Using molecular dynamics with an accurate many-body potential, we studied the rapid expansion of Ta metal following the high compression (50 to 100 GPa) induced by high velocity (2 to 4 km/s) impact. We find that catastrophic failure in this system coincides with a critical behavior characterized by a void distribution of the form N(v)∝V-τ, with τ∼2.2. This corresponds to a threshold in which percolation of the voids results in tensile failure. We define an order parameter (φ, the ratio of the volume of the largest void to the total void volume) which changes rapidly from ∼0 to ∼1 when the metal fails and scales with as φ∝(ρ-ρc)β with exponent β∼0.4, where ρ is the total void fraction. We found similar behavior for FCC Ni suggesting that this critical behavior is a universal characteristic for failure of solids in rapid expansion

Goddard, William A., III

Strachan, Alejandro

Çağin, Tahir

English

Caltech Authors - Main

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PHYSICAL REVIEW B, VOLUME 63, 060103~R!Critical behavior in spallation failure of metals
Alejandro Strachan, Tahir C¸ag˘ın, and William A. Goddard III*
Materials and Process Simulation Center, Beckman Institute (139-74), California Institute of Technology, Pasadena, California 91125
~Received 30 October 2000; published 23 January 2001!
Using molecular dynamics with an accurate many-body potential, we studied the rapid expansion of Ta
metal following the high compression ~50 to 100 GPa! induced by high velocity (2 to 4 km/s! impact. We find
that catastrophic failure in this system coincides with a critical behavior characterized by a void distribution of
the form N(v)}V2t, with t;2.2. This corresponds to a threshold in which percolation of the voids results in
tensile failure. We define an order parameter (f , the ratio of the volume of the largest void to the total void
volume! which changes rapidly from ;0 to ;1 when the metal fails and scales with as f}(r2rc)b with
exponent b;0.4, where r is the total void fraction. We found similar behavior for FCC Ni suggesting that this
critical behavior is a universal characteristic for failure of solids in rapid expansion.
DOI: 10.1103/PhysRevB.63.060103 PACS number~s!: 62.50.1p, 62.20.Fe, 62.20.Mk, 64.60.HtDespite much progress in characterizing and analyzing
the mechanical processes controlling the strength of materi-
als, there are still many uncertainties regarding the dynamic
failure of metals, particularly for the atomistic phenomena
underlying macroscopic failure. To study such process at the
atomistic level, we developed a first principles-based force
field ~FF! for Ta. We then applied this FF to examine dy-
namic failure using a high velocity plate colliding with a
static target. Upon impact compressive waves travel both
into the target and projectile. When these waves reach the
free surfaces they reflect, becoming tensile, and propagate
back into the material. When the two tensile waves meet, the
material is subjected to a high tension, and for sufficiently
high impact velocity the material will fail dynamically. This
process, known as spallation, leads to failure of the material
through formation, growth, and coalescence of microvoids or
cracks.
Spallation has been studied extensively both experimen-
tally and theoretically. The most widely used experimental
methods are plate impact1 and, more recently, high power
pulsed laser shock generators.2–6 Strain rates up to 108 sec21
can be achieved with laser shocks, higher than those in plate
impact experiments.
A complete description of spallation involves understand-
ing and linking processes taking place at very different time
and space scales, ranging from atomic size ~vacancy coales-
cence and initial void formation! to microns ~plastic defor-
mation and linkage of micron-sized voids!. Consequently
theoretical studies of spallation have used methods ranging
from microscopic molecular-dynamics ~MD! simulations,7 to
mesoscale micromechanical models and continuum
descriptions.1,8,9
We report here a MD study of the initial stages of spalla-
tion ~length scales of nm and time scales of ps! that provides
a detailed atomistic description for the time evolution of the
process. We focus on characterizing the dynamical evolution
of the void distribution.
To simulate high velocity impact, we assign a specific
