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Plastic deformation of copper thin foils
Thin Solid Films 424 (2003) 88–92
Plastic deformation of copper thin foils N. Nozaki*, Masao Doyama, Y. Kogure Teikyo University of Science and Technology, Uenohara, Yamanashi 409-0193, Japan
Abstract According to one suggested model, bending of a single crystal introduces edge dislocations of the same sign. In the present study, this model is examined by computer simulation using molecular dynamics. When a notch is present on the tension surface, Heidenreich–Shockley partial dislocations are created near the tip of the notch. In the compression surface, partial dislocations are created due to wrinkling of the crystal plane. The results of simulation shows that dislocations are more easily created in a compressive bending region than in a tension bending region or simple tension region. For shear deformation, partial dislocations are created on the highest resolved shear stress slip plane {1 1 1} and slip in the direction of highest resolved shear stress. 䊚 2002 Elsevier Science B.V. All rights reserved. Keywords: Bending, shearing, and compression of single crystals; Copper; Partial dislocations; Creation of dislocations; Motion of dislocations; Molecular dynamics; Embedded atom potential; Computer simulation
1. Introduction
2. Potential functions
Creation, motion, and interaction of dislocations are well known to play an important role in the plastic deformation of crystalline solids. Determining how the dislocations are created and move in the process of bending and shearing of a metallic single crystal are important. In this paper, plastic deformation is simulated by use of molecular dynamics. Copper is chosen for the simulation, because it has a face centered cubic lattice and is one of the most common metals. Shear is a combination of bending and stretching. According to one theory, pure bending creates edge dislocations from the inner surface and outer surface and move to the median plane; one of the purposes of this paper is to determine how this model changes with elongation, by way of computer simulation. In metals, conduction electrons travel from one atom to another. Accordingly, the interaction cannot be represented by a pairwise potential, and instead must be represented 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 on other factors. The embedded function w1–5x can also be used to treat surface problems.
For the n-body embedded function proposed in this paper, total energy is given by
*Corresponding author.
Etotals8Ei
(1)
rijsriyrj
(2)
Eis(1y2)FŽrij.qFŽri.
(3)
FŽri.sDrilnri
(4)
risfŽrij.
(5)
Here Etotal is total internal energy, Ei is the internal energy associated with atom i, ri is the electron density at atom i due to all other atoms, F(ri) is the energy to embed an atom into an electron gas density ri, F(rij) is the two body central potential between atoms i and j separated by rij, and f(rij) is the contribution to the electron density at atom i due to atom j at the distance rij from atom i. F(ri) is an attractive term. For the functional forms, FŽrij.sA1Žrc1yrij.2expŽyc1rij.
(6)
fŽrij.sA2Žrc2yrij.2expŽyc2rij.
(7)
are assumed. f(rij) and F(rij) are smoothly truncated at rc1 and rc2, respectively. rc1 was chosen to be 1.65d (d
0040-6090/03/$ - see front matter 䊚 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 0 - 6 0 9 0 Ž 0 2 . 0 0 9 2 3 - 9
N. Nozaki et al. / Thin Solid Films 424 (2003) 88–92
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Fig. 2. Projection of all atoms of Specimen 1 on the (y1 1 0) plane. (a) Before bending; (b) at 1=105 time steps; (c) at 1.2=105 time steps; (d) at 1.5=105 time steps; and (e) at 2=105 time steps. Fig. 1. (a) Specimens 1, 2, and 3 are rectangular parallelepipeds having the faces of (0 0 y1), (0 0 1), (y1 1 0), (1 y1 0), (1 1 0) and (y1 y1 0). The x-, y-, and z-directions are defined as the w0 0 y1x, wy1 1 0x and w1 1 0x directions, respectively. (b) Specimen 4 is a rectangular parallelepiped having the faces of (1 y1 0), (y1 1 0), (y 1 1 2), (1 y1 2), (1 1 1) and (y1 y1 y1). The x-, y-, and z-directions are defined as the w1 y1 0x, wy1 1 2x and w1 1 1x directions, respectively.
