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Molecular dynamics simulation of healing of an ellipsoid crack in copper under compressive stress
Materials Letters 58 (2004) 543 – 546 www.elsevier.com/locate/matlet
Molecular dynamics simulation of healing of an ellipsoid crack in copper under compressive stress M. Li *, W.Y. Chu, K.W. Gao, L.J. Qiao University of Science and Technology Beijing, Beijing 100083, China Received 27 March 2003; accepted 25 June 2003
Abstract The molecular dynamics method is used to simulate healing of an ellipsoid crack inside copper under compressive stress with embedded atom method (EAM) potential. The result shows that dislocations are emitted firstly from the ellipsoid crack and move along the (11¯1) and (1¯11) planes under a constant compressive stress of 0.34 GPa. The ellipsoid crack becomes smaller and smaller until it is healed through dislocation emission, motion, and annihilation on the surfaces. After crack healing, there are a residual dislocation net and some vacancy sites. It seems that the cavity inside the ellipsoid crack is transferred to the surfaces through dislocation emission, motion, and annihilation. At the same time, the crystal undergoes large plastic deformation and the surface become rough because of dislocation annihilation. D 2003 Published by Elsevier B.V. Keywords: Crack healing; Copper; Molecular dynamics simulation; Compressive stress
1. Introduction Crack healing is now considered to show promise in the recovery of the mechanical properties of the cracked materials. Microcracks in ceramics [1 –3], Plexiglas [4], and ice [5] can be healed through atom diffusion during heating. Microcracks in hot-rolled 600 alloy were closed during heating over 870 jC with a large compressive stress by the moving of grain boundaries because of recrystallization [6]. Microcracks in cold-rolled Cu30Fe duplex alloy could be sealed after rolling exceeded 30% [7]. Microcracks in 20 MnMo steel disappeared during hot deformation [8]. In situ observation in TEM showed that the microcracks or microvoids in single crystal iron could be healed completely upon heating [9]. The molecular dynamics method was used to simulate microcrack healing during heating or/and under compressive stress [10,11]. The simulation results showed that a center penetrating microcrack in Cu or Al crystal could be healed under a compressive stress or by heating, and the role of compressive stress and heating in crack healing was
* Corresponding author. 0167-577X/$ - see front matter D 2003 Published by Elsevier B.V. doi:10.1016/j.matlet.2003.06.016
additive [10,11]. During microcrack healing, dislocation emission and motion occurred and pre-existed dislocation could decrease the critical temperature or compressive stress necessary for microcrack healing [10,11]. In these simulations, however, a periodic boundary condition was used, and there was a quasi-three-dimensional simulation involving only 6 103 atoms [10,11]. Whether healing of an ellipsoid crack in three-dimensional crystal can be simulated and whether dislocation emission and motion are preconditions for crack healing or accompanying conditions during healing remain a question. In the present paper, the molecular dynamics method is used to simulate healing of an ellipsoid crack in a crystal containing 4 105 atoms during pressing and to exhibit the role of dislocation emission and motion in crack healing.
2. Simulation procedure Copper is chosen as the simulated material. The lengths of the crystal along the x = [11¯2], y =[11¯1] and z = [110] directions are 17.9, 16.9, and 15.6 nm, respectively. The number of atoms is about 4 105. A flat ellipsoid hole with the sizes of 4.42, 4.42, and 0.494 nm is input in the center of the crystal, and then the configuration is relaxed
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M. Li et al. / Materials Letters 58 (2004) 543 – 546
for 1 ps at 40 K until equilibrium is reached, as shown in Fig. 1(a). A compressive stress along the z = [110] direction is applied with a loading rate of 57 MPa/ps. The time step is 10 3 ps. The inter-atomic potential used here is the N-body potential proposed by Finnis and Sinclair [12] according to the embedded atom method (EAM) and constructed by Ackland et al. [13]. The inner atoms follow the law of Newton, and a leapfrog algorithm [14] is applied to calculate positions and velocities of atoms. The initial velocity is
the Maxwell –Boltzmann distribution corresponding to a given temperature. The system temperature is maintained at 40 K by scaling the atom velocities during the simulation. The potential energy of all lattice atoms distributes between 3.55 and 3.51 eV, and the energy of the atoms around dislocation core is higher than that of the lattice atoms and distributes between 3.49 and 3.45 eV after dislocation emission. These atoms with higher energy compose dislocation lines. The atoms with various energy distributions are distinguished using various colours. There-
Fig. 1. Healing of an ellipsoid crack under pressing, (a) an ellipsoid crack inside crystal, (b) dislocation emitting under compressive stress of 0.342 GPa, (c) dislocation emission and motion along the (11¯1) and (1¯11) planes after keeping the constant compressive stress for 7 ps, (d) dislocation annihilation on the surfaces after relaxing for 10 ps under the compressive stress of 0.342 GPa, (e) crack being healed, and a dislocation net and vacancies resided within after relaxing for 37.5 ps under the compressive stress of 0.342 GPa, and (f) crystal deformation after the crack is healed.
