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Atomistic simulation of crack cleavage and blunting in bcc-Fe
Materials Science and Engineering A349 (2003) 29 /35 www.elsevier.com/locate/msea
Atomistic simulation of crack cleavage and blunting in bcc-Fe Ya-Fang Guo a,b,*, Chong-Yu Wang b,c, Dong-Liang Zhao b a
b
Northern JiaoTong University, Beijing 100044, People’s Republic of China Central Iron and Steel Research Institute, Functional Materials Division, Beijing 100081, People’s Republic of China c Tsinghua University, Beijing 100084, People’s Republic of China Received 29 October 2001; received in revised form 1 May 2002
Abstract The behavior of crack propagation at different strain rate and temperature in bcc-Fe has been studied by molecular dynamic simulation method with fixed-displacement boundary condition. The results show that at low temperature, the cleavage fracture and the twin (or stacking fault) formation are cooperative processes in brittle fracture, and the shear process induced by the twin formation is favorable for bond breakage at the crack tip. At higher temperature, the twinning becomes weakened, and vanishes at the brittle-to-ductile transition temperature accompanied dislocation nucleation, which is perpendicular to the crack surface. The crack tip is blunted by the plastic deformation due to dislocation nucleation and emission. It can be concluded that the formation of stacking fault and twin at crack tip is particularly important for brittle cleavage. The shear process and the plastic deformation can be taken as, respectively, a feature of brittle fracture and a feature of ductile fracture in bcc-Fe. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Crack propagation; bcc-Fe; Twin formation; Dislocation nucleation; Brittle-to-ductile transition
1. Introduction The strength of body centered cubic (bcc) metals usually dependents strongly on the temperature, which is displayed by the well known brittle-to-ductile transition (BDT) phenomenon. The BDT mechanism has been an important subject in the materials strength, and attracted much interests and attentions from the experimental and theoretical aspects. In 1966, Kelly pointed out that a solid material would be either ductile or truly brittle, which depends on the ratio of theoretical shear strength to theoretical tensile strength [1]. Afterwards, Rice and Thomson distinguished the intrinsic ductile materials from the brittle materials according to whether the dislocation is emitted from crack tip or not when the stress intensity is lower than the KIC [2]. However, dislocation nucleation at the crack tip in brittle fracture has been observed in many experiments [3 /7]. These studies suggest that the BDT is most likely controlled by either dislocation nucleation or dislocation mobility,
* Corresponding author. Fax: /86-10-62182756. E-mail address: [email protected] (Y.-F. Guo).
which is known as nucleation-controlled BDT [2,8,9] or mobility-controlled BDT [10 /12]. Obviously, people still lack of the knowledge on the basic physical process of BDT even for the simplest and well-characterized materials such as transition metal bcc-Fe. Since the fracture strength is governed eventually by the competition of the dynamic atomic bond breaking with the dislocation mobility, it is meaningful to use the molecular dynamics (MD) simulation technique to study the crack propagation behavior. Some atomistic models of pre-existing crack propagating under imposed loading have been used to simulate brittle cleavage and dislocation emission at the highly nonlinear region of crack tip, and different kinds of atomistic potentials are adopted in these simulations [13 /21]. In addition, twin was clearly observed in the simulations and experiments [13,22/26], which indicates that the twinning and the fracture at low temperature or at high strain rate are cooperating processes in bcc-Fe. But as we know, the emission of partial dislocation can go with the formation of stacking fault or twin. Thus the brittle fracture cannot be taken as a simple cleavage process of the two opposite crack surfaces. It is valuable to study theoretically and experimentally the effects of the formation of
0921-5093/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-5093(02)00287-3
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Y.-F. Guo et al. / Materials Science and Engineering A349 (2003) 29 /35
stacking fault and twin on the brittle fracture, and to find out the factors affecting the BDT formation. In this work, we use MD method to simulate the crack propagation at different strain rate and temperature in bcc-Fe. The aim of the work is to find out the effect of twining on cleavage fracture and the effect of temperature on crack propagation. As we can see in the following, the simple fixed-displacement boundary condition can be used to obtain reasonable result if the selected MD relaxation region is large enough.
