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Computer simulation of creation and motion of dislocations during plastic deformation in copper
Materials Science and Engineering A309–310 (2001) 451–455
Computer simulation of creation and motion of dislocations during plastic deformation in copper Masao Doyama∗ , Y. Kogure Teikyo University of Science and Technology, Uenohara, Yamanashi 409-0193, Japan
Abstract ¯ (1 1¯ 0), (1¯ 1 0), Dislocations were created near the center of the surface (1 1¯ 0) of a copper small crystal whose surfaces are (1 1 1), (1¯ 1¯ 1), ¯ and (1¯ 1¯ 2) by use of n-body embedded atom potential and molecular dynamics. A Heidenreich–Shockley partial dislocation, not a (1 1 2), perfect dislocation, was created. As the partial dislocation proceeds, the partial dislocation and the surface were connected with a stacking fault until the next Heidenreich–Shockley partial dislocation was created at the surface. Just before the creation of a partial dislocation the stress was the highest. A very sharp yield stress was observed. Copper small crystals with a notch were pulled or compressed. Again Heidenreich–Shockley partial dislocations were created near the notch. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Partial dislocations; Creation of dislocations; Motion of dislocations; Molecular dynamics; Embedded atom potential; Computer simulation
1 Φ(rij ) + F (ρi ) 2
1. Introduction
Ei =
It is well known that creation, motion and interaction of dislocations play an important role in the plastic deformation of crystalline solids. It is important to know how the dislocations are created and move in metals. In this paper creation and motion of dislocations during plastic deformation were simulated by use of molecular dynamics. Copper was chosen as an example, because it has a face centered cubic lattice and is one of the most common metals. In metals, the conduction electrons travel from one atom to another atom and the interaction cannot be represented by a pairwise potential but by many body potentials. Embedded atom potentials of metals have been developed. The interaction between the ith atom and jth atom depends not only on the distance between them but also other factors. By the embedded function [1–5], surface problems can also be treated.
F (ρi ) = Dρi ln ρi
2. Potential functions For the n-body embedded function proposed here in this paper, the total energy is given by Etotal = Ei (1) rij = |ri − rj |
(2)
∗ Corresponding author. Tel.: +81-554-63-4411; fax: +81-554-63-4431. E-mail address: [email protected] (M. Doyama).
ρi =
f (rij )
(3) (4) (5)
Here Etotal is the total internal energy, Ei the internal energy associated with atom i, ρ i the electron density at atom i due to all other atoms, F(ρ i ) the energy to embed an atom into an electron gas density ρ i , Φ(rij ) the two body central potential between atoms i and j separated by rij , f(rij ) the contribution to the electron density at atom i due to atom j at the distance rij from atom i, F(ρ i ) an attractive term. For the functional forms Φ(rij ) = A1 (rc1 − rij )2 exp(1 − c1 rij )
(6)
f (rij ) = A2 (rc2 − rij )2 exp(−c2 rij )
(7)
are assumed. f(rij ) and Φ(rij ) are smoothly truncated at rc1 and rc2 , respectively. rc1 was chosen to be 1.65d (d is the nearest neighbor distance). Φ(rij ) was chosen to be 1.95d. The potential functions described in Eqs. (1)–(7) contain five parameters, A1 , A2 , C1 , C2 and D. These are determined to reproduce the Born stability, cohesive energy, elastic constants c11 , c12 and c44 , the formation energy of a vacancy and stacking fault energy. For copper, A1 = 8.28945997705 × 103 , A2 = 1.83251035107 × 10−2 , C1 = 10.72729128641, C2 = 3.19759369823 × 10−1 and D = 13.07921251628.
0921-5093/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 0 ) 0 1 7 0 0 - 7
452
M. Doyama, Y. Kogure / Materials Science and Engineering A309–310 (2001) 451–455
Fig. 1. (a) Specimen 1; (b) Specimen 2; (c) Specimen 3.