M. Li et al. / Materials Letters 58 (2004) 543 – 546
545
Fig. 1 (continued).
fore, the positions of all dislocations at any instant can be determined.
3. Result and discussion When the applied compressive stress is below the critical value necessary for dislocation emission, the energy of all atoms distributes between 3.55 and 3.51 eV, and there are no high-energy atoms. When compressive stress increases to 0.342 GPa, high-energy atoms appear ahead of the ellipsoid crack, as shown in Fig. 1(b). Fig. 1(b) shows
that dislocations begin to emit from the ellipsoid crack tip, and the sites of the dislocations can be determined by the high-energy atoms with deep colour. Keeping the constant compressive stress, dislocations are emitted and move continuously along the (11¯1) and (1¯11) planes, as shown in Fig. 1(c). When dislocations move close to the surfaces of the crystal, they are drawn toward the surfaces by the image force and annihilated on the surfaces, as shown in Fig. 1(d). During dislocation emission and motion, the ellipsoid crack becomes smaller and smaller, and it seems that the cavity inside the ellipsoid crack is transferred to the surfaces through dislocation emission, motion, and annihilation.
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M. Li et al. / Materials Letters 58 (2004) 543 – 546
After relaxing for 37.5 ps at the constant compressive stress, the ellipsoid crack has been healed completely and there exist a dislocation net and some vacancy sites, as shown in Fig. 1(e). At the same time, the crystal containing the ellipsoid crack generates large plastic deformation, and the surfaces become rough because of annihilation of dislocation on the surfaces, as shown in Fig. 1(f). The simulation result shows that crack healing under a compressive stress is preceded by dislocation emission and motion. The crack become smaller and smaller until it is healed completely through successive dislocation emission and motion. It seems that the cavity inside the crack transferred to the surfaces through dislocation emission and motion. When temperature exceeds 200 K, the energy distribution of the lattice atoms widens greatly and overlays with that of dislocation core. In this case, the site of dislocation core can not be determined according to energy distribution, i.e., the colour of the atoms. Therefore, we cannot simulate three-dimensional crack healing during heating. For a penetrating crack in quasi-three-dimensional simulation using a periodic boundary condition, all dislocations emitted from crack tip are straight lines penetrating the thickness, and can be determined according to the site of the half atomic plane [10,11]. Quasi-three-dimensional simulations indicated that the process of crack healing and dislocation emission and motion during heating were the same as that under compressive stress or during combining of compressive stress and heating [10,11]. It may well be that crack healing during heating is preceded also by dislocation emission and motion, as that under compressive stress. For a penetrating crack in a small crystal containing about 6 103 atoms, the critical compressive stress necessary for crack healing was very high, e.g., 6.5 GPa for Al [11] and 7.26 GPa for Cu [10]. The critical compressive stress for crack healing of the ellipsoid crack in the crystal containing 4 105 atoms is 0.34 GPa, which is one order of magnitude smaller than that for the penetrating crack. It is possible that an ellipsoid crack is easier to be healed than a penetrating crack.