2. Simulation details for crack tip Fig. 2. Geometry of {0 1 0} 1 0 1 crack in bcc-Fe.
As shown in Fig. 1, the model used in the simulation includes three regions: the interior atomistic region I containing the crack tip, the exterior continuum region II and the boundary region III. The boundary region is a common zone between atomistic region and continuum region, and the continuum region can be taken into account in MD simulation through its effect on the common boundary region. We have used Finnis / Sinclair potential [27,28], a well known N-body for bcc-Fe in the simulation, and the boundary region consists of two layers of atoms, which is equal to the cut-off distance of the Finnis /Sinclair potential. In the atomistic region, the initial configuration of the pre-existing crack is determined by shifting the atom from the position of perfect crystal to the position that is specified by the anisotropic elasticity continuum mechanics equations [29] for a desired value of stress intensity KI. As shown in Fig. 2, the crack surface is (0 1 0) plane and the crack front is oriented along [1 0 1] direction. We consider the plane strain problem and apply the periodic boundary condition to [1 0 1] direction. We test several systems of different size beforehand, and find that the results, which will be considered in this work, such as the twining process under loading and the dislocation nucleation at the crack tip, are insensitive to the system size if the atomistic region is large enough. In the present simulation, we have selected a system that the whole atomistic region consists of 180 ¯ direction, 256 atomic layers atomic layers along [1 0 1] along [0 1 0] direction, and 6 atomistic layers along
Fig. 1. Scheme of the calculation model: region I is the atomistic regions, region II is the continuum region, and region III is the common zone between atomistic and continuum.
[1 0 1] direction, includes totally 138 240 atoms. The ˚. initial crack length is about 37 A Fig. 3 illustrates the crack loading process. Before applying the external loading, the atoms of the initial crack are relaxed for removing the effect of surface relaxation on the crack tip process. The external loading is applied increasingly according to the anisotropic linear-elastic solution, and the atoms in the boundary region shift accordingly, which will be further used as the fixed-displacement boundary for the interior atomistic region. For a specified stress intensity KI, the system is relaxed under fixed-displacement boundary condition for hundreds time steps by a magnitude 5 / 10 15 s. The temperature of the system keeps invariant during the loading process, which is obtained by scaling
Fig. 3. Flow chart for the crack loading processes.
Y.-F. Guo et al. / Materials Science and Engineering A349 (2003) 29 /35 Table 1 Stress intensity factor KI/KIC for bcc-Fe determined by MD using F-S N-body potential at fixed-displacement border condition Temperature (K) {0 1 0} 1 0 1 {0 1 0} 1 0 1 {0 1 0} 1 0 0
K K K
5
100
200
300
400
600
1.11 0.89 2.00
1.09 0.90 1.80
1.07 0.94 1.72
1.04 0.98 1.71
1.04 0.99 1.65
1.05 1.01 1.64
all atoms instantaneous velocities with the appropriate Maxwell /Boltzmann distribution at a specified temperature. The strain rate can be expressed as ‘KIC/time steps’, and the crack system is simulated at different strain rate and at different temperature respectively. One must keep in mind that the stress intensity factor KI decreases with the movement of crack tip because KI is related to the length of the crack. So following the movement of crack tip, the KI used in the current relaxation step must be recalculated according to continuum mechanics solution for the updated length
31
of the crack. Moreover, the fixed-displacement boundary model can ensure the equality of displacement field, and thus the strain between the descriptions of the lattice and the continuum in the common boundary region. In order to obtain the critical stress intensity factor KIC [30], the free surface energy is calculated by using Finnis /Sinclair N-body potential. The calculated surface energy of (0 1 0) surface is 1.90 J m 2, which is very close to the value given by previous calculation [13], but smaller than the experimental value 2.32 J m 2. For the consistency of result, we use it to calculate the critical stress intensity factor in the present simulation.
3. Results and discussion 3.1. Critical stress intensity factor As well known, the lattice trapping owing to the discrete atomic bond will cause the crack to be stable
Fig. 4. Atomic configuration of crack at 5 K: (a) is the initial crack under KI /1.0KIC, (b) is the equilibrium configuration after 4000 steps relaxation of crack under KI /1.0KIC, (c) and (d) are the crack configurations, separately after 700 and 900 steps relaxation under KI /1.2KIC. Cross indicates the initial position of crack tip.