is the nearest neighbor distance). F(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, A1s8.28945997705=103, A2s1.83251035107 =10y2, C1s10.72729128641, C2s0.319759369823, and Ds13.07921251628 w6x. 3. Specimens Four simulated copper specimens were prepared. Specimens 1, 2, and 3 (Fig. 1a) are rectangular parallelepipeds having the faces of (0 0 y1), (0 0 1), (y1 1 0), (1 y1 0), (1 1 0), and (y1 y1 0). The x-, y-, and z-directions were defined as the w0 0 y1x, wy1 1 0x, and w1 1 0x directions, respectively. The length of Specimen 1 in the x-direction is 8.5a, where a is the lattice parameter. The length in the y-direction is 9.5d and the length in the z-direction is 28.5d, where d is the nearest neighbor distance. Specimen 1 has a notch at its center on (1 y1 0), oriented in the direction of wy1 1 0x. Specimen 1 contains 5720 atoms. Specimens 2 and 3 are rectangular parallelepipeds. A notch was introduced to Specimen 2 on (0 0 1) in the direction of w1 y1 0x. The x-axis was defined as the w0 0 y1x, the y-axis as wy1 1 0x, and the z-axis as w1 1 0x. The size of Specimen 3 is 6.5a=9.5d=32.5d, but no notch is provided. Specimens 2 and 3 contain 5060 and 5082 atoms, respectively. Specimen 4 (Fig. 1b) is a rectangular parallelepiped having the faces of (1 y1 0), (y1 1 2), (1 1 1),
(y1 1 0), (1 y1 y2), and (y1 y1 y1). The x-, y-, and z-directions were defined as the w1 y1 0x, wy1 1 2x, and w1 1 1x directions, respectively. The length of Specimen 4 in the x-direction is 8.5d, where d is the nearest neighbor distance. The length in the y-direction is 7(2)0.5d and the length in the z-direction is 16=(2y 3)0.5d, where d is the nearest neighbor distance. No notch is provided to Specimen 4. Specimen 4 contains 2646 atoms. The surfaces of all specimens are free and periodic boundary conditions are not used. 4. Deformation The specimens were relaxed using molecular dynamics and the time step was 10y15 s. Every 100 time steps, small bending was introduced. When the total time steps are 5=105 steps, the total time required is 5=10y10 s, which is very short time and corresponds to a very fast bending and shearing. Specimens 1–3 were uniformly bent. The center or axis of the bending was taken to be parallel to the y-axis on the x-axis. Let the distance between the bending axis and the median plane be r, radius of curvature. As the bending proceeds, the bending axis approaches to the y-axis, which passes the median plane. For a uniform bending, the length of the
Fig. 3. Projection of atoms on the center two atomic planes of Specimen 2 with a notch, (a) at 2.2=105 time steps; (b) at 3=105 time steps; and (c) at 5=105 time steps. All are projected on the yz plane.
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parallel to the shear as shown in Fig. 6. The bent positions were deformed following a sine function. 5. Results and discussion
Fig. 4. Projection of atoms on the center two atomic planes of Specimen 3 (a) at 2=105 time steps; (b) at 2.2=105 time steps; (c) at 3=105 time steps; and (d) at 5=105 time steps. All are projected on the yz plane.
