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Abraham2002-Fracture.pdf
Simulating materials failure by using up to one billion
atoms and the world’s fastest computer:
Brittle fracture
Farid F. Abraham*†, Robert Walkup*‡, Huajian Gao*§, Mark Duchaineau¶, Tomas Diaz De La Rubia¶, and Mark Seager¶
*IBM Research Division, Almaden Research Center, San Jose, CA 95120; ‡T. J. Watson Research Laboratory, Yorktown Heights, NY 10598; §Max-Planck
Institut für Metallforschung, Stuttgart, D-70569 Germany; and ¶Lawrence Livermore National Laboratory, Livermore, CA 94550
Communicated by L. B. Freund, Brown University, Providence, RI, January 8, 2002 (received for review September 3, 2001)
physical systems and not governed by the particular complexities
of a unique molecular interaction. It is very important to
emphasize that this is a conscious choice because it is not
uncommon to hear others object that one is not studying ‘‘real’’
materials when using simple potentials. Feynman (3) summarized this viewpoint well on page 2, volume I of his famous
three-volume series Feynman’s Lectures In Physics:
Molecular Dynamics (MD) Experiments on ASCI (Accelerated
Strategic Computing Initiative) White Computer
D
uring September of the new millennium, IBM delivered the
world’s fastest computer to the Lawrence Livermore National Laboratory. Scientific American calls it ‘‘the computer of
tomorrow’’ (1). For the first 4 months, the ASCI White computer
was tuned, and scientists from a variety of fields were privileged
to play with it. We were some of the lucky players. An important
goal of the Department of Energy materials science program is
to gain better understandings of materials failure under extreme
conditions. We accomplished two atomistic simulation projects.
The first was a multimillion-atom simulation study of crack
propagation in rapid brittle fracture, and the second investigated
ductile failure by using over one billion atoms. What we accomplished in a frantic few months of intense effort will be described.
We present a brief discussion about simulating the dynamics of
many atoms by using big computers.
With the present-day supercomputers, simulation is becoming a very powerful tool for providing important insights into
the nature of materials failure. Atomistic simulations yield ab
initio information about materials deformation at length and
time scales unattainable by experimental measurement and
unpredictable by continuum elasticity theory. Using our ‘‘computational microscope,’’ we can see what is happening at the
atomic scale. Our simulation tool is computational MD (2),
and it is very easy to describe. MD predicts the motion of a
large number of atoms governed by their mutual interatomic
interaction, and it requires the numerical integration of the
equations of motion, force equals mass times acceleration or
F ⫽ ma. We learn in beginning physics that the dynamics of two
atoms can be solved exactly. Beyond two atoms, this is
impossible except for a few very special cases, and we must
resort to numerical methods. A simulation study is defined by
a model created to incorporate the important features of the
physical system of interest. These features may be external
forces, initial conditions, boundary conditions, and the choice
of the interatomic force law.
In the present simulations, we adopt simple interatomic force
laws because we want to investigate the generic features of a
particular many-body problem common to a large class of real
www.pnas.org兾cgi兾doi兾10.1073兾pnas.062012699
A simple interatomic potential may be thought of as a ‘‘model
potential,’’ and the model potentials for the present studies are
harmonic and anharmonic springs and the Lennard-Jones (LJ)
12:6 potential. Complaints of model approximations are not new.
In his book entitled The New Science of Strong Materials or Why
Don’t You Fall Through the Floor, Gordon (4) comments on
Griffith’s desire to have a simpler experimental material that
would have an uncomplicated brittle fracture. He writes, ‘‘In
those days, models were all very well in the wind tunnel for
aerodynamic experiments but, damn it, who ever heard of a
model material?’’
In the mid-1960s, a few hundred atoms could be treated. In
1984, we reached 100,000 atoms. Before that time, computational scientists were concerned that the speed of scientific
computers could not go much beyond 4 Gigaf lops, or 4 billion
arithmetic operations per second and that this plateau would
be reached by the year 2000! That became forgotten history
with the introduction of concurrent computing. A modern
parallel computer is made up of several (tens, hundreds, or
thousands) small computers working simultaneously on different portions of the same problem and sharing information
by communicating with one another. The communication is
done through message passing procedures. The present record
is well over a few Teraf lops for optimized performance, and we
have now simulated over 1,000,000,000 atoms in a workAbbreviations: ASCI, Accelerated Strategic Computing Initiative; MD, molecular dynamics;
FCC, face-center-cubic; LJ, Lennard-Jones.
†To
whom reprint requests should be addressed.
The publication costs of this article were defrayed in part by page charge payment. This
article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C.
§1734 solely to indicate this fact.