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Y.-F. Guo et al. / Materials Science and Engineering A349 (2003) 29 /35
and not to advance until a load K/I is reached, which is somewhat larger than the critical Griffith load KIC. Similarly, the crack will not heal until a load K/I / B/KIC is reached. The critical loads K/I and K/I are called respectively, the upper and the lower trapping limit. In the previous studies, it has been found that the upper and the lower trapping limits are sensitive to the simulation conditions [13,15,18] such as the type of atomistic potential, the boundary condition, the crystallography and the temperature. Table 1 lists our simulation results. It can be found that the calculated K/I and K/I for {0 1 0} 1 0 1 crack in bcc-Fe by using Finnis /Sinclair potential are very close to the theoretical critical stress intensity factor KIC according to the Griffith criterion, where cleavage occurs along {1 0 0} planes. Comparison with the results obtained by stress-boundary condition and flexible-boundary condition [13], we can see that the fixed-displacement boundary condition will be valid for simulating crack propagation if the system size is selected large enough. But when the crack front lies on 1 0 0 direction, the K/I value increases greatly. The differences between K/I ; K/I and KIC at high temperature are smaller than those at lower temperature, which indicate the trapping barrier can be overcome by thermal activation. The results also show that the upper and the lower trapping limits are almost invariant with the temperature when the temperature of the crack system is higher than 200 K.
stacking faults formation. At 900 steps as shown in Fig. 4(d), the twin transformation from stacking fault is observed in crack tip when the total displacement along the 1 1 1 slip direction reaches the Burgers vector of b . At the same time, the system potential energy also reaches the maximum and will decreases rapidly once the twin is formed, whereas the kinetic energy will increase. Thus we can observe from Fig. 4 that, although the cracks at low temperature display a brittle character of extension, an obvious elastic twinning process is observed in the 1 1 1 {1 1 2} slip system after the emission of one a61 1 1 partial dislocation or more. It can be concluded that the brittle cleavage of crack is a process accompanied bond breakage and twin growth along the crack tip. No perfect dislocation has been observed in our simulation, which indicates that instead of the formation of a complete edge dislocation, the formations of stacking fault and twin are more favorable in the 1 1 1 {1 1 2} slip system of bcc-Fe at the
3.2. Formation of SF and twin Many experiments and simulations reveal that twinning and fracture at low temperature or at high strain rate are cooperating processes in bcc-Fe [22,26]. From Fig. 4(b), we can find that although the bond rupture is not evident at the crack tip, the stacking fault has been formed after 4000 steps relaxation of the initial crack at critical stress intensity factor KIC, which indicates that the stress in crack tip is relaxed by the formation of stacking fault. When the imposed loading increases to 2.2KIC, some long twins have been formed along the crack tip at 5 K accompanied with the crack extension, as shown in Fig. 6(a). Fig. 5 shows the changes of the system energies, including the kinetic energy EK, the potential energy EP and the total energy ETOT, during the relaxation of crack under 1.2KIC at 5 K. Y-axis denotes the energy per atom in the simulation system and X-axis denotes the time step. Three high steps can be seen in the potential energy curve, at 700, 900 and 1000 steps respectively, which correspond to the atomic bond breaking at the crack tip. Comparing the atomic configuration in Fig. 4(b) with that in Fig. 4(c), we can see that the stacking faults extend after 700 steps relaxation, thus the imposing energy due to the external loading is released by
Fig. 5. Energetic curves of crack relaxation at 5 K with KI /1.2KIC: (a) is kinetic energy per atom, (b) is potential energy per atom and (c) is total energy per atom.
Y.-F. Guo et al. / Materials Science and Engineering A349 (2003) 29 /35
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Fig. 6. Atomic configuration of crack with KI /2.2KIC at (a) 5 K (b) 100 K (c) 200 K (d) 300 K.
temperature near 0 K. Additionally, it is found that the average velocity of twin extension under higher strain rate is greater than that under lower strain rate, consistent with the experimental observation. 3.3. Temperature effect on crack propagation Fig. 6(a) /(c) which correspond respectively to the temperature of 5, 100 and 200 K, show clearly that the crack propagation along with the twin formation in crack tip is intrinsically brittle in bcc-Fe. Whereas Fig. 6(d) shows that the ductile extension appears under 300 K accompanied with branching at the crack tip. In order to exhibit the width and the length of the twin, there are more atoms displayed in Fig. 6 than in Fig. 4 or in Fig. 7. It can be observed that the width and the length of twin at lower temperature are apparently larger than that at higher temperature. The average velocities of twin extension under strain rate of 0.1KIC/400 steps at 5, 100 and 200 K are about 500, 250 and 120 m s 1 respectively. We can conclude that the twinning process
at the crack tip will be weakened at higher temperature, and will disappear eventually near the BDT temperature. In general, the increased energy due to the loading of the system will transform into surface energy and plastic strain energy during the crack propagation. We can see in Fig. 6 that the increased energy due to loads transform into surface energy and twin or stacking fault shear deformation energy below BDT temperature, whereas the shear deformation energy is substituted by plastic strain energy due to the dislocation nucleation and emission after BDT. Therefore, we can take the occurrence of the shear deformation as a notable feature for distinguishing the brittle and the ductile fractures for bcc-Fe, and make an assumption that stacking fault or twin plays an important role on crack cleavage, while the shear process at the crack tip is favorable to the atomic bond breakage. Comparing the results of the upper trapping limits K/I ; we find that the K/I value of {0 1 0} 1 0 0 crack is much higher than that of {0 1 0} 1 0 1 crack because there is no twinning
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Y.-F. Guo et al. / Materials Science and Engineering A349 (2003) 29 /35
Fig. 7. Atomic configuration of crack: (a) at 200 K with KI /1.4KIC, (b) at 300 K with KI /1.4KIC, (c) at 300 K with KI /1.6KIC and (d) at 300 K with KI /2.0KIC.