median plane does not change before and after bending. All the atoms above the median plane are stretched and all the atoms below the median plane are compressed. Let the glancing angle be u. Then before bending, u is zero and r is infinite. rusL, where L is the length of the specimen in the z-direction. When the radius of curvature is r, the outer surface extends (rqDy2)uy rusDuy2sDLy(2r). The inner surface is compressed by (ryDy2)uyrusyDuy2sDLy(2r). The amount of elongation and compression on curved surfaces Dy2 from the median plane is Dy(2r) for a unit length. The specimens were relaxed by use of molecular dynamics, and a time step of 10y15 s was employed. Every 100 time steps, small bending or shearing or compression was introduced. When 5=105 time steps are employed, the total time required is 5=10y10 s, which is a very short time and corresponds to a very fast plastic deformation. Both ends of the specimens were transferred parallel. Specimens 4 and 5 were used for shearing. Every 100 time steps, small shear was introduced. Twelve layers from both the ends of Specimen 3 were only shifted
Fig. 2 shows projection of all atoms of Specimen 1 on the (y1 1 0) plane. (a) is the state before bending, (b) is at 1=105 time steps, (c) is at 1.2=105 time steps, (d) is at 1.5=106 time steps, and (e) is at 2=105 time steps. At approximately 105 time steps, some indication of deformation near the tip of the notch and irregularity in the second layer from the inner surface can be seen (Fig. 2b). These are clearer at 1.2=105 time steps (Fig. 2c). At approximately 1.5=105 time steps, bending of the inner surface is no longer smooth; two partial dislocations have started from the inner surface (Fig. 2d). At approximately 5=105 time steps, several dislocations are present in the specimen (Fig. 2e). These are clearly noticed when the figure is viewed from a small angle from the paper. Fig. 3 shows projection of all atoms of Specimen 2 on the (y1 1 0) plane. (a) is the state at 2.2=105 time steps, (b) is at 3=105 time steps, (c) is at 5=105 time steps, some indication of irregularity can be noticed near the tip of the notch (Fig. 3b and c). At approximately 2.2=105, some irregularities are observed on the inner surface. Four partial dislocations are observed. At 3=105 time steps, nearly all atoms near the center of inner surface are out of atomic alignment (Fig. 3c). Fig. 4 shows projection of all atoms of Specimen 3 (without notch) on the (y1 1 0) plane. (a) is the state at 2=105 time steps, (b) is at 2.2=105 time steps, (c) is at 3=105 time steps, and (d) at 5=105 time steps. At 3=105 time steps, some irregularity can be noticed on the inner surface. Five partial dislocations are observed. Even at 5=105 time steps, not much irregularities are noticed except one partial dislocation (Fig. 4d).
Fig. 5. The relation between crystal energy and number of time steps for Specimens 1, and 2.
N. Nozaki et al. / Thin Solid Films 424 (2003) 88–92
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Fig. 7. The relation between crystal energy and number of time steps for Specimen 4.
Fig. 6. Projection of atoms on the center two atomic planes of Specimen 4 (a) at 5=104 time steps; (b) at 6=104 time steps; (c) at 8=104 time steps; (d) at 1=105 time steps; and (e) at 1.5=105 time steps. All are projected on the yz plane.
At equilibrium, the interatomic potential rises more sharply for compression than for tension. For the same deformation, or the same bending, the inner surface has higher energy or stress than does the outer surface (0 0 y1). Therefore the inner surface (0 0 1) wrinkles, so that partial dislocations are introduced on the inner surface in the specimen and move toward the median plane. For Specimen 3 having no notch, no dislocations were introduced on the outer surface (0 0 y1), but a half dislocation was introduced on the inner surface (0 0 1). When a notch is provided on the outer surface (0 0 y1), partial dislocations were introduced near the tip of the notch, as we had expected. At approximately 1=105 time steps, which corresponds to approximately 10% elongation on the outer surface, a half dislocation was initiated near the tip of the notch in Specimen 1 (Fig. 2b). For Specimen 3 provided with no notch, no dislocations were initiated through 2.2=105 time steps, which corresponds to approximately 7.5% compression on the inner surface and approximately 7.5% elongation on the outer surface. At approximately 3=105 time steps, which corresponds to approximately 16.7% compression on the inner surface and approximately 16.7% elongation on the outer surface, several partial dislocations were initiated at the inner surface, but no dislocations were initiated on the outer surface. Fig. 5 shows the relation between crystal energies and time step. Because of the surface relaxation, the energies of the crystals decrease first 2–3=104 time steps. Then,
due to the effect of bending, the crystal energies go up; subsequently, sharp bending is observed in the crystal energy vs. time steps curve was observed, which corresponds to the creation of dislocations. After dislocations are created in the specimens, the energies of the crystals increase, but not as sharply as the initial increase, which corresponds to the elastic bending region. Fig. 6 shows projection of all atoms on the (1–10) plane. (a) At the situation of 5=104 time steps, (b) is at 6=104 time steps, (c) is at 8=104 time steps, (d) is at 1=105 time steps, and (e) is at 1.5=105 time steps. At approximately 6=104 time steps, some indication of deformation in the deformed shear region can be seen (Fig. 6b). These are clearer at 8=104 and 1=105 time steps (Fig. 6c and d). Fig. 7 shows the relation between crystal energies and time step. Because of the surface relaxation, the energies of the crystals decrease first up to 5=103 time steps. Then, due to the effect of shearing, the crystal energies go up. Subsequently, sharp bending is observed in the crystal energy vs. time steps curve, which corresponds to the creation of dislocations. After dislocations are created in the specimens, the energies of the crystals increase, but not as sharply as the initial increase, which corresponds to the elastic bending region. 6. Conclusions When a copper single crystal without a notch is bent, partial dislocations are initiated on the inner, or compressive plane, because of wrinkling of the crystal planes due to compression. When a copper single crystal with a notch on the tension surface is bent, partial dislocations are initiated near the tip of the notch as well as on the inner, or compressive plane. When a copper single crystal without notch is sheared, partial dislocations are created at the sheared region.