relative velocity to previously thermalized ~perfect crystals at
300 K! projectile and target materials. We impose periodic
boundary conditions to our system in the two directions per-
pendicular to the impact velocity and free boundary in the0163-1829/2001/63~6!/060103~4!/$15.00 63 0601load direction, in this way we simulate two ‘‘infinite’’ plates.
We then describe the collisional process using adiabatic MD
simulations. We studied spallation in bcc Ta and fcc Ni. For
Ta the projectile contains Np58192 atoms per periodic
simulation cell @16316316 unit cells, ;(53.253 Å)3] and
the target Nt516 384 (32316316 unit cells!. For Ni we
considered Np58788 @13313313 unit cells, (;46 Å)3]
and Nt517 576 (26313313 unit cells!. We considered a
variety of impact velocities (v imp) ~relative velocity between
target and projectile! from 2 to 4 km/s ~the sound velocity in
Ta is ;3.4 km/s!. For Ta, we use an embedded atom model
FF fitted to accurate quantum mechanical calculations
~QM!.10 For Ni we used an empirical many-body QSC
~quantum Sutton-Chen! FF ~Ref. 11! used previously for
simulations of plasticity in Ni, and also a simple two-body
Morse FF.12
Figure 1 shows snapshots of the process at different times
for Ta with impact velocity of 2 km/s. The dots represent
atomic positions projected on a @100# plane. Figure 1~a!
shows the initial state of the system. Figure 1~b! shows the
system at 2 ps when the system is compressed; Fig. 1~c!
shows time 5.25 ps, at this time the system is expanding.
Figures 1~d! and 1~e! show times t56.5 ps and t57.25 ps,
where we can see the spall plane and void growth.
The aim of this work is to understand the dynamical evo-
lution of voids in spallation and its relation to such macro-
scopic quantities as stress and failure. We consider a void as
a connected empty region in the material which is separated
from other voids by atoms. We define ‘‘connected’’ and
‘‘separated by atoms’’ as follows: ~i! we map the space con-
taining the atoms onto a cubic grid, with the grid spacing
(Rg50.52 Å for Ta! about five times smaller than the dis-
tance between nearest-neighbor atoms (Rnn;2.88 Å for Ta!.
~ii! We ‘‘fill’’ the grid points which contain an atom, by
marking all the grid points closer to an atom than some
atomic radius Rat ~1.56 Å;0.54Rnn for Ta!. ~iii! we define
voids as connected clusters of empty sites: two empty sites
belong to the same cluster if it is possible to go from one to
the other through empty sites with ~simple cubic! nearest-
neighbors jumps. ~iv! After filling the sites corresponding to
the positions of all atoms, we also fill all the empty sites
which contain more than one filled nearest-neighbor site; this©2001 The American Physical Society03-1
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STRACHAN, C¸AG˘ IN, AND GODDARD PHYSICAL REVIEW B 63 060103~R!is done to ensure that no voids are found when the atoms are
in their perfect crystal positions.
We tested this procedure by analyzing equilibrium ~300
K, 432 atoms! MD runs for three different cases: i! perfect
crystal, ii! crystal with one vacancy, and iii! crystal with a
divacancy ~two vacancies in nearest-neighbor sites!. For Ta
with lattice parameter a53.328 Å ~atomic volume V
;18.43 Å3) we found: ~i! perfect case: no volumes larger
than ;0.6 Å3; ~ii! monovacancy case: we found one void of
;15 Å3; and ~iii! divacancy case: one void of ;27 Å3. Thus
we conclude that the void definition matches physical expec-
tations.
Figures 2~a! and 2~b! shows the void volume distribution
N(V) ~the number of voids as a function of volume! for Ta
with v imp52 km/s at different times during the initial stage
of the process. The void distributions shown were averaged
over bins with bin limits given by Vg2 i with Vg5Rg
3 and i
50,1,2 . . . . In the initial stage of void nucleation, the dis-
tribution decays rapidly with volume ~negative curvature!,
leading only to small voids @see t54 ps in Fig. 2~a!#. This
distribution broadens as voids coalesce and deform plasti-
cally, see times 4.25 and 4.5 ps in Fig. 2~a!. At time tc
55.25 ps @filled squares in Fig. 2~a!# the void distribution
has a power law form for a wide volume range:
N~V !5N0V2t, ~1!
with an exponent t52.260.05 @shown by the thick line in
Fig. 2~a!#. The deviations from the power law for small and