process in {0 1 0} 1 0 0 crack system. Meanwhile, we can observe dislocation nucleation at the crack tip in {0 1 0} 1 0 0 crack system during the crack extension, which is an evidence for our assumption: the twin formation is favorable for crack brittle cleavage, whereas the dislocation nucleation perpendicular to crack surface blunt the crack tip. Fig. 7 indicates the atomistic process of crack tip under 300 K. Two dislocations can be seen obviously, which is nucleated along 0 1 0 direction at the stress intensity factor of 1.4KIC and finally blunt the crack tip. In addition, the crack propagates along 0 1 0 direction as the load increases, and then turns to 1 1 1 direction when plastic deformation occurs at the crack tip. However, no full dislocation is observed in our simulation while the periodic boundary condition is applied in [1 0 1] direction. Meanwhile, no dislocation emission is observed due to the fixed-displacement boundary condition. Thus we assume that it is the dislocation nucleation and plastic deformation at high
temperature that cause the crack blunted in our simulation.
4. Conclusions Under the fixed-displacement boundary condition, the stacking fault and twin are observed in brittle fracture at low temperature. The brittle cleavage fracture is a process accompanied with the bonds breakage and the twins growth at the crack tip. Therefore, we can assume that the shear process due to the twin formation at the crack tip is favorable for bond breakage, and stacking fault or twin formation plays an important role on brittle cleavage. But at high temperature, while the twin disappears in crack tip, the dislocation is nucleated to be perpendicular to crack surface and the crack tip is blunted. We can conclude that the shear process and the plastic deformation at the crack tip are, respectively, the feature of brittle and ductile fracture in bcc-Fe.
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Acknowledgements The research was supported by ‘973’ Project from the Ministry of Science and Technology of China (Grant No. G2000067102), Chinese Nature Science Foundation (Grant No. 59971041) and NJTU Pandeng project. The authors thank Dr Ngan, Dr Jin Zhaohui and Dr Wang Shanying for their helpful discussions.
References [1] A. Kelly, Strong Solids, Oxford University Press, London, 1966, p. 22. [2] J.R. Rice, R. Thomson, Phil. Mag. A29 (1974) 73. [3] M.J. Lii, X.F. Chen, Y. Katz, W.W. Gerberich, Acta Metall. Matter. 38 (1990) 2435. [4] H. Huang, W.W. Gerberich, Acta Metall. Matter. 40 (1992) 2873. [5] W. Zielinski, M.J. Lii, W.W. Gerberich, Acta Metall. Matter. 40 (1992) 2861. [6] P.G. Marsh, W. Zielinski, H. Huang, W.W. Gerberich, Acta Metall. Matter. 40 (1992) 2883. [7] N. Narita, K. Higashida, S. Kitano, Scripta Metall. 21 (1987) 1273. [8] M. Khantha, D. Pope, V. Vitek, Phys. Rev. Lett. 73 (1994) 684. [9] G. Xu, A.S. Argon, M. Ortiz, Phil. Mag. A75 (1997) 341. [10] P.B. Hirsch, S.G. Roberts, J. Samuels, Proc. R. Soc. Lond. Ser. A 25 (1989) 421.