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92
Acknowledgments This work is supported by the Ministry of Education, Sports, Culture and Science and Technology. References w1x S.M. Foiles, M.I. Baskes, M.S. Daw, Phys. Rev. B 33 (1986) 7983.
w2x D.J. Oh, R.A. Johnson, J. Mater. Res. 3 (1986) 471. w3x D.J. Oh, R.A. Johnson, in: V. Vitek, D.J. Srolovitz (Eds.), Atomic Simulation of Materials, Plenum Press, New York, 1989, p. 233. w4x G.J. Ackland, G. Tichy, M.W. Finnis, Phil. Mag. A 56 (1987) 735. w5x M.J. Baskes, Phys. Rev. B 46 (1992) 2727. w6x M. Doyama, Y. Kogure, Radiation effects and defects in solids 142 (1997) 107.
Plastic deformation of copper thin foils N. Nozaki*, Masao Doyama, Y. Kogure Teikyo University of Science and Technology, Uenohara, Yamanashi 409-0193, Japan
Abstract According to one suggested model, bending of a single crystal introduces edge dislocations of the same sign. In the present study, this model is examined by computer simulation using molecular dynamics. When a notch is present on the tension surface, Heidenreich–Shockley partial dislocations are created near the tip of the notch. In the compression surface, partial dislocations are created due to wrinkling of the crystal plane. The results of simulation shows that dislocations are more easily created in a compressive bending region than in a tension bending region or simple tension region. For shear deformation, partial dislocations are created on the highest resolved shear stress slip plane {1 1 1} and slip in the direction of highest resolved shear stress. 䊚 2002 Elsevier Science B.V. All rights reserved. Keywords: Bending, shearing, and compression of single crystals; Copper; Partial dislocations; Creation of dislocations; Motion of dislocations; Molecular dynamics; Embedded atom potential; Computer simulation
1. Introduction
2. Potential functions
Creation, motion, and interaction of dislocations are well known to play an important role in the plastic deformation of crystalline solids. Determining how the dislocations are created and move in the process of bending and shearing of a metallic single crystal are important. In this paper, plastic deformation is simulated by use of molecular dynamics. Copper is chosen for the simulation, because it has a face centered cubic lattice and is one of the most common metals. Shear is a combination of bending and stretching. According to one theory, pure bending creates edge dislocations from the inner surface and outer surface and move to the median plane; one of the purposes of this paper is to determine how this model changes with elongation, by way of computer simulation. In metals, conduction electrons travel from one atom to another. Accordingly, the interaction cannot be represented by a pairwise potential, and instead must be represented 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 on other factors. The embedded function w1–5x can also be used to treat surface problems.
For the n-body embedded function proposed in this paper, total energy is given by
*Corresponding author.
Etotals8Ei
(1)
rijsriyrj
(2)
Eis(1y2)FŽrij.qFŽri.