4. Summary An ellipsoid crack becomes smaller and smaller until it is healed through dislocation emission, motion, and annihilation on the surfaces under a compressive stress. Therefore, crack healing under compressive stress is preceded by dislocation emission and motion.
Acknowledgements The present work was supported by the special Funds for the Major State Basic Research Projects (No G19990650) and by the NNSF of China (No. 50171012).
References [1] Z. Wang, Y.Z. Li, M.P. Harmer, Y.T. Chou, J. Am. Ceram. Soc. 74 (1992) 1596. [2] T.K. Gupta, J. Am. Ceram. Soc. 59 (1976) 259. [3] A.M. Thompson, H.M. Chan, M.P. Harmer, R.F. Cook, J. Am. Ceram. Soc. 78 (1995) 567. [4] M.K.C. Holden, V.D. Frchette, J. Am. Ceram. Soc. 729 (1989) 2189. [5] S.C. Colbeck, Acta Metall. 34 (1) (1986) 89. [6] E. Shapiro, G.E. Dietar, Metall. Trans. 1 (1970) 1711. [7] J. Foct, N. Akdut, G. Gottstein, Scr. Metall. Mater. 27 (1992) 1033. [8] J.T. Han, G. Zhao, Q.X. Cao, Sci. China 27E (1997) 23. [9] K.W. Gao, L.J. Qiao, W.Y. Chu, Scr. Mater. 44 (2001) 1055. [10] S. Li, K.W. Gao, L.J. Qiao, F.X. Zhou, W.Y. Chu, Comput. Mater. Sci. 20 (2001) 143. [11] G.H. Zhou, K.W. Gao, L.J. Qiao, Y.B. Wang, W.Y. Chu, Model. Simul. Mater. Sci. Eng. 8 (2000) 603. [12] M.W. Finnis, J.E. Sinclair, Phila. Mag., A 50 (1984) 50. [13] G.J. Ackland, G. Tichy, V. Vitek, M.W. Finnis, Phila. Mag., A. 56 (1987) 56. [14] D.W. Heerman, Computer Simulation Methods in Theoretic Physics, 2nd ed., Springer, Berlin, 1990.
Molecular dynamics simulation of healing of an ellipsoid crack in copper under compressive stress M. Li *, W.Y. Chu, K.W. Gao, L.J. Qiao University of Science and Technology Beijing, Beijing 100083, China Received 27 March 2003; accepted 25 June 2003
Abstract The molecular dynamics method is used to simulate healing of an ellipsoid crack inside copper under compressive stress with embedded atom method (EAM) potential. The result shows that dislocations are emitted firstly from the ellipsoid crack and move along the (11¯1) and (1¯11) planes under a constant compressive stress of 0.34 GPa. The ellipsoid crack becomes smaller and smaller until it is healed through dislocation emission, motion, and annihilation on the surfaces. After crack healing, there are a residual dislocation net and some vacancy sites. It seems that the cavity inside the ellipsoid crack is transferred to the surfaces through dislocation emission, motion, and annihilation. At the same time, the crystal undergoes large plastic deformation and the surface become rough because of dislocation annihilation. D 2003 Published by Elsevier B.V. Keywords: Crack healing; Copper; Molecular dynamics simulation; Compressive stress
1. Introduction Crack healing is now considered to show promise in the recovery of the mechanical properties of the cracked materials. Microcracks in ceramics [1 –3], Plexiglas [4], and ice [5] can be healed through atom diffusion during heating. Microcracks in hot-rolled 600 alloy were closed during heating over 870 jC with a large compressive stress by the moving of grain boundaries because of recrystallization [6]. Microcracks in cold-rolled Cu30Fe duplex alloy could be sealed after rolling exceeded 30% [7]. Microcracks in 20 MnMo steel disappeared during hot deformation [8]. In situ observation in TEM showed that the microcracks or microvoids in single crystal iron could be healed completely upon heating [9]. The molecular dynamics method was used to simulate microcrack healing during heating or/and under compressive stress [10,11]. The simulation results showed that a center penetrating microcrack in Cu or Al crystal could be healed under a compressive stress or by heating, and the role of compressive stress and heating in crack healing was
* Corresponding author. 0167-577X/$ - see front matter D 2003 Published by Elsevier B.V. doi:10.1016/j.matlet.2003.06.016
additive [10,11]. During microcrack healing, dislocation emission and motion occurred and pre-existed dislocation could decrease the critical temperature or compressive stress necessary for microcrack healing [10,11]. In these simulations, however, a periodic boundary condition was used, and there was a quasi-three-dimensional simulation involving only 6 103 atoms [10,11]. Whether healing of an ellipsoid crack in three-dimensional crystal can be simulated and whether dislocation emission and motion are preconditions for crack healing or accompanying conditions during healing remain a question. In the present paper, the molecular dynamics method is used to simulate healing of an ellipsoid crack in a crystal containing 4 105 atoms during pressing and to exhibit the role of dislocation emission and motion in crack healing.
