the Regression Analysis Test parameteres Specimen A B C
Parameters in Equation 3 ⫺1
⌬p
dp/dt (s )
⬁ (MPa)
G (MPa)
M (MPa)
G
M
5 ⫻ 10⫺3 5 ⫻ 10⫺3 5 ⫻ 10⫺3
5 ⫻ 10⫺5 5 ⫻ 10⫺4 5 ⫻ 10⫺3
35.9 53.3 71.0
5.6 10.2 8.1
16.0 22.0 24.3
0.04 0.15 0.11
1.96 4.52 1.67
density occurs generally at the early stage of the fatigue (11). For single crystalline silicon fatigued in tension-compression, due to its intrinsic brittleness, only a few slip systems can be activated and the number of dislocations in each active slip system is limited. So the contribution of the dislocation generation to the cyclic deformation resistance would tend to saturation faster than that of the dislocation motion. Denoting
冉 冊 冉 冊
V G ⫽ G exp ⫺
␥ pc G
(4a)
V M ⫽ M exp ⫺
␥ pc M
(4b)
Using the best-fit values of the parameters listed in Table I, the values of VG and VM for specimen A are calculated and plotted as functions of the cumulative shear-strain in Figure 2, respectively. It can be seen that, with the increasing cumulative shear-strain, the VG-value decreases sharply at first and then tends to invariable, while the VM-value decreases smoothly and continuously. It can be concluded immediately from Figure 2 that the VG term in Equation 3 may represent the effect of the dislocation generation on the cyclic deformation resistance, while the VM term represents the effect of the dislocation motion. The same conclusions can also be obtained for specimens B and C.
Figure 2. The variations of VG (Equation 4a) and VM (Equation 4b) with the cumulative shear-strain, ␥pc, for specimen A.
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Discussion Having established the empirical equation for describing the variation of the cyclic deformation resistance with the cumulative shear-strain, it is necessary to attribute some physical meanings to each of the parameters included in this equation. (1) Cyclic Hardening The essence of the present new approach for studying the cyclic deformation behavior is that the cyclic deformation resistance is composed of two components, i.e., resistance to dislocation generation, RG, and resistance to dislocation motion, RM:
⫽ RG ⫹ RM
(5)
The following relationships can be obtained directly from Equation 3:
冉 冊 冉 冊
RG ⫽ SG ⫺ G exp ⫺
␥ pc G
(6a)
RM ⫽ SM ⫺ M exp ⫺
␥ pc M
(6b)
where SG and SM are the saturation values of RG and RM, respectively; SG ⫹ SM ⫽ ⬁. The similarity between Equations 6 and 2 makes it easier to discuss the cyclic hardening behavior shown as Figure 1, in the light of the work of Ramachandran and Shetty (10) on the crack-growthresistance curves of ceramics. Obviously, the cyclic hardening results from the increasing tendency of the resistances to both dislocation generation and dislocation motion during cyclic fatigue. As can be predicted from Figure 2, the resistance to dislocation generation would achieve its saturation state more rapidly than that to dislocation motion. Accordingly, the cyclic hardening can be divided into three stages. At the first stage, both RG and RM increase sharply with the cumulative shear-strain and a rapid hardening is resulted. Since RG reaches its saturation value after only a few cycles while RM increases continuously, the cyclic hardening in the second stage is dominated mainly by the dislocation motion. Finally, a transition from the hardening to the saturation for the cyclic deformation occurs at the third stage as a result of the formation of a well-developed dislocation structure. Certainly, the characteristics of such a well-developed dislocation structure is dependent on the test conditions, such as temperature, loading rate, resolved shear-strain amplitude, etc. (2) Cyclic Saturation The present new approach provides a simple and convenient method to determine the states of cyclic saturation characterized by the saturation value of the cyclic deformation resistance, S, and the corresponding value of cycle number, NS. In fact, the value of parameter ⬁ listed in Table I can be treated directly as the saturation value of the cyclic deformation resistance, S. The increase saturation resistance to deformation, ⬁ or S, with the increasing strain rate, as indicated in Table I, seems to be reasonable, for it has been concluded that the average dislocation kinetics is rate-controlled for silicon cyclically fatigued in tension-compression (5) and a higher loading rate would impede the dislocation motion.