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[11] A. Hartmaier, P. Gumbsch, Phys. States Solid B R1 (1997) 202. [12] P. Gumbsch, J. Riedle, A. Hartmaier, H.F. Fischmeister, Science 282 (1998) 1293. [13] B. DeCelis, A.S. Argon, S. Yip, J. Appl. Phys. 54 (1983) 4864. [14] K.S. Cheng, S. Yip, Phys. Rev. Lett. 65 (1990) 2804. [15] S. Kohlhoff, P. Gumbsch, H.F. Fischmeister, Phil. Mag. A64 (1991) 851. [16] C.S. Becquart, D. Kim, J.A. Rifkin, P.C. Clapp, Mater. Sci. Eng. A170 (1993) 87. [17] F.F. Abraham, Phys. Rev. Lett. 77 (1996) 869. [18] P. Gumbsch, S.J. Zhou, B.L. Holian, Phys. Rev. B55 (1997) 3445. [19] L. Hua, H. Rafii-Tabar, M. Cross, Phil. Mag. Lett. 75 (1997) 237. [20] F. Cleri, S. Yip, D. Wolf, S.R. Phillpot, Phys. Rev. Lett. 79 (1997) 1309. [21] S.J. Zhou, D.L. Preston, P.S. Lomdahl, D.M. Beazley, Science 279 (1998) 1525. [22] K. Ogawa, Phil. Mag. 11 (1965) 217. [23] V. Vitek, Scr. Metall. 4 (1970) 725. [24] X. Wei, J.A. Moriarty, Phys. Rev. B54 (1996) 6941. [25] V. Shastry, D. Farkas, Mater. Res. Soc. Symp. Proc. 409 (1996) 75. [26] A. Machova´, G.E. Beltz, M. Chang, Modelling Simul. Mater. Sci. Eng. 7 (1999) 949. [27] M.W. Finnis, J.E. Sinclair, Phil. Mag. A50 (1984) 45. [28] M.W. Finnis, J.E. Sinclair, Phil. Mag. A53 (1986) 161. [29] G.C. Sih, H. Liebowitz (Eds.), Fracture: An Advanced Treatise, vol. 2, Academic, New York, 1968, p. 67. [30] A.A. Griffith, Philos. Trans. R. Soc. A221 (1920) 163.
Atomistic simulation of crack cleavage and blunting in bcc-Fe Ya-Fang Guo a,b,*, Chong-Yu Wang b,c, Dong-Liang Zhao b a
b
Northern JiaoTong University, Beijing 100044, People’s Republic of China Central Iron and Steel Research Institute, Functional Materials Division, Beijing 100081, People’s Republic of China c Tsinghua University, Beijing 100084, People’s Republic of China Received 29 October 2001; received in revised form 1 May 2002
Abstract The behavior of crack propagation at different strain rate and temperature in bcc-Fe has been studied by molecular dynamic simulation method with fixed-displacement boundary condition. The results show that at low temperature, the cleavage fracture and the twin (or stacking fault) formation are cooperative processes in brittle fracture, and the shear process induced by the twin formation is favorable for bond breakage at the crack tip. At higher temperature, the twinning becomes weakened, and vanishes at the brittle-to-ductile transition temperature accompanied dislocation nucleation, which is perpendicular to the crack surface. The crack tip is blunted by the plastic deformation due to dislocation nucleation and emission. It can be concluded that the formation of stacking fault and twin at crack tip is particularly important for brittle cleavage. The shear process and the plastic deformation can be taken as, respectively, a feature of brittle fracture and a feature of ductile fracture in bcc-Fe. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Crack propagation; bcc-Fe; Twin formation; Dislocation nucleation; Brittle-to-ductile transition
1. Introduction The strength of body centered cubic (bcc) metals usually dependents strongly on the temperature, which is displayed by the well known brittle-to-ductile transition (BDT) phenomenon. The BDT mechanism has been an important subject in the materials strength, and attracted much interests and attentions from the experimental and theoretical aspects. In 1966, Kelly pointed out that a solid material would be either ductile or truly brittle, which depends on the ratio of theoretical shear strength to theoretical tensile strength [1]. Afterwards, Rice and Thomson distinguished the intrinsic ductile materials from the brittle materials according to whether the dislocation is emitted from crack tip or not when the stress intensity is lower than the KIC [2]. However, dislocation nucleation at the crack tip in brittle fracture has been observed in many experiments [3 /7]. These studies suggest that the BDT is most likely controlled by either dislocation nucleation or dislocation mobility,
* Corresponding author. Fax: /86-10-62182756. E-mail address: [email protected] (Y.-F. Guo).