2. Simulation procedure Copper is chosen as the simulated material. The lengths of the crystal along the x = [11¯2], y =[11¯1] and z = [110] directions are 17.9, 16.9, and 15.6 nm, respectively. The number of atoms is about 4 105. A flat ellipsoid hole with the sizes of 4.42, 4.42, and 0.494 nm is input in the center of the crystal, and then the configuration is relaxed
544
M. Li et al. / Materials Letters 58 (2004) 543 – 546
for 1 ps at 40 K until equilibrium is reached, as shown in Fig. 1(a). A compressive stress along the z = [110] direction is applied with a loading rate of 57 MPa/ps. The time step is 10 3 ps. The inter-atomic potential used here is the N-body potential proposed by Finnis and Sinclair [12] according to the embedded atom method (EAM) and constructed by Ackland et al. [13]. The inner atoms follow the law of Newton, and a leapfrog algorithm [14] is applied to calculate positions and velocities of atoms. The initial velocity is
the Maxwell –Boltzmann distribution corresponding to a given temperature. The system temperature is maintained at 40 K by scaling the atom velocities during the simulation. The potential energy of all lattice atoms distributes between 3.55 and 3.51 eV, and the energy of the atoms around dislocation core is higher than that of the lattice atoms and distributes between 3.49 and 3.45 eV after dislocation emission. These atoms with higher energy compose dislocation lines. The atoms with various energy distributions are distinguished using various colours. There-
Fig. 1. Healing of an ellipsoid crack under pressing, (a) an ellipsoid crack inside crystal, (b) dislocation emitting under compressive stress of 0.342 GPa, (c) dislocation emission and motion along the (11¯1) and (1¯11) planes after keeping the constant compressive stress for 7 ps, (d) dislocation annihilation on the surfaces after relaxing for 10 ps under the compressive stress of 0.342 GPa, (e) crack being healed, and a dislocation net and vacancies resided within after relaxing for 37.5 ps under the compressive stress of 0.342 GPa, and (f) crystal deformation after the crack is healed.
M. Li et al. / Materials Letters 58 (2004) 543 – 546
545
Fig. 1 (continued).
fore, the positions of all dislocations at any instant can be determined.
3. Result and discussion When the applied compressive stress is below the critical value necessary for dislocation emission, the energy of all atoms distributes between 3.55 and 3.51 eV, and there are no high-energy atoms. When compressive stress increases to 0.342 GPa, high-energy atoms appear ahead of the ellipsoid crack, as shown in Fig. 1(b). Fig. 1(b) shows
that dislocations begin to emit from the ellipsoid crack tip, and the sites of the dislocations can be determined by the high-energy atoms with deep colour. Keeping the constant compressive stress, dislocations are emitted and move continuously along the (11¯1) and (1¯11) planes, as shown in Fig. 1(c). When dislocations move close to the surfaces of the crystal, they are drawn toward the surfaces by the image force and annihilated on the surfaces, as shown in Fig. 1(d). During dislocation emission and motion, the ellipsoid crack becomes smaller and smaller, and it seems that the cavity inside the ellipsoid crack is transferred to the surfaces through dislocation emission, motion, and annihilation.
