- Email: [email protected]
Molecular dynamics simulations of spallation in metals and alloys
COMPUTATIONAL MATERIALS SCIENCE ELSEVIER
Computational
Materials
Science
IO ( 1998) 432435
Molecular dynamics simulations of spallation in metals and alloys W.C. Morrey, L.T. Wille * Deparimenf
of Physics, Florida Atlantic
Universin
777 Glades Road, Boca Raton, FL 33431. USA
Abstract We report on the results of large-scale molecular dynamics simulations of the mechanical behavior of two-dimensional metallic systems. The specific impact phenomenon studied is that in which a flyer of mass M moving with x-velocity v impacts a target of mass 2h4 moving with x-velocity -v/2. Simulations of such a spallation experiment have been performed for a generic metal, modelled with an embedded atom potential and also for a Cu-Ni alloy system, modelled with truncated Lennard-Jones potentials. Our simulations indicate cold-welding upon impact, and shock wave generation, followed by rebound from the boundaries. The alloy was less ductile than the generic metal and consequently the system came apart due to the cooperative effect of the reflected shock waves. Copyright 0 1998 Elsevier Science B.V. Keywords:
Fracture; Spallation; Molecular dynamics; Parallel computing
in metals has been a held of study for many
years because of the availability
of scalable parallel
years [ 11. Interest in this area has been sustained by the pervasive economic impact of even the smallest ad-
computers. In general,
consist of a multi-
vances. Understanding metallic fracture is at the heart of improvements in safety, reliability, and cost of a
plicity of processors, which have interconnecting
vast range of human endeavor. Tremendous benefits accrue to manufacturing, transportation, and construction, among others, by improving a metal’s resistance
These computers are used to tackle a problem by hav-
Fracture
to fracture. Understanding
fracture mechanisms
is the
key to devising that improvement. The path to illumination of mechanisms underlying fracture is microscopic simulations. Molecular dynamics (MD) simulations using realistic potentials are now performed on a scale that is approaching the actual dimensions of the microscopic features of interest [2]. This breakthrough has only become possible in recent * Corresponding author. Tel.: +1 561 297 3379; fax: + I 561 297 2662; e-mail:
[email protected].
munications,
parallel computers
and some overall controlling
ing different elements of the processing
com-
element.
section work
on different parts of the problem. There are two basic architectural approaches to the design of parallel computers [3]. In order to break up the logic of the problem and dispense different tasks to different processors, each processor has to be powerful enough to handle its own task independently. This requirement gives rise to MIMD (multiple-instruction multiple-data) parallel computation. On the other hand, if the size and complexity of the problem arises from the number of objects in the problem, not by the variations of the actions of each object, the same logic can be used by each processor. This results in keeping the logic in the
0927-0256/98/$19.00 Copyright 0 1998 Elsevier Science B.V. All rights reserved PII SO927-0256(97)00135-3
WC. Morrey, LT. Wille/Computational
controlling portion, leaving each processor as a simple calculating engine, with communications. This structure is SIMD (single-instruction multiple-data) parallel computation. Various strategies for parallelizing MD simulations of large systems have been described elsewhere [4]. The simulation of fracture in materials at the atomic level lends itself very nicely to an SIMD approach. Each particle is in a potential due to surrounding particles. These potentials result in applied forces which move the particle. Each step in the algorithms to calculate these potentials, forces, and motions can be applied in lock-step to a different particle by each processor. Several investigators have used SIMD machines to study fracture problems [5-81. The present calculations were performed on one such computer, the MasPar MP- 1. A plane-wave impact experiment simulation using a square grid of -200000atoms was conducted in a manner analogous to that employed in [5]. We took a two-dimensional array of generic atoms arranged on a hexagonal lattice. The atomic locations were laid out so that not quite all of the PEs on the MasPar were used. Sufficient free space was left around the grid to allow for expansion in all directions, the surfaces were relaxed and the velocities were quenched until the system was within a few Kelvin of absolute zero. The idea of the simulation was to have a segment of the particles, the flyer, impact the remainder of the particles, the target. The atoms were not physically separated between the flyer and the target: the start of the run represented the moment of impact. A velocity V was added to the left l/3 of the atoms representing the flyer, while -V/2 was added to the right 2/3 of the atoms, which were the target. This provided a centerof-mass velocity of zero. From the start of the simulation shock waves move out from the impact plane both forward toward the opposite end of the target and backward into the flyer. These waves reflect off the free ends and return, broadening due to nonlinear effects. Since the target is two flyer-widths (FWs), and assuming constant wave velocities, the waves travel three FWs and meet at a plane one FW from the end of the target. We chose the where E represents relative velocity to be - 1.2m,
Materials Science IO (1998) 432435 500
I
I
I
I
433 I
I
I
I
450 400 350 300 250 200 150 100 50 50
I
I
1
I
1
I
I
I
I
100
150
200
250
300
350
400
450
500
Fig. 1. Spallation in a generic metal using EAM potential.