large volumes will be explained below. In Fig. 2~b! we show
FIG. 1. Ta with v imp52. Snapshots of the spall process at dif-
ferent times. The dots represent atomic positions projected in a
@100# plane. ~a! Initial configuration; ~b! t52 ps when the system is
compressed after the collision; ~c! t55.25 ps expansion stage; ~d!
and ~e! t56.5, and t57.5 ps, respectively, the spall plane and void
growth can be seen.06010the void volume distribution for the late stage of the process,
times 5.25, 5.75, 6.5, and 7.25 ps. We can see for times
longer than 5.25 ps, a large void appears, leading to positive
curvature in the void distribution for large voids, but nega-
tive curvature for small and intermediate volume voids
~similar to the initial stages!. It is clear from Figs. 2~a! and
2~b! that the void distribution has a special character at time
tc;5.25 ps.
Figure 3~a! shows the time evolution of the volume of the
largest void, Vmax , and Fig. 3~b! shows the stress in the load
direction as a function of time for Ta with v imp52 km/s.
During the initial stages, for which the target and projectile
FIG. 2. Ta with v imp52. ~a! Void volume distribution for the
initial stages of the process. t54 ps ~empty circles!, t54.25 ps
~filled circles!, t54.5 ps ~empty squares! and tc55.25 ps ~filled
squares!. We see that at t5tc the volume distribution follows a
power law ~the thick line shows a power law with exponent t
52.2). ~b! Void volume distribution for the late stages of the pro-
cess. tc55.25 ps ~filled squares!, t55.75 ps ~empty circles!, t
56.5 ps ~filled circles!, and t57.5 ps ~empty squares!. We see the
growth of the largest void at the expense of the intermediate and
low volume ones.3-2
RAPID COMMUNICATIONS
CRITICAL BEHAVIOR IN SPALLATION FAILURE OF METALS PHYSICAL REVIEW B 63 060103~R!are under compression, the stress attains large negative val-
ues (;250 GPa!. During this stage the volume of the voids
is almost zero. As the system expands the compressive stress
relaxes and becomes tensile ~positive stress or negative pres-
sure!, until the metal starts to fail ~at t f;5.25 ps! and the
stress drops from its maximum of ;12.6 GPa. Figure 3~a!
shows that the behavior of the maximum void volume exhib-
its two different regimes; it remains very small during the
initial stages of the process until at a critical time tc its slope
increases abruptly as the largest void starts growing very
fast. Figures 3~a! and 3~b! show the clear relation between
the presence of a large void and the failure of the material.
We define ~i! the strength of the largest void as f
5Vmax /Vtot , where Vmax is the volume of the largest void
and Vtot is the total volume contained in voids; and ~ii! the
total void concentration as r5Vtot /V , where V is the total
volume of the system.
The inset of Fig. 3~a! shows f as a function of r . It is
clear that there is a critical void concentration rc at which
the strength of the largest void starts to grow steeply. For r
→rc1 ~from above!, f follows:
f}~r2rc!
b
, ~2!
with b50.460.04 ~the dash-dot line in the inset of Fig. 3~a!
shows b50.4).
FIG. 3. Ta with v imp52. ~a! Time evolution of the volume of
the largest void and; inset: strength of the largest cluster as a func-
tion of total void concentration. ~b! Stress in the load direction as a
function of time. The maximum stress in compression ~not shown!
is ;246 GPa. The failure time t f;5.3 ps is very similar to the
time tc55.25 ps at which the volume of the largest void starts
growing.06010For Ta with v imp53 km/s, the void volume distribution
shows the same behavior; it follows a power law at tc
54.125 ps with the same exponent t;2.260.05. This time
correspond to the point of largest void growth, and is very
close to the failure point t f;4.2 with a maximum stress of
9.45 GPa. The strength of the largest void, f also follows
Eq. ~2! with b;0.460.04.
We find similar void distribution behavior for Ni de-
scribed with either the QSC FF or the Morse FF. For Morse
Ni with v imp52 km/s we find power law behavior of the
void volume distribution for t;3.1 ps, the associated critical
exponent is t522.260.8. In this case the exponent associ-
ated with f @Eq. ~2!# is b50.460.08. We found the same
behavior for Morse Ni with v imp53 km/s and v imp54 km/s
and for QSC FF for Ni at different impact velocities. In all