PNAS 兩 April 30, 2002 兩 vol. 99 兩 no. 9 兩 5777–5782
SCIENCES
‘‘If in some cataclysm all scientific knowledge were to be
destroyed and only one sentence passed on to the next
generation of creatures, what statement would contain
the most information in the fewest words? I believe it is
the atomic hypothesis that all things are made of
atoms—little particles that move around in perpetual
motion, attracting each other when they are a little
distance apart, but repelling upon being squeezed into
one another. In that one sentence, you will see there is
an enormous amount of information about the world, if
just a little imagination and thinking are applied.’’
APPLIED PHYSICAL
We describe the first of two large-scale atomic simulation projects
on materials failure performed on the 12-teraflop ASCI (Accelerated Strategic Computing Initiative) White computer at Lawrence
Livermore National Laboratory. This is a multimillion-atom simulation study of crack propagation in rapid brittle fracture where the
cracks travel faster than the speed of sound. Our finding centers on
a bilayer solid that behaves under large strain like an interface
crack between a soft (linear) material and a stiff (nonlinear)
material. We verify that the crack behavior is dominated by the
local (nonlinear) wave speeds, which can be in excess of the
conventional sound speeds of a solid.
hardening study at the Lawrence Livermore National Laboratory by using the ASCI White 12-teraf lop computer.
Moore’s Law states that computer speed doubles every 11⁄2
years. For 35 years, that translates into a computer speed
increase of 10 million. This is exactly the increase in the
number of atoms that we could simulate over the last 35
years. For a visual description of ASCI White, we refer the
reader to Scientific American’s special issue entitled Extreme
Engineering (1).
First Study: Supersonic Crack Propagation In Brittle Fracture
How Fast Can Cracks Move? The first of our two simulation studies
addressed the important question ‘‘how fast can cracks propagate?’’
In this study, we used system sizes of about 20 million atoms, very
large by present-day standards but modest compared with our
second study on ductile failure (23). Based on our current simulation model, we develop our earlier studies on transonic crack
propagation in linear materials and supersonic crack propagation in
nonlinear solids. Our finding centers on a bilayer solid that behaves
under large strain like an interface crack between a soft (linear)
material and a stiff (nonlinear) material. In this mixed case, we
observe that the initial mother crack propagating at the Rayleigh
sound speed gives birth to a transonic daughter crack. Then, quite
unexpectedly, we observe the birth of a supersonic granddaughter
crack. We verify that the crack behavior is dominated by the local
(nonlinear) wave speeds, which can be in excess of the conventional
sound speeds of a solid.
In this problem, there are three important wave speeds in the
solid. In order of increasing magnitude, they are the Rayleigh
wave speed, or the speed of sound on a solid surface, the shear
(transverse) wave speed, and the longitudinal wave speed.
Predictions of continuum mechanics (5, 6) suggest that a brittle
crack cannot propagate faster than the Rayleigh wave speed.
For a mode I (tensile) crack, the energy release rate vanishes
for all crack velocities in excess of the Rayleigh wave speed,
implying that the crack cannot propagate at a velocity greater
than the Rayleigh wave speed. A mode II (shear) crack behaves
similarly to a mode I crack in the subsonic velocity range; i.e.,
the energy release rate monotonically decreases to zero at the
Rayleigh wave speed and remains zero between the Rayleigh
and shear wave speeds. However, the predictions for the two
loading modes differ for crack velocities greater than the shear
wave speed. Whereas the energy release rate remains zero for
a mode I crack, it is positive for a mode II crack over the entire
range of intersonic velocities. From these theoretical solutions,
it has been concluded that a mode I crack’s limiting speed is
clearly the Rayleigh speed. The same conclusion also has been
suggested for a mode II crack’s limiting speed because the
‘‘forbidden velocity zone’’ between the Rayleigh and shear
wave speeds acts as an impenetrable barrier for the shear crack
to go beyond the Rayleigh wave speed.
The first direct experimental observation of cracks moving
faster than the shear wave speed has been reported by Rosakis
et al. (7). They investigated shear dominated crack growth
along weak planes in a brittle polyester resin under dynamic
loading. Around the same time, we per formed twodimensional MD simulations of crack propagation along a
weak interface joining two strong crystals (8). We assumed
that the interatomic forces are harmonic except for those pairs
of atoms with a separation cutting the centerline of the
simulation slab. For these pairs, the interatomic potential is
taken to be a simple model potential that allows the atomic
bonds to break. Our simulations demonstrated intersonic
crack propagation and the existence of a mother-daughter
crack mechanism for a subsonic shear crack to jump over the
forbidden velocity zone. We have since discovered that a crack
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Fig. 1. Variation of elastic modulus as a function of strain under uniaxial
stretching in the (110) direction of a FCC crystal formed by the anharmonic
(tethered repulsive LJ) potential.
cannot only travel supersonically (9) but a mother-daughtergranddaughter crack mechanism exists in bilayer slabs.