3. Specimens
4. Deformation
Three copper specimens were prepared. Specimen 1 (Fig. 1(a)) was rectangular parallelepiped having the faces ¯ (1 1 2), ¯ (1¯ 1¯ 2), (1 1¯ 0) and (1¯ 1 0). x, y of (1 1 1), (1¯ 1¯ 1), and z directions were taken in [1 1 0], [1¯ 1¯ 2] and [1 1 1] directions, respectively. Specimen 2 (Fig. 1(b)) contained 4338 atoms having the faces of (1¯ 0 0), (1 0 0), (0 1¯ 0), ¯ x, y and z directions were taken (0 1 0), (0 0 1) and (0 0 1). to be in the direction of [1 0 0], [0 1 0] and [0 0 1], respectively. The size of Specimen 2 was 8a × 8a × 26a, where a is the lattice parameter and a notch was made in y direction at the center on (1 0 0) (Fig. 1(b)). Specimen 3 was a rectangular parallelepiped having the faces ¯ A of (1¯ 1 0), (1 1¯ 0), (1¯ 1¯ 0), (1 1 0), (0 0 1) and (0 0 1). notch was introduced near the center of (1 1 0). The x-axis was taken to be [1¯ 1 0], y[1¯ 1¯ 0] and z[0 0 1]. The size of Specimen 3 was 5d × 7d × 31a, where d is the nearest neighbor distance and a is the lattice parameter. Specimen 3 contained 2225 atoms. The surfaces were free and periodic boundary conditions were not used.
For Specimen 1, a step (height = 0.01d, where d is the nearest neighbor distance) was made on the (1 1 0), then all the atoms in the crystal was relaxed 50 cycles by the molecular dynamics. The step height was again increased by 0.01d. This process was repeated. Specimen 2 was pulled in the direction by 0.5%, all the atoms were deformed uniformly, then all the atoms except the one layer of each end were relaxed 25 cycles, and again pulled 0.5% in the [0 0 1] direction, and all atoms were relaxed 25 cycles. The atoms in the end layers were relaxed only in the x and y directions not in z direction. This procedure was repeated. The time step of the molecular dynamics was 5 × 10−15 s. Specimen 3 was uniformly compressed in z direction [0 0 1] 1% and relaxed all atoms. Every 150 time cycles uniform compression was given. After giving a uniform deformation, all atoms in the specimen were relaxed 150 time cycles and then compressed again. This process was repeated. The time step of the molecular dynamics used was 3 × 10−15 s.
Fig. 2. Projection of atoms on (1¯ 1¯ 2). Creation of dislocations.
M. Doyama, Y. Kogure / Materials Science and Engineering A309–310 (2001) 451–455
453
¯ Two half dislocations are connected with a stacking fault. (a) 3500 time steps (0.7d step height); (b) at 5000 time Fig. 3. Projection of atoms on (1¯ 1¯ 1). steps (1.0d step height); (c) at 6000 time steps (1.2d step height); (d) at 7000 time steps (1.4d step height); (e) at 7500 time steps (1.5d step height); (f) 9500 time steps (1.9d step height).
5. Results and discussion Atomic positions of Specimen 1 are projected on the (1¯ 1¯ 2) plane (Fig. 2). A Heidenreich–Shockly partial dislocation was created at the step on (1 1¯ 0) leaving a stacking fault to the surface, then the second Heidenreich–Shockley partial dislocation was created, joined with the stacking fault, moved as a pair of dislocations, ejected from the other side of the crystal. This can be compared with the experimental value between 163 and 30 erg/cm2 . If the readers look at Fig. 2 from the bottom of the page with a low angle, the reader can see half dislocations. Each time the half dislocation was introduced, stress relaxation and a sharp yield point
was observed. Atomic positions just below and just above the slip plane is plotted in Fig. 3. Two half dislocations are connected with the stacking fault (Fig. 3). The largest open circles are two layers above the slip plane. The medium size open circles are just above the slip plane. The smallest circles are just below the slip plane. In the perfect crystal largest and medium size circles form hexagons and the smallest circles are at the center of the haxagons. These can be seen in the right-hand side of Fig. 3(a) and the left-hand side of Fig. 3(c). In the stacking fault region, the largest and smallest circles come at the same point, so that no atoms can be seen in the hexagons. Such a plane can be found at the left-hand side of Fig. 3(a), right-hand side of Fig. 3(b) and
Fig. 4. Speciemen 2: (a) 300 steps (6% elongation); (b) 375 steps (7.5% elongation); (c) 500 steps (10% elongation); (d) 675 steps (13.5% elongation). All are projected on xz plane.
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M. Doyama, Y. Kogure / Materials Science and Engineering A309–310 (2001) 451–455
Fig. 5. Stress–strain curve for Specimen 2 (tensile test).
right-hand side of Fig. 3(c). There is a half dislocation in the transient region from the perfect region to the stacking fault region. The splitting of dislocations can also be seen, if the readers look at Fig. 4 from the right-hand side with a low angle. The most clear case is Fig. 3(c). A half dislocation is near the center of the figure.
Fig. 6. Schematic diagram of creation and motion of half dislocations and motion of half dislocations in Specimen 2.