cases we found t;2.2 and b;0.4. Thus, the general behav-
ior for Ni ~even described by a simple two-body force field!
is the same as for Ta.
The power law behavior of the void distribution at the
critical point for failure is very similar to the distribution of
clusters in the percolation model at the critical probability13
and the distribution of liquid drops in a liquid-vapor phase
transition at the critical point.14 The nonequilibrium process
of fragmentation also shows power-law mass distribution
and signals of critical behavior, see Refs. 15–17 and refer-
ences therein. The mass distributions deviate from the power
law for small and large masses, due to failure of scaling for
small masses and finite-size effects for large masses.
The exponent t is one of the critical exponents which
determine the behavior of the system near the second-order
phase transition. For three dimensional percolation t52.2,13
and for the three-dimensional liquid-gas phase transition the
exponent is t;2.33.14
An essential characteristic to describe second-order phase
transitions is an ‘‘order parameter’’ which is zero on one side
of the phase transition ~usually the high-temperature phase!,
and grows very rapidly ~but continuously! to a finite value on
the other side. The order parameter measures the difference
between the system at opposite sides of the phase transition.
In the case of the ferromagnetic-paramagnetic phase transi-
tion the order parameter is the spontaneous magnetization,
@zero for T.Tc ~in the absence of external magnetic field!,
and nonzero at lower T, denoting spontaneous magnetiza-
tion#. For liquid-gas phase transition the order parameter is
the density difference between the coexisting liquid and gas.
In these cases the order parameter follows Eq. ~2! with ex-
ponent b;1/3.
Our simulations show that the order parameter for spalla-
tion should be defined as the strength of the largest void, f .
The order parameter in percolation is defined in a similar
way and its associated critical exponent is also b50.4.13
From Fig. 3 we see that a nonzero order parameter indicates
the failure of the material. A phenomenological microme-
chanical model developed to describe the void-link up ef-
fects in ductile fracture8 and analysis of recent experiments
also suggest similar percolationlike behavior at high strain
rates.1
Summarizing, using MD simulations of high velocity im-
pact, we find that spallation failure of the metals at high3-3
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STRACHAN, C¸AG˘ IN, AND GODDARD PHYSICAL REVIEW B 63 060103~R!strain rates corresponds to a critical point in which the dis-
tribution of voids follows a power law behavior with expo-
nent t;2.2. Our results suggest that the order parameter is
defined by the ratio of the volume of the largest void to the
total volume contained in voids. This order parameter fol-
lows the expected power law behavior near the critical con-
centration of voids with exponent b;0.4 similar to the val-
ues found in percolation and the liquid-gas phase transition.
We obtained similar results for Ta ~represented by a many
body force field! and Ni using both a simple two-body and a
many body potential. The implication is that spallation fail-
ure of metals exhibits critical behavior. This is relevant for060103spall failure at high strain rates that can be obtained via laser
shocks; in this regime pre-existing defects in the material
play a minor role in spall failure.5 The similarity of critical
exponents for different metals and force fields suggests the
universality of this phenomenon of failure under tension at
high strain rates, which may also apply to cavitation.18
This research was funded by Grant No. DOE W-7405-
ENG-48 from DOE-ASCI-ASAP. The facilities of the MSC
are also supported by grants from NSF ~MRI CHE 99!, ARO
~MURI!, ARO ~DURIP!, NASA, BP Amoco, Exxon, Dow
Chemical, Seiko Epson, Avery Dennison, Chevron Corp.,
Asahi Chemical, 3M, and Beckman Institute.*Author to whom correspondence should be addressed.
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Critical behavior in spallation failure of metals

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Critical behavior in spallation failure of metals

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