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Although the saturation value of the cyclic deformation resistance is calculated as the cyclic deformation resistance for a cycle number of infinite, a quasi-steady state of deformation can still be expected to occur after a finite numbers of cycles. Rewriting Equation 3 as:
冉 冊
⌬ ⫽ ⬁ ⫺ ⫽ G exp ⫺
冉 冊
␥ pc ␥ pc ⫹ M exp ⫺ G M
(7)
it can be seen that the absolute value of ⌬ in Equation 7 decreases with increasing ␥pc, thereby the cycle number, N. When the cycle number increases to a characteristic value, the value of ⌬ would become so small that it can be neglected compared with the value of the saturation resistance to deformation, ⬁. Thus, the measured R would remain nearly constant and a quasi-steady state of deformation would be observed. The characteristic value of the cycle number corresponding to such a quasi-steady state of deformation can be treated approximately as the value of NS. In continuation of this idea, the values of NS are calculated to be about 820 for specimen A, about 2000 for specimen B, and about 760 for C, respectively, by reasonably assuming ⌬ ⫽ 0.05 GPa based on the consideration of the measurement precision of the test system. Besides the parameters S and NS discussed above, G and M in Equation 3 are also important parameters for describing the cyclic saturation behavior. According to the analysis of Ramachandran and Shetty (10), the parameter in Equation 2 is related to the dimension of the well-developed shielding zone around the crack tip. Similarly, the parameters G and M in Equation 3 can be related (but not equal) to the cumulative shear-strain corresponding to the saturation states of the resistances to dislocation generation and dislocation motion, respectively. Because of the rate-controlled nature of the dislocation motion in silicon, a higher loading rate would result in large values of G and M. This seems to be the reason why the values of G and M for specimen B are larger than those for specimen A, respectively. One can predict that, among all the three specimens tested, the largest values of G and M would be observed in specimen C. The inconsistency of this prediction with the experimental results listed in Table I can be explained as a result of the competition between brittle fracture and plastic deformation. Generally, two different physical processes exist in silicon cyclically fatigued in tension-compression (4): one being the generation and the motion of dislocation, which may result in an increase in the material resistance to cyclic deformation, hence the maximum stress in each cycle; the other being the increase in the stress intensity at the crack tips of pre-existing defects, which results from the increase in the applied stress, i.e., the maximum stress in each cycle. At a lower level of loading rate, such as those used for specimens A and B, even the maximum applied stress resulted from the cyclic hardening, i.e., the saturation stress, may not result in a stress intensity higher than its critical value, the fracture toughness, at the crack tip. So a quasi-steady state of deformation can be achieved. At a higher level of loading rate, however, a relatively higher applied stress is needed for the generation and the motion of dislocation, thus the stress intensity at the crack tip can reach its critical value more easily. As a result, the brittle fracture may occur usually before the achievement of the quasi-steady state of cyclic deformation in this case. Since the brittle fracture of specimen C occurs after only 125 cycles, it is reasonable to conclude that a real quasi-steady state of the dislocation motion may not achieve and the best-fit values of G and M may be unreliable due to the uncompleted experimental data. (3) The Initial State of Test Specimen Setting ␥pc ⫽ 0 in Equation 6 would yield the values of the cyclic deformation resistance corresponding to the initial state for a specimen. In other words, the values of (SG ⫺ G) and (SM ⫺ M) are the intrinsic values of the material resistances to dislocation generation and dislocation motion, respec-
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tively. Note that, according to the description of Degli-Esposti et al. (5), the specimens were prestrained at 1373K for a few cycles before the deformation measurement at 1173K to introduce dislocation with a sufficient density in order to prevent specimens from brittle fracture during the subsequent tests. Such a pre-straining treatment may result in some uncertainties in the determination of the values of G and M. Furthermore, the values of SG and SM in Equation 6 can also not be determined with the present new approach. So it seems to be impossible at present to discuss the physical meanings of G and M more clearly. Acknowledgments One of the authors, J. Gong, wishes to thank the Center National de la Recherche Scientifique (CNRS, France) for the award of a CNRS-K.C.WONG post-doctoral fellowship. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
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