which is known as nucleation-controlled BDT [2,8,9] or mobility-controlled BDT [10 /12]. Obviously, people still lack of the knowledge on the basic physical process of BDT even for the simplest and well-characterized materials such as transition metal bcc-Fe. Since the fracture strength is governed eventually by the competition of the dynamic atomic bond breaking with the dislocation mobility, it is meaningful to use the molecular dynamics (MD) simulation technique to study the crack propagation behavior. Some atomistic models of pre-existing crack propagating under imposed loading have been used to simulate brittle cleavage and dislocation emission at the highly nonlinear region of crack tip, and different kinds of atomistic potentials are adopted in these simulations [13 /21]. In addition, twin was clearly observed in the simulations and experiments [13,22/26], which indicates that the twinning and the fracture at low temperature or at high strain rate are cooperating processes in bcc-Fe. But as we know, the emission of partial dislocation can go with the formation of stacking fault or twin. Thus the brittle fracture cannot be taken as a simple cleavage process of the two opposite crack surfaces. It is valuable to study theoretically and experimentally the effects of the formation of
0921-5093/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-5093(02)00287-3
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Y.-F. Guo et al. / Materials Science and Engineering A349 (2003) 29 /35
stacking fault and twin on the brittle fracture, and to find out the factors affecting the BDT formation. In this work, we use MD method to simulate the crack propagation at different strain rate and temperature in bcc-Fe. The aim of the work is to find out the effect of twining on cleavage fracture and the effect of temperature on crack propagation. As we can see in the following, the simple fixed-displacement boundary condition can be used to obtain reasonable result if the selected MD relaxation region is large enough.
2. Simulation details for crack tip Fig. 2. Geometry of {0 1 0} 1 0 1 crack in bcc-Fe.
As shown in Fig. 1, the model used in the simulation includes three regions: the interior atomistic region I containing the crack tip, the exterior continuum region II and the boundary region III. The boundary region is a common zone between atomistic region and continuum region, and the continuum region can be taken into account in MD simulation through its effect on the common boundary region. We have used Finnis / Sinclair potential [27,28], a well known N-body for bcc-Fe in the simulation, and the boundary region consists of two layers of atoms, which is equal to the cut-off distance of the Finnis /Sinclair potential. In the atomistic region, the initial configuration of the pre-existing crack is determined by shifting the atom from the position of perfect crystal to the position that is specified by the anisotropic elasticity continuum mechanics equations [29] for a desired value of stress intensity KI. As shown in Fig. 2, the crack surface is (0 1 0) plane and the crack front is oriented along [1 0 1] direction. We consider the plane strain problem and apply the periodic boundary condition to [1 0 1] direction. We test several systems of different size beforehand, and find that the results, which will be considered in this work, such as the twining process under loading and the dislocation nucleation at the crack tip, are insensitive to the system size if the atomistic region is large enough. In the present simulation, we have selected a system that the whole atomistic region consists of 180 ¯ direction, 256 atomic layers atomic layers along [1 0 1] along [0 1 0] direction, and 6 atomistic layers along
Fig. 1. Scheme of the calculation model: region I is the atomistic regions, region II is the continuum region, and region III is the common zone between atomistic and continuum.
[1 0 1] direction, includes totally 138 240 atoms. The ˚. initial crack length is about 37 A Fig. 3 illustrates the crack loading process. Before applying the external loading, the atoms of the initial crack are relaxed for removing the effect of surface relaxation on the crack tip process. The external loading is applied increasingly according to the anisotropic linear-elastic solution, and the atoms in the boundary region shift accordingly, which will be further used as the fixed-displacement boundary for the interior atomistic region. For a specified stress intensity KI, the system is relaxed under fixed-displacement boundary condition for hundreds time steps by a magnitude 5 / 10 15 s. The temperature of the system keeps invariant during the loading process, which is obtained by scaling
Fig. 3. Flow chart for the crack loading processes.