(3)
FŽri.sDrilnri
(4)
risfŽrij.
(5)
Here Etotal is total internal energy, Ei is the internal energy associated with atom i, ri is the electron density at atom i due to all other atoms, F(ri) is the energy to embed an atom into an electron gas density ri, F(rij) is the two body central potential between atoms i and j separated by rij, and f(rij) is the contribution to the electron density at atom i due to atom j at the distance rij from atom i. F(ri) is an attractive term. For the functional forms, FŽrij.sA1Žrc1yrij.2expŽyc1rij.
(6)
fŽrij.sA2Žrc2yrij.2expŽyc2rij.
(7)
are assumed. f(rij) and F(rij) are smoothly truncated at rc1 and rc2, respectively. rc1 was chosen to be 1.65d (d
0040-6090/03/$ - see front matter 䊚 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 0 - 6 0 9 0 Ž 0 2 . 0 0 9 2 3 - 9
N. Nozaki et al. / Thin Solid Films 424 (2003) 88–92
89
Fig. 2. Projection of all atoms of Specimen 1 on the (y1 1 0) plane. (a) Before bending; (b) at 1=105 time steps; (c) at 1.2=105 time steps; (d) at 1.5=105 time steps; and (e) at 2=105 time steps. Fig. 1. (a) Specimens 1, 2, and 3 are rectangular parallelepipeds having the faces of (0 0 y1), (0 0 1), (y1 1 0), (1 y1 0), (1 1 0) and (y1 y1 0). The x-, y-, and z-directions are defined as the w0 0 y1x, wy1 1 0x and w1 1 0x directions, respectively. (b) Specimen 4 is a rectangular parallelepiped having the faces of (1 y1 0), (y1 1 0), (y 1 1 2), (1 y1 2), (1 1 1) and (y1 y1 y1). The x-, y-, and z-directions are defined as the w1 y1 0x, wy1 1 2x and w1 1 1x directions, respectively.
is the nearest neighbor distance). F(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, A1s8.28945997705=103, A2s1.83251035107 =10y2, C1s10.72729128641, C2s0.319759369823, and Ds13.07921251628 w6x. 3. Specimens Four simulated copper specimens were prepared. Specimens 1, 2, and 3 (Fig. 1a) are rectangular parallelepipeds having the faces of (0 0 y1), (0 0 1), (y1 1 0), (1 y1 0), (1 1 0), and (y1 y1 0). The x-, y-, and z-directions were defined as the w0 0 y1x, wy1 1 0x, and w1 1 0x directions, respectively. The length of Specimen 1 in the x-direction is 8.5a, where a is the lattice parameter. The length in the y-direction is 9.5d and the length in the z-direction is 28.5d, where d is the nearest neighbor distance. Specimen 1 has a notch at its center on (1 y1 0), oriented in the direction of wy1 1 0x. Specimen 1 contains 5720 atoms. Specimens 2 and 3 are rectangular parallelepipeds. A notch was introduced to Specimen 2 on (0 0 1) in the direction of w1 y1 0x. The x-axis was defined as the w0 0 y1x, the y-axis as wy1 1 0x, and the z-axis as w1 1 0x. The size of Specimen 3 is 6.5a=9.5d=32.5d, but no notch is provided. Specimens 2 and 3 contain 5060 and 5082 atoms, respectively. Specimen 4 (Fig. 1b) is a rectangular parallelepiped having the faces of (1 y1 0), (y1 1 2), (1 1 1),
(y1 1 0), (1 y1 y2), and (y1 y1 y1). The x-, y-, and z-directions were defined as the w1 y1 0x, wy1 1 2x, and w1 1 1x directions, respectively. The length of Specimen 4 in the x-direction is 8.5d, where d is the nearest neighbor distance. The length in the y-direction is 7(2)0.5d and the length in the z-direction is 16=(2y 3)0.5d, where d is the nearest neighbor distance. No notch is provided to Specimen 4. Specimen 4 contains 2646 atoms. The surfaces of all specimens are free and periodic boundary conditions are not used. 4. Deformation The specimens were relaxed using molecular dynamics and the time step was 10y15 s. Every 100 time steps, small bending was introduced. When the total time steps are 5=105 steps, the total time required is 5=10y10 s, which is very short time and corresponds to a very fast bending and shearing. Specimens 1–3 were uniformly bent. The center or axis of the bending was taken to be parallel to the y-axis on the x-axis. Let the distance between the bending axis and the median plane be r, radius of curvature. As the bending proceeds, the bending axis approaches to the y-axis, which passes the median plane. For a uniform bending, the length of the
Fig. 3. Projection of atoms on the center two atomic planes of Specimen 2 with a notch, (a) at 2.2=105 time steps; (b) at 3=105 time steps; and (c) at 5=105 time steps. All are projected on the yz plane.