the depth of the potential well and m is the atomic mass. This gave a high enough impact velocity that the superimposition of rarefaction wave reflections from the free surfaces led to spallation. A snapshot of this generic metal spallation run is shown in Fig. 1. In particular, we note the necking at the top and bottom of the material centered around location 300. This is due to the bulk tensile strain in the X direction. There is also a concavity at the right and left around location 250. This is a rebound from the expansion caused when the shock waves hit the ends. Next we took the step of expanding the applicability of the simulation to alloys. In doing this, it was decided to eliminate the EAM segment of the calculations, because we were eventually to use this model to study fracture and Holian and Ravelo [7] have shown that the LJ potential alone could be used in fracture studies. They discuss the differences between brittle and ductile materials as being characterized by the strength of the attractive portion of the potential at the point of critical strain. The higher the potential at that point, the more likely that the material will be brittle. The LJ potential itself is inherently ductile, and retaining that shape of the potential in the repulsive
434
WC. Morrey L.T. Wille/Computational
region out to the minimum
of the potential will result
in keeping the correct anharmonic
behavior
Materials Science 10 (19981 432435 300 I-
in com-
pression. Adding a cubic spline cutoff from the inflec-
250
tion point of the potential retains the ductile dynamic fracture behavior
of the full LJ potential
in tension.
Starting the spline cutoff at the minimum results in brittle fracture behavior. Since both brittle and ductile behavior can be accommodated
200
by a change in the
shape of the LJ cutoff tail, it seems reasonable
to start
150
the study of alloy fracture with the LJ potential alone. Further, with the chosen material, Cu-Ni, being ductile, we will remain with the same cubic spline cutoff, starting at the inflection
100
point (as used in the mono-
atomic simulation). We used this model to simulate spallation
50
in an al-
loy. Since this model did not include the many-body EAM portion of the code, the material was much less
0 0
100
200
300
400
500
600
ductile than in our mono-atomic simulation. At the same relative velocities as used in the mono-atomic spallation, this material exploded at the point of impact, as shown in Fig. 2. Particles were blown out
Fig. 3. Flyer/target velocity of impact
of the lower edge of the PE range and disappeared
is extruded outward at high velocity. The sample is
from the simulation. What occurred at the point of impact can be seen at the top of the sample. Material
burst apart internally, and atoms can be seen boiling off from the surfaces. Reduction of the impact velocity by a factor of 4
impact in Cu-Ni using LJ potential. Relative one-fourth mono-atomic EAM simulation.
put the collision in the range where the elastic material response dominated the shock propagation. However, the pure LJ alloy does not spa11 at the 2/3 line as the mono-atomic simulation with EAM did. As can be seen in Fig. 3, this material eventually fails under tension. The forward and backward shock waves from the impact have both had time to reflect from the front and back surfaces, and the material is unable to sustain the subsequent elongation. Wagner et al. [5] attribute similar results for a pure metal in part to an expected decrease in total strain at the spa11 plane compared to that calculated for an EAM material.
0
0
100
Fig. 2. Flyer/target velocity of impact
200
300
400
500
600
impact in Cu-Ni using LJ potential. Relative same as mono-atomic EAM simulation.
The authors wish to thank Dr. B. Holian at Los Alamos National Laboratory for extremely helpful discussions and Dr. C. Halloy and the Joint Institute for Computational Science at the University of
WC. Morrey, L.Z Wille/Computational
Tennessee, Knoxville, for the generous use of time on their MasPar MP-2.
References [l] L.B. Freund, Dynamical Fracture Mechanics (Cambridge University Press, Cambridge, 1990). [2] N. Gronbech-Jensen, T. Germann, P.S. Lomdahl and D.M. Beazly, IEEE Comput. Sci. Eng. 2 (2) (1995) 4.
Materials Science IO (1998) 432-435
435
[3] K. Hwang, Advanced Computer Architecture - Parallelism,
Scalability, Programmability (McGraw-Hill, New York, 1993). [4] L.T. ‘Wille, C.F. Comwell and W.C. Money, Mater. Res. Sot. Symp. Proc. 409 (1996) 81. [5] N.J. Wagner, B.L. Holian and A.F. Voter, Phys. Rev. A 45 (1992) 8457. [6] F.F. Abraham, D. B&beck, R.A. Rafey and W.E. Rudge, Phys. Rev. Lett. 73 (1994) 272. [7] B.L. Holian and R. Ravelo, Phys. Rev. B 51 (1995) 11275. [8] S.J. Zhou, P.S. Lomdahl, R. Thomson and B.L. Holian, Phys. Rev. Lett. 76 (1996) 13.