Fig. 7. The projection of atom positions of Specimen 3 on xz-plane (1 1 0): (a) at 500 time cycles; (b) 2000 time cycles; (c) at 2500 time cycles; (d) at 3500 time cycles; (e) at 4500 time cycles; (f) at 7500 cycles; (g) at 9000 time cycles; (h) at 15,000 time cycles.
M. Doyama, Y. Kogure / Materials Science and Engineering A309–310 (2001) 451–455
Atomic positions near the center of Specimen 2 projected on the xz plane (0 1 0) are plotted in Fig. 4(a)–(d). In Fig. 4(b), near the tip of the crack, a half-dislocation was created. Two other half-dislocations are started near the grip of Specimen 2. These can be clearly seen when the readers look at the figures from the bottom of the page with a small angle from the paper. In Fig. 5, the stress–strain curve is plotted. A very sharp yield point was found. It was found that a notch is normally a source of dis¯ 111) ¯ locations due to the stress concentration. [2¯ 11]( and ¯ ¯ [112](111) half dislocations were created leaving stacking faults. Schematic diagram is given in Fig. 6 for Specimen 2. The initial stage of deformation was surely conducted by the creation and motion of dislocations. In the middle stage, dislocations on different slip planes cross each other. The work hardening occurs. In the final stage, in the case of copper, crystalline state with many dislocations and the inhomogeneous deformation is normal. Specimen 3 was uniformly compressed. Partial dislocations or twining of three atomic layers started near the tip of
455
the notch above 1000 time cycles (Fig. 7(b)). Bulging, particularly near the notch, can be observed above 3500 time cycles (Fig. 7(d)). Fig. 7(f) is at 7500 time cycles, Fig. 7(g) at 9000 time cycles and Fig. 7(h) is at 15,000 cycles. Severe bulging is observed.
Acknowledgements This work is supported by the Ministry of Education, Science, Sports and Culture. References [1] S.M. Foiles, M.I. Baskes, M.S. Daw, Phys. Rev. B 33 (1986) 7983. [2] D.J. Oh, R.A. Johnson, J. Mater. Res. 3 (1986) 471. [3] D.J. Oh, R.A. Johnson, in: V. Vitek, D.J. Srolovitz (Eds.), Atomic Simulation of Materials, Plenum Press, New York, 1989, p. 233. [4] G.J. Ackland, G. Tichy, M.W. Finnis, Phil. Mag. A 56 (1987) 735. [5] M.J. Baskes, Phys. Rev. B 46 (1992) 2727.
Computer simulation of creation and motion of dislocations during plastic deformation in copper Masao Doyama∗ , Y. Kogure Teikyo University of Science and Technology, Uenohara, Yamanashi 409-0193, Japan
Abstract ¯ (1 1¯ 0), (1¯ 1 0), Dislocations were created near the center of the surface (1 1¯ 0) of a copper small crystal whose surfaces are (1 1 1), (1¯ 1¯ 1), ¯ and (1¯ 1¯ 2) by use of n-body embedded atom potential and molecular dynamics. A Heidenreich–Shockley partial dislocation, not a (1 1 2), perfect dislocation, was created. As the partial dislocation proceeds, the partial dislocation and the surface were connected with a stacking fault until the next Heidenreich–Shockley partial dislocation was created at the surface. Just before the creation of a partial dislocation the stress was the highest. A very sharp yield stress was observed. Copper small crystals with a notch were pulled or compressed. Again Heidenreich–Shockley partial dislocations were created near the notch. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Partial dislocations; Creation of dislocations; Motion of dislocations; Molecular dynamics; Embedded atom potential; Computer simulation
1 Φ(rij ) + F (ρi ) 2
1. Introduction
Ei =
It is well known that creation, motion and interaction of dislocations play an important role in the plastic deformation of crystalline solids. It is important to know how the dislocations are created and move in metals. In this paper creation and motion of dislocations during plastic deformation were simulated by use of molecular dynamics. Copper was chosen as an example, because it has a face centered cubic lattice and is one of the most common metals. In metals, the conduction electrons travel from one atom to another atom and the interaction cannot be represented by a pairwise potential but by many body potentials. Embedded atom potentials of metals have been developed. The interaction between the ith atom and jth atom depends not only on the distance between them but also other factors. By the embedded function [1–5], surface problems can also be treated.
F (ρi ) = Dρi ln ρi
2. Potential functions For the n-body embedded function proposed here in this paper, the total energy is given by Etotal = Ei (1) rij = |ri − rj |
(2)
∗ Corresponding author. Tel.: +81-554-63-4411; fax: +81-554-63-4431. E-mail address: [email protected] (M. Doyama).