Y.-F. Guo et al. / Materials Science and Engineering A349 (2003) 29 /35 Table 1 Stress intensity factor KI/KIC for bcc-Fe determined by MD using F-S N-body potential at fixed-displacement border condition Temperature (K) {0 1 0} 1 0 1 {0 1 0} 1 0 1 {0 1 0} 1 0 0
K K K
5
100
200
300
400
600
1.11 0.89 2.00
1.09 0.90 1.80
1.07 0.94 1.72
1.04 0.98 1.71
1.04 0.99 1.65
1.05 1.01 1.64
all atoms instantaneous velocities with the appropriate Maxwell /Boltzmann distribution at a specified temperature. The strain rate can be expressed as ‘KIC/time steps’, and the crack system is simulated at different strain rate and at different temperature respectively. One must keep in mind that the stress intensity factor KI decreases with the movement of crack tip because KI is related to the length of the crack. So following the movement of crack tip, the KI used in the current relaxation step must be recalculated according to continuum mechanics solution for the updated length
31
of the crack. Moreover, the fixed-displacement boundary model can ensure the equality of displacement field, and thus the strain between the descriptions of the lattice and the continuum in the common boundary region. In order to obtain the critical stress intensity factor KIC [30], the free surface energy is calculated by using Finnis /Sinclair N-body potential. The calculated surface energy of (0 1 0) surface is 1.90 J m 2, which is very close to the value given by previous calculation [13], but smaller than the experimental value 2.32 J m 2. For the consistency of result, we use it to calculate the critical stress intensity factor in the present simulation.
3. Results and discussion 3.1. Critical stress intensity factor As well known, the lattice trapping owing to the discrete atomic bond will cause the crack to be stable
Fig. 4. Atomic configuration of crack at 5 K: (a) is the initial crack under KI /1.0KIC, (b) is the equilibrium configuration after 4000 steps relaxation of crack under KI /1.0KIC, (c) and (d) are the crack configurations, separately after 700 and 900 steps relaxation under KI /1.2KIC. Cross indicates the initial position of crack tip.
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and not to advance until a load K/I is reached, which is somewhat larger than the critical Griffith load KIC. Similarly, the crack will not heal until a load K/I / B/KIC is reached. The critical loads K/I and K/I are called respectively, the upper and the lower trapping limit. In the previous studies, it has been found that the upper and the lower trapping limits are sensitive to the simulation conditions [13,15,18] such as the type of atomistic potential, the boundary condition, the crystallography and the temperature. Table 1 lists our simulation results. It can be found that the calculated K/I and K/I for {0 1 0} 1 0 1 crack in bcc-Fe by using Finnis /Sinclair potential are very close to the theoretical critical stress intensity factor KIC according to the Griffith criterion, where cleavage occurs along {1 0 0} planes. Comparison with the results obtained by stress-boundary condition and flexible-boundary condition [13], we can see that the fixed-displacement boundary condition will be valid for simulating crack propagation if the system size is selected large enough. But when the crack front lies on 1 0 0 direction, the K/I value increases greatly. The differences between K/I ; K/I and KIC at high temperature are smaller than those at lower temperature, which indicate the trapping barrier can be overcome by thermal activation. The results also show that the upper and the lower trapping limits are almost invariant with the temperature when the temperature of the crack system is higher than 200 K.
stacking faults formation. At 900 steps as shown in Fig. 4(d), the twin transformation from stacking fault is observed in crack tip when the total displacement along the 1 1 1 slip direction reaches the Burgers vector of b . At the same time, the system potential energy also reaches the maximum and will decreases rapidly once the twin is formed, whereas the kinetic energy will increase. Thus we can observe from Fig. 4 that, although the cracks at low temperature display a brittle character of extension, an obvious elastic twinning process is observed in the 1 1 1 {1 1 2} slip system after the emission of one a61 1 1 partial dislocation or more. It can be concluded that the brittle cleavage of crack is a process accompanied bond breakage and twin growth along the crack tip. No perfect dislocation has been observed in our simulation, which indicates that instead of the formation of a complete edge dislocation, the formations of stacking fault and twin are more favorable in the 1 1 1 {1 1 2} slip system of bcc-Fe at the
3.2. Formation of SF and twin Many experiments and simulations reveal that twinning and fracture at low temperature or at high strain rate are cooperating processes in bcc-Fe [22,26]. From Fig. 4(b), we can find that although the bond rupture is not evident at the crack tip, the stacking fault has been formed after 4000 steps relaxation of the initial crack at critical stress intensity factor KIC, which indicates that the stress in crack tip is relaxed by the formation of stacking fault. When the imposed loading increases to 2.2KIC, some long twins have been formed along the crack tip at 5 K accompanied with the crack extension, as shown in Fig. 6(a). Fig. 5 shows the changes of the system energies, including the kinetic energy EK, the potential energy EP and the total energy ETOT, during the relaxation of crack under 1.2KIC at 5 K. Y-axis denotes the energy per atom in the simulation system and X-axis denotes the time step. Three high steps can be seen in the potential energy curve, at 700, 900 and 1000 steps respectively, which correspond to the atomic bond breaking at the crack tip. Comparing the atomic configuration in Fig. 4(b) with that in Fig. 4(c), we can see that the stacking faults extend after 700 steps relaxation, thus the imposing energy due to the external loading is released by
Fig. 5. Energetic curves of crack relaxation at 5 K with KI /1.2KIC: (a) is kinetic energy per atom, (b) is potential energy per atom and (c) is total energy per atom.