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parallel to the shear as shown in Fig. 6. The bent positions were deformed following a sine function. 5. Results and discussion
Fig. 4. Projection of atoms on the center two atomic planes of Specimen 3 (a) at 2=105 time steps; (b) at 2.2=105 time steps; (c) at 3=105 time steps; and (d) at 5=105 time steps. All are projected on the yz plane.
median plane does not change before and after bending. All the atoms above the median plane are stretched and all the atoms below the median plane are compressed. Let the glancing angle be u. Then before bending, u is zero and r is infinite. rusL, where L is the length of the specimen in the z-direction. When the radius of curvature is r, the outer surface extends (rqDy2)uy rusDuy2sDLy(2r). The inner surface is compressed by (ryDy2)uyrusyDuy2sDLy(2r). The amount of elongation and compression on curved surfaces Dy2 from the median plane is Dy(2r) for a unit length. The specimens were relaxed by use of molecular dynamics, and a time step of 10y15 s was employed. Every 100 time steps, small bending or shearing or compression was introduced. When 5=105 time steps are employed, the total time required is 5=10y10 s, which is a very short time and corresponds to a very fast plastic deformation. Both ends of the specimens were transferred parallel. Specimens 4 and 5 were used for shearing. Every 100 time steps, small shear was introduced. Twelve layers from both the ends of Specimen 3 were only shifted
Fig. 2 shows projection of all atoms of Specimen 1 on the (y1 1 0) plane. (a) is the state before bending, (b) is at 1=105 time steps, (c) is at 1.2=105 time steps, (d) is at 1.5=106 time steps, and (e) is at 2=105 time steps. At approximately 105 time steps, some indication of deformation near the tip of the notch and irregularity in the second layer from the inner surface can be seen (Fig. 2b). These are clearer at 1.2=105 time steps (Fig. 2c). At approximately 1.5=105 time steps, bending of the inner surface is no longer smooth; two partial dislocations have started from the inner surface (Fig. 2d). At approximately 5=105 time steps, several dislocations are present in the specimen (Fig. 2e). These are clearly noticed when the figure is viewed from a small angle from the paper. Fig. 3 shows projection of all atoms of Specimen 2 on the (y1 1 0) plane. (a) is the state at 2.2=105 time steps, (b) is at 3=105 time steps, (c) is at 5=105 time steps, some indication of irregularity can be noticed near the tip of the notch (Fig. 3b and c). At approximately 2.2=105, some irregularities are observed on the inner surface. Four partial dislocations are observed. At 3=105 time steps, nearly all atoms near the center of inner surface are out of atomic alignment (Fig. 3c). Fig. 4 shows projection of all atoms of Specimen 3 (without notch) on the (y1 1 0) plane. (a) is the state at 2=105 time steps, (b) is at 2.2=105 time steps, (c) is at 3=105 time steps, and (d) at 5=105 time steps. At 3=105 time steps, some irregularity can be noticed on the inner surface. Five partial dislocations are observed. Even at 5=105 time steps, not much irregularities are noticed except one partial dislocation (Fig. 4d).
Fig. 5. The relation between crystal energy and number of time steps for Specimens 1, and 2.
N. Nozaki et al. / Thin Solid Films 424 (2003) 88–92
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Fig. 7. The relation between crystal energy and number of time steps for Specimen 4.