ρi =
f (rij )
(3) (4) (5)
Here Etotal is the total internal energy, Ei the internal energy associated with atom i, ρ i the electron density at atom i due to all other atoms, F(ρ i ) the energy to embed an atom into an electron gas density ρ i , Φ(rij ) the two body central potential between atoms i and j separated by rij , f(rij ) the contribution to the electron density at atom i due to atom j at the distance rij from atom i, F(ρ i ) an attractive term. For the functional forms Φ(rij ) = A1 (rc1 − rij )2 exp(1 − c1 rij )
(6)
f (rij ) = A2 (rc2 − rij )2 exp(−c2 rij )
(7)
are assumed. f(rij ) and Φ(rij ) are smoothly truncated at rc1 and rc2 , respectively. rc1 was chosen to be 1.65d (d is the nearest neighbor distance). Φ(rij ) was chosen to be 1.95d. The potential functions described in Eqs. (1)–(7) contain five parameters, A1 , A2 , C1 , C2 and D. These are determined to reproduce the Born stability, cohesive energy, elastic constants c11 , c12 and c44 , the formation energy of a vacancy and stacking fault energy. For copper, A1 = 8.28945997705 × 103 , A2 = 1.83251035107 × 10−2 , C1 = 10.72729128641, C2 = 3.19759369823 × 10−1 and D = 13.07921251628.
0921-5093/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 0 ) 0 1 7 0 0 - 7
452
M. Doyama, Y. Kogure / Materials Science and Engineering A309–310 (2001) 451–455
Fig. 1. (a) Specimen 1; (b) Specimen 2; (c) Specimen 3.
3. Specimens
4. Deformation
Three copper specimens were prepared. Specimen 1 (Fig. 1(a)) was rectangular parallelepiped having the faces ¯ (1 1 2), ¯ (1¯ 1¯ 2), (1 1¯ 0) and (1¯ 1 0). x, y of (1 1 1), (1¯ 1¯ 1), and z directions were taken in [1 1 0], [1¯ 1¯ 2] and [1 1 1] directions, respectively. Specimen 2 (Fig. 1(b)) contained 4338 atoms having the faces of (1¯ 0 0), (1 0 0), (0 1¯ 0), ¯ x, y and z directions were taken (0 1 0), (0 0 1) and (0 0 1). to be in the direction of [1 0 0], [0 1 0] and [0 0 1], respectively. The size of Specimen 2 was 8a × 8a × 26a, where a is the lattice parameter and a notch was made in y direction at the center on (1 0 0) (Fig. 1(b)). Specimen 3 was a rectangular parallelepiped having the faces ¯ A of (1¯ 1 0), (1 1¯ 0), (1¯ 1¯ 0), (1 1 0), (0 0 1) and (0 0 1). notch was introduced near the center of (1 1 0). The x-axis was taken to be [1¯ 1 0], y[1¯ 1¯ 0] and z[0 0 1]. The size of Specimen 3 was 5d × 7d × 31a, where d is the nearest neighbor distance and a is the lattice parameter. Specimen 3 contained 2225 atoms. The surfaces were free and periodic boundary conditions were not used.
For Specimen 1, a step (height = 0.01d, where d is the nearest neighbor distance) was made on the (1 1 0), then all the atoms in the crystal was relaxed 50 cycles by the molecular dynamics. The step height was again increased by 0.01d. This process was repeated. Specimen 2 was pulled in the direction by 0.5%, all the atoms were deformed uniformly, then all the atoms except the one layer of each end were relaxed 25 cycles, and again pulled 0.5% in the [0 0 1] direction, and all atoms were relaxed 25 cycles. The atoms in the end layers were relaxed only in the x and y directions not in z direction. This procedure was repeated. The time step of the molecular dynamics was 5 × 10−15 s. Specimen 3 was uniformly compressed in z direction [0 0 1] 1% and relaxed all atoms. Every 150 time cycles uniform compression was given. After giving a uniform deformation, all atoms in the specimen were relaxed 150 time cycles and then compressed again. This process was repeated. The time step of the molecular dynamics used was 3 × 10−15 s.
Fig. 2. Projection of atoms on (1¯ 1¯ 2). Creation of dislocations.
M. Doyama, Y. Kogure / Materials Science and Engineering A309–310 (2001) 451–455
453
¯ Two half dislocations are connected with a stacking fault. (a) 3500 time steps (0.7d step height); (b) at 5000 time Fig. 3. Projection of atoms on (1¯ 1¯ 1). steps (1.0d step height); (c) at 6000 time steps (1.2d step height); (d) at 7000 time steps (1.4d step height); (e) at 7500 time steps (1.5d step height); (f) 9500 time steps (1.9d step height).