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Fig. 6. Atomic configuration of crack with KI /2.2KIC at (a) 5 K (b) 100 K (c) 200 K (d) 300 K.
temperature near 0 K. Additionally, it is found that the average velocity of twin extension under higher strain rate is greater than that under lower strain rate, consistent with the experimental observation. 3.3. Temperature effect on crack propagation Fig. 6(a) /(c) which correspond respectively to the temperature of 5, 100 and 200 K, show clearly that the crack propagation along with the twin formation in crack tip is intrinsically brittle in bcc-Fe. Whereas Fig. 6(d) shows that the ductile extension appears under 300 K accompanied with branching at the crack tip. In order to exhibit the width and the length of the twin, there are more atoms displayed in Fig. 6 than in Fig. 4 or in Fig. 7. It can be observed that the width and the length of twin at lower temperature are apparently larger than that at higher temperature. The average velocities of twin extension under strain rate of 0.1KIC/400 steps at 5, 100 and 200 K are about 500, 250 and 120 m s 1 respectively. We can conclude that the twinning process
at the crack tip will be weakened at higher temperature, and will disappear eventually near the BDT temperature. In general, the increased energy due to the loading of the system will transform into surface energy and plastic strain energy during the crack propagation. We can see in Fig. 6 that the increased energy due to loads transform into surface energy and twin or stacking fault shear deformation energy below BDT temperature, whereas the shear deformation energy is substituted by plastic strain energy due to the dislocation nucleation and emission after BDT. Therefore, we can take the occurrence of the shear deformation as a notable feature for distinguishing the brittle and the ductile fractures for bcc-Fe, and make an assumption that stacking fault or twin plays an important role on crack cleavage, while the shear process at the crack tip is favorable to the atomic bond breakage. Comparing the results of the upper trapping limits K/I ; we find that the K/I value of {0 1 0} 1 0 0 crack is much higher than that of {0 1 0} 1 0 1 crack because there is no twinning
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Fig. 7. Atomic configuration of crack: (a) at 200 K with KI /1.4KIC, (b) at 300 K with KI /1.4KIC, (c) at 300 K with KI /1.6KIC and (d) at 300 K with KI /2.0KIC.
process in {0 1 0} 1 0 0 crack system. Meanwhile, we can observe dislocation nucleation at the crack tip in {0 1 0} 1 0 0 crack system during the crack extension, which is an evidence for our assumption: the twin formation is favorable for crack brittle cleavage, whereas the dislocation nucleation perpendicular to crack surface blunt the crack tip. Fig. 7 indicates the atomistic process of crack tip under 300 K. Two dislocations can be seen obviously, which is nucleated along 0 1 0 direction at the stress intensity factor of 1.4KIC and finally blunt the crack tip. In addition, the crack propagates along 0 1 0 direction as the load increases, and then turns to 1 1 1 direction when plastic deformation occurs at the crack tip. However, no full dislocation is observed in our simulation while the periodic boundary condition is applied in [1 0 1] direction. Meanwhile, no dislocation emission is observed due to the fixed-displacement boundary condition. Thus we assume that it is the dislocation nucleation and plastic deformation at high
temperature that cause the crack blunted in our simulation.
4. Conclusions Under the fixed-displacement boundary condition, the stacking fault and twin are observed in brittle fracture at low temperature. The brittle cleavage fracture is a process accompanied with the bonds breakage and the twins growth at the crack tip. Therefore, we can assume that the shear process due to the twin formation at the crack tip is favorable for bond breakage, and stacking fault or twin formation plays an important role on brittle cleavage. But at high temperature, while the twin disappears in crack tip, the dislocation is nucleated to be perpendicular to crack surface and the crack tip is blunted. We can conclude that the shear process and the plastic deformation at the crack tip are, respectively, the feature of brittle and ductile fracture in bcc-Fe.
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Acknowledgements The research was supported by ‘973’ Project from the Ministry of Science and Technology of China (Grant No. G2000067102), Chinese Nature Science Foundation (Grant No. 59971041) and NJTU Pandeng project. The authors thank Dr Ngan, Dr Jin Zhaohui and Dr Wang Shanying for their helpful discussions.
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