Computational
Materials
Science
IO ( 1998) 432435
Molecular dynamics simulations of spallation in metals and alloys W.C. Morrey, L.T. Wille * Deparimenf
of Physics, Florida Atlantic
Universin
777 Glades Road, Boca Raton, FL 33431. USA
Abstract We report on the results of large-scale molecular dynamics simulations of the mechanical behavior of two-dimensional metallic systems. The specific impact phenomenon studied is that in which a flyer of mass M moving with x-velocity v impacts a target of mass 2h4 moving with x-velocity -v/2. Simulations of such a spallation experiment have been performed for a generic metal, modelled with an embedded atom potential and also for a Cu-Ni alloy system, modelled with truncated Lennard-Jones potentials. Our simulations indicate cold-welding upon impact, and shock wave generation, followed by rebound from the boundaries. The alloy was less ductile than the generic metal and consequently the system came apart due to the cooperative effect of the reflected shock waves. Copyright 0 1998 Elsevier Science B.V. Keywords:
Fracture; Spallation; Molecular dynamics; Parallel computing
in metals has been a held of study for many
years because of the availability
of scalable parallel
years [ 11. Interest in this area has been sustained by the pervasive economic impact of even the smallest ad-
computers. In general,
consist of a multi-
vances. Understanding metallic fracture is at the heart of improvements in safety, reliability, and cost of a
plicity of processors, which have interconnecting
vast range of human endeavor. Tremendous benefits accrue to manufacturing, transportation, and construction, among others, by improving a metal’s resistance
These computers are used to tackle a problem by hav-
Fracture
to fracture. Understanding
fracture mechanisms
is the
key to devising that improvement. The path to illumination of mechanisms underlying fracture is microscopic simulations. Molecular dynamics (MD) simulations using realistic potentials are now performed on a scale that is approaching the actual dimensions of the microscopic features of interest [2]. This breakthrough has only become possible in recent * Corresponding author. Tel.: +1 561 297 3379; fax: + I 561 297 2662; e-mail:
[email protected].
munications,
parallel computers
and some overall controlling
ing different elements of the processing
com-
element.
section work
on different parts of the problem. There are two basic architectural approaches to the design of parallel computers [3]. In order to break up the logic of the problem and dispense different tasks to different processors, each processor has to be powerful enough to handle its own task independently. This requirement gives rise to MIMD (multiple-instruction multiple-data) parallel computation. On the other hand, if the size and complexity of the problem arises from the number of objects in the problem, not by the variations of the actions of each object, the same logic can be used by each processor. This results in keeping the logic in the
0927-0256/98/$19.00 Copyright 0 1998 Elsevier Science B.V. All rights reserved PII SO927-0256(97)00135-3
WC. Morrey, LT. Wille/Computational
controlling portion, leaving each processor as a simple calculating engine, with communications. This structure is SIMD (single-instruction multiple-data) parallel computation. Various strategies for parallelizing MD simulations of large systems have been described elsewhere [4]. The simulation of fracture in materials at the atomic level lends itself very nicely to an SIMD approach. Each particle is in a potential due to surrounding particles. These potentials result in applied forces which move the particle. Each step in the algorithms to calculate these potentials, forces, and motions can be applied in lock-step to a different particle by each processor. Several investigators have used SIMD machines to study fracture problems [5-81. The present calculations were performed on one such computer, the MasPar MP- 1. A plane-wave impact experiment simulation using a square grid of -200000atoms was conducted in a manner analogous to that employed in [5]. We took a two-dimensional array of generic atoms arranged on a hexagonal lattice. The atomic locations were laid out so that not quite all of the PEs on the MasPar were used. Sufficient free space was left around the grid to allow for expansion in all directions, the surfaces were relaxed and the velocities were quenched until the system was within a few Kelvin of absolute zero. The idea of the simulation was to have a segment of the particles, the flyer, impact the remainder of the particles, the target. The atoms were not physically separated between the flyer and the target: the start of the run represented the moment of impact. A velocity V was added to the left l/3 of the atoms representing the flyer, while -V/2 was added to the right 2/3 of the atoms, which were the target. This provided a centerof-mass velocity of zero. From the start of the simulation shock waves move out from the impact plane both forward toward the opposite end of the target and backward into the flyer. These waves reflect off the free ends and return, broadening due to nonlinear effects. Since the target is two flyer-widths (FWs), and assuming constant wave velocities, the waves travel three FWs and meet at a plane one FW from the end of the target. We chose the where E represents relative velocity to be - 1.2m,
Materials Science IO (1998) 432435 500
I
I
I
I
433 I
I
I
I
450 400 350 300 250 200 150 100 50 50
I
I
1
I
1
I
I
I
I
100
150
200
250
300
350
400
450
500
Fig. 1. Spallation in a generic metal using EAM potential.