Fig. 6. Projection of atoms on the center two atomic planes of Specimen 4 (a) at 5=104 time steps; (b) at 6=104 time steps; (c) at 8=104 time steps; (d) at 1=105 time steps; and (e) at 1.5=105 time steps. All are projected on the yz plane.
At equilibrium, the interatomic potential rises more sharply for compression than for tension. For the same deformation, or the same bending, the inner surface has higher energy or stress than does the outer surface (0 0 y1). Therefore the inner surface (0 0 1) wrinkles, so that partial dislocations are introduced on the inner surface in the specimen and move toward the median plane. For Specimen 3 having no notch, no dislocations were introduced on the outer surface (0 0 y1), but a half dislocation was introduced on the inner surface (0 0 1). When a notch is provided on the outer surface (0 0 y1), partial dislocations were introduced near the tip of the notch, as we had expected. At approximately 1=105 time steps, which corresponds to approximately 10% elongation on the outer surface, a half dislocation was initiated near the tip of the notch in Specimen 1 (Fig. 2b). For Specimen 3 provided with no notch, no dislocations were initiated through 2.2=105 time steps, which corresponds to approximately 7.5% compression on the inner surface and approximately 7.5% elongation on the outer surface. At approximately 3=105 time steps, which corresponds to approximately 16.7% compression on the inner surface and approximately 16.7% elongation on the outer surface, several partial dislocations were initiated at the inner surface, but no dislocations were initiated on the outer surface. Fig. 5 shows the relation between crystal energies and time step. Because of the surface relaxation, the energies of the crystals decrease first 2–3=104 time steps. Then,
due to the effect of bending, the crystal energies go up; subsequently, sharp bending is observed in the crystal energy vs. time steps curve was observed, which corresponds to the creation of dislocations. After dislocations are created in the specimens, the energies of the crystals increase, but not as sharply as the initial increase, which corresponds to the elastic bending region. Fig. 6 shows projection of all atoms on the (1–10) plane. (a) At the situation of 5=104 time steps, (b) is at 6=104 time steps, (c) is at 8=104 time steps, (d) is at 1=105 time steps, and (e) is at 1.5=105 time steps. At approximately 6=104 time steps, some indication of deformation in the deformed shear region can be seen (Fig. 6b). These are clearer at 8=104 and 1=105 time steps (Fig. 6c and d). Fig. 7 shows the relation between crystal energies and time step. Because of the surface relaxation, the energies of the crystals decrease first up to 5=103 time steps. Then, due to the effect of shearing, the crystal energies go up. Subsequently, sharp bending is observed in the crystal energy vs. time steps curve, which corresponds to the creation of dislocations. After dislocations are created in the specimens, the energies of the crystals increase, but not as sharply as the initial increase, which corresponds to the elastic bending region. 6. Conclusions When a copper single crystal without a notch is bent, partial dislocations are initiated on the inner, or compressive plane, because of wrinkling of the crystal planes due to compression. When a copper single crystal with a notch on the tension surface is bent, partial dislocations are initiated near the tip of the notch as well as on the inner, or compressive plane. When a copper single crystal without notch is sheared, partial dislocations are created at the sheared region.
N. Nozaki et al. / Thin Solid Films 424 (2003) 88–92
92
Acknowledgments This work is supported by the Ministry of Education, Sports, Culture and Science and Technology. References w1x S.M. Foiles, M.I. Baskes, M.S. Daw, Phys. Rev. B 33 (1986) 7983.
w2x D.J. Oh, R.A. Johnson, J. Mater. Res. 3 (1986) 471. w3x D.J. Oh, R.A. Johnson, in: V. Vitek, D.J. Srolovitz (Eds.), Atomic Simulation of Materials, Plenum Press, New York, 1989, p. 233. w4x G.J. Ackland, G. Tichy, M.W. Finnis, Phil. Mag. A 56 (1987) 735. w5x M.J. Baskes, Phys. Rev. B 46 (1992) 2727. w6x M. Doyama, Y. Kogure, Radiation effects and defects in solids 142 (1997) 107.