5. Results and discussion Atomic positions of Specimen 1 are projected on the (1¯ 1¯ 2) plane (Fig. 2). A Heidenreich–Shockly partial dislocation was created at the step on (1 1¯ 0) leaving a stacking fault to the surface, then the second Heidenreich–Shockley partial dislocation was created, joined with the stacking fault, moved as a pair of dislocations, ejected from the other side of the crystal. This can be compared with the experimental value between 163 and 30 erg/cm2 . If the readers look at Fig. 2 from the bottom of the page with a low angle, the reader can see half dislocations. Each time the half dislocation was introduced, stress relaxation and a sharp yield point
was observed. Atomic positions just below and just above the slip plane is plotted in Fig. 3. Two half dislocations are connected with the stacking fault (Fig. 3). The largest open circles are two layers above the slip plane. The medium size open circles are just above the slip plane. The smallest circles are just below the slip plane. In the perfect crystal largest and medium size circles form hexagons and the smallest circles are at the center of the haxagons. These can be seen in the right-hand side of Fig. 3(a) and the left-hand side of Fig. 3(c). In the stacking fault region, the largest and smallest circles come at the same point, so that no atoms can be seen in the hexagons. Such a plane can be found at the left-hand side of Fig. 3(a), right-hand side of Fig. 3(b) and
Fig. 4. Speciemen 2: (a) 300 steps (6% elongation); (b) 375 steps (7.5% elongation); (c) 500 steps (10% elongation); (d) 675 steps (13.5% elongation). All are projected on xz plane.
454
M. Doyama, Y. Kogure / Materials Science and Engineering A309–310 (2001) 451–455
Fig. 5. Stress–strain curve for Specimen 2 (tensile test).
right-hand side of Fig. 3(c). There is a half dislocation in the transient region from the perfect region to the stacking fault region. The splitting of dislocations can also be seen, if the readers look at Fig. 4 from the right-hand side with a low angle. The most clear case is Fig. 3(c). A half dislocation is near the center of the figure.
Fig. 6. Schematic diagram of creation and motion of half dislocations and motion of half dislocations in Specimen 2.
Fig. 7. The projection of atom positions of Specimen 3 on xz-plane (1 1 0): (a) at 500 time cycles; (b) 2000 time cycles; (c) at 2500 time cycles; (d) at 3500 time cycles; (e) at 4500 time cycles; (f) at 7500 cycles; (g) at 9000 time cycles; (h) at 15,000 time cycles.
M. Doyama, Y. Kogure / Materials Science and Engineering A309–310 (2001) 451–455
Atomic positions near the center of Specimen 2 projected on the xz plane (0 1 0) are plotted in Fig. 4(a)–(d). In Fig. 4(b), near the tip of the crack, a half-dislocation was created. Two other half-dislocations are started near the grip of Specimen 2. These can be clearly seen when the readers look at the figures from the bottom of the page with a small angle from the paper. In Fig. 5, the stress–strain curve is plotted. A very sharp yield point was found. It was found that a notch is normally a source of dis¯ 111) ¯ locations due to the stress concentration. [2¯ 11]( and ¯ ¯ [112](111) half dislocations were created leaving stacking faults. Schematic diagram is given in Fig. 6 for Specimen 2. The initial stage of deformation was surely conducted by the creation and motion of dislocations. In the middle stage, dislocations on different slip planes cross each other. The work hardening occurs. In the final stage, in the case of copper, crystalline state with many dislocations and the inhomogeneous deformation is normal. Specimen 3 was uniformly compressed. Partial dislocations or twining of three atomic layers started near the tip of
455
the notch above 1000 time cycles (Fig. 7(b)). Bulging, particularly near the notch, can be observed above 3500 time cycles (Fig. 7(d)). Fig. 7(f) is at 7500 time cycles, Fig. 7(g) at 9000 time cycles and Fig. 7(h) is at 15,000 cycles. Severe bulging is observed.
Acknowledgements This work is supported by the Ministry of Education, Science, Sports and Culture. References [1] S.M. Foiles, M.I. Baskes, M.S. Daw, Phys. Rev. B 33 (1986) 7983. [2] D.J. Oh, R.A. Johnson, J. Mater. Res. 3 (1986) 471. [3] D.J. Oh, R.A. Johnson, in: V. Vitek, D.J. Srolovitz (Eds.), Atomic Simulation of Materials, Plenum Press, New York, 1989, p. 233. [4] G.J. Ackland, G. Tichy, M.W. Finnis, Phil. Mag. A 56 (1987) 735. [5] M.J. Baskes, Phys. Rev. B 46 (1992) 2727.