the depth of the potential well and m is the atomic mass. This gave a high enough impact velocity that the superimposition of rarefaction wave reflections from the free surfaces led to spallation. A snapshot of this generic metal spallation run is shown in Fig. 1. In particular, we note the necking at the top and bottom of the material centered around location 300. This is due to the bulk tensile strain in the X direction. There is also a concavity at the right and left around location 250. This is a rebound from the expansion caused when the shock waves hit the ends. Next we took the step of expanding the applicability of the simulation to alloys. In doing this, it was decided to eliminate the EAM segment of the calculations, because we were eventually to use this model to study fracture and Holian and Ravelo [7] have shown that the LJ potential alone could be used in fracture studies. They discuss the differences between brittle and ductile materials as being characterized by the strength of the attractive portion of the potential at the point of critical strain. The higher the potential at that point, the more likely that the material will be brittle. The LJ potential itself is inherently ductile, and retaining that shape of the potential in the repulsive
434
WC. Morrey L.T. Wille/Computational
region out to the minimum
of the potential will result
in keeping the correct anharmonic
behavior
Materials Science 10 (19981 432435 300 I-
in com-
pression. Adding a cubic spline cutoff from the inflec-
250
tion point of the potential retains the ductile dynamic fracture behavior
of the full LJ potential
in tension.
Starting the spline cutoff at the minimum results in brittle fracture behavior. Since both brittle and ductile behavior can be accommodated
200
by a change in the
shape of the LJ cutoff tail, it seems reasonable
to start
150
the study of alloy fracture with the LJ potential alone. Further, with the chosen material, Cu-Ni, being ductile, we will remain with the same cubic spline cutoff, starting at the inflection
100
point (as used in the mono-
atomic simulation). We used this model to simulate spallation
50
in an al-
loy. Since this model did not include the many-body EAM portion of the code, the material was much less
0 0
100
200
300
400
500
600
ductile than in our mono-atomic simulation. At the same relative velocities as used in the mono-atomic spallation, this material exploded at the point of impact, as shown in Fig. 2. Particles were blown out
Fig. 3. Flyer/target velocity of impact
of the lower edge of the PE range and disappeared
is extruded outward at high velocity. The sample is
from the simulation. What occurred at the point of impact can be seen at the top of the sample. Material
burst apart internally, and atoms can be seen boiling off from the surfaces. Reduction of the impact velocity by a factor of 4
impact in Cu-Ni using LJ potential. Relative one-fourth mono-atomic EAM simulation.
put the collision in the range where the elastic material response dominated the shock propagation. However, the pure LJ alloy does not spa11 at the 2/3 line as the mono-atomic simulation with EAM did. As can be seen in Fig. 3, this material eventually fails under tension. The forward and backward shock waves from the impact have both had time to reflect from the front and back surfaces, and the material is unable to sustain the subsequent elongation. Wagner et al. [5] attribute similar results for a pure metal in part to an expected decrease in total strain at the spa11 plane compared to that calculated for an EAM material.
0
0
100
Fig. 2. Flyer/target velocity of impact
200
300
400
500
600
impact in Cu-Ni using LJ potential. Relative same as mono-atomic EAM simulation.
The authors wish to thank Dr. B. Holian at Los Alamos National Laboratory for extremely helpful discussions and Dr. C. Halloy and the Joint Institute for Computational Science at the University of
WC. Morrey, L.Z Wille/Computational
Tennessee, Knoxville, for the generous use of time on their MasPar MP-2.
References [l] L.B. Freund, Dynamical Fracture Mechanics (Cambridge University Press, Cambridge, 1990). [2] N. Gronbech-Jensen, T. Germann, P.S. Lomdahl and D.M. Beazly, IEEE Comput. Sci. Eng. 2 (2) (1995) 4.
Materials Science IO (1998) 432-435
435
[3] K. Hwang, Advanced Computer Architecture - Parallelism,
Scalability, Programmability (McGraw-Hill, New York, 1993). [4] L.T. ‘Wille, C.F. Comwell and W.C. Money, Mater. Res. Sot. Symp. Proc. 409 (1996) 81. [5] N.J. Wagner, B.L. Holian and A.F. Voter, Phys. Rev. A 45 (1992) 8457. [6] F.F. Abraham, D. B&beck, R.A. Rafey and W.E. Rudge, Phys. Rev. Lett. 73 (1994) 272. [7] B.L. Holian and R. Ravelo, Phys. Rev. B 51 (1995) 11275. [8] S.J. Zhou, P.S. Lomdahl, R. Thomson and B.L. Holian, Phys. Rev. Lett. 76 (1996) 13.