546
M. Li et al. / Materials Letters 58 (2004) 543 – 546
After relaxing for 37.5 ps at the constant compressive stress, the ellipsoid crack has been healed completely and there exist a dislocation net and some vacancy sites, as shown in Fig. 1(e). At the same time, the crystal containing the ellipsoid crack generates large plastic deformation, and the surfaces become rough because of annihilation of dislocation on the surfaces, as shown in Fig. 1(f). The simulation result shows that crack healing under a compressive stress is preceded by dislocation emission and motion. The crack become smaller and smaller until it is healed completely through successive dislocation emission and motion. It seems that the cavity inside the crack transferred to the surfaces through dislocation emission and motion. When temperature exceeds 200 K, the energy distribution of the lattice atoms widens greatly and overlays with that of dislocation core. In this case, the site of dislocation core can not be determined according to energy distribution, i.e., the colour of the atoms. Therefore, we cannot simulate three-dimensional crack healing during heating. For a penetrating crack in quasi-three-dimensional simulation using a periodic boundary condition, all dislocations emitted from crack tip are straight lines penetrating the thickness, and can be determined according to the site of the half atomic plane [10,11]. Quasi-three-dimensional simulations indicated that the process of crack healing and dislocation emission and motion during heating were the same as that under compressive stress or during combining of compressive stress and heating [10,11]. It may well be that crack healing during heating is preceded also by dislocation emission and motion, as that under compressive stress. For a penetrating crack in a small crystal containing about 6 103 atoms, the critical compressive stress necessary for crack healing was very high, e.g., 6.5 GPa for Al [11] and 7.26 GPa for Cu [10]. The critical compressive stress for crack healing of the ellipsoid crack in the crystal containing 4 105 atoms is 0.34 GPa, which is one order of magnitude smaller than that for the penetrating crack. It is possible that an ellipsoid crack is easier to be healed than a penetrating crack.
4. Summary An ellipsoid crack becomes smaller and smaller until it is healed through dislocation emission, motion, and annihilation on the surfaces under a compressive stress. Therefore, crack healing under compressive stress is preceded by dislocation emission and motion.
Acknowledgements The present work was supported by the special Funds for the Major State Basic Research Projects (No G19990650) and by the NNSF of China (No. 50171012).
References [1] Z. Wang, Y.Z. Li, M.P. Harmer, Y.T. Chou, J. Am. Ceram. Soc. 74 (1992) 1596. [2] T.K. Gupta, J. Am. Ceram. Soc. 59 (1976) 259. [3] A.M. Thompson, H.M. Chan, M.P. Harmer, R.F. Cook, J. Am. Ceram. Soc. 78 (1995) 567. [4] M.K.C. Holden, V.D. Frchette, J. Am. Ceram. Soc. 729 (1989) 2189. [5] S.C. Colbeck, Acta Metall. 34 (1) (1986) 89. [6] E. Shapiro, G.E. Dietar, Metall. Trans. 1 (1970) 1711. [7] J. Foct, N. Akdut, G. Gottstein, Scr. Metall. Mater. 27 (1992) 1033. [8] J.T. Han, G. Zhao, Q.X. Cao, Sci. China 27E (1997) 23. [9] K.W. Gao, L.J. Qiao, W.Y. Chu, Scr. Mater. 44 (2001) 1055. [10] S. Li, K.W. Gao, L.J. Qiao, F.X. Zhou, W.Y. Chu, Comput. Mater. Sci. 20 (2001) 143. [11] G.H. Zhou, K.W. Gao, L.J. Qiao, Y.B. Wang, W.Y. Chu, Model. Simul. Mater. Sci. Eng. 8 (2000) 603. [12] M.W. Finnis, J.E. Sinclair, Phila. Mag., A 50 (1984) 50. [13] G.J. Ackland, G. Tichy, V. Vitek, M.W. Finnis, Phila. Mag., A. 56 (1987) 56. [14] D.W. Heerman, Computer Simulation Methods in Theoretic Physics, 2nd ed., Springer, Berlin, 1990.