The classical theories of fracture (5, 6) are largely based on linear
elastic solutions to the stress fields near cracks. An implicit assumption in such theories is that the dynamic behavior of cracks is
determined by the linear elastic properties of a material. We have
found (10) that the MD simulation results for harmonic atomic
forces are indeed well interpreted by elasticity theories. However,
the effects of anharmonic material properties on dynamic behaviors
of cracks are not clearly understood. This is partly because of the
general difficulties in obtaining nonlinear elastic solutions to dynamic crack problems. MD simulations can be easily adapted to the
anharmonic case so that nonlinear effects can be thoroughly
investigated. We now discuss the anharmonic simulations.
The Computer Experiments Setup. We consider a strongly nonlinear
elastic solid described by a tethered LJ potential where the compressive part of this potential is identical to that of the usual LJ 12:6
function and the tensile part is the reflection of the compressive part
with respect to the potential minimum. A face-center-cubic (FCC)
crystal formed by this potential exhibits a strongly nonlinear
stress-strain behavior resulting in elastic stiffening and an increase
of the elastic modulus with strain, as shown in Fig. 1. Note that the
elastic modulus increases by a factor of 10 at 13% of elastic strain,
indicating that the material properties of such a solid is strongly
nonlinear in the hyperelastic regime.
We performed three-dimensional MD simulations of two
FCC crystals joined by a weak interface. In this study, we used
system sizes of about 20 million atoms, although a billion atoms
were used in a preliminary simulation (9). For comparison, we
consider the anharmonic tethered potential together with the
harmonic potential where the spring constant is equal to the
tethered potential at equilibrium. The interatomic force bonding the two crystals is given by the LJ 12:6 potential. The
simulation results are expressed in reduced units: lengths are
scaled by the value of the interatomic separation where the LJ
potential is zero and energies are scaled by the depth of the
minimum of the LJ potential. Atoms bond only with their
original nearest neighbors. Hence, rebonding of displaced
atoms caused by applied loading does not occur. For the
adjoining two crystal slabs, we consider the following three
simulation cases: (i) harmonic case— both crystal slabs are
Abraham et al.
Table 1. Calculated wave speeds for the harmonic wave speeds,
bulk anharmonic wave speeds caused by applied loading, and
local wave speeds
Bulk anharmonic
Local
9.49
4.24
10.36
5.95
13.4
10.4
Fig. 2. The space-time history of the crack tip for the three different
simulations described in the text (reduced units are used).
different from the harmonic properties. In an earlier attempt
to explain why the highest crack velocities recorded for mode
I crack propagation in a homogeneous body are significantly
lower than the Rayleigh wave speed, Broberg (5, 6, 11) has
suggested that the reason could be because of some kind of a
‘‘local’’ Rayleigh wave speed in the highly strained region near
the crack tip, rather than the Rayleigh wave speed in the
undisturbed material. Because the local strain is an extremely
strong function of position from the crack tip, one might think
that the local wave speed should depend on distance from the
crack tip and cannot have one single value. It was not until 30
years after Broberg’s suggestion that this issue was finally
quantitatively studied by Gao (12, 13) who made use of the
Barenblatt cohesive model of a mode I crack and, for the first
time, defined the local wave speed unambiguously as the wave
speed at the location where stress is exactly equal to the
cohesive strength of the material, i.e., the true point of fracture
nucleation. The local wave speed characterizes how fast elastic
APPLIED PHYSICAL
characterized by the harmonic potential; this is used as a
control for the anharmonic studies; (ii) anharmonic case—
both crystal slabs are characterized by the anharmonic potential; and (iii) mixed case— one crystal slab is characterized by
the anharmonic potential, and the other is characterized by the
harmonic potential. In each case, a shear crack lies along the
(110) plane and oriented toward the [110] direction. The crack
front is parallel to the (001) direction. The applied loading is
dominated by shear.
To interpret the results of MD simulation, we need the
following wave speeds in the FCC crystals formed by harmonic
and兾or anharmonic potentials: the conventional longitudinal
and shear wave speeds in the harmonic and anharmonic
crystals in the direction of crack propagation; the longitudinal
and shear wave speeds under applied loading in the anharmonic crystal in the direction of crack propagation; and the
local longitudinal and shear wave speeds near the crack tip in
the anharmonic crystal in the direction of crack propagation.
The calculations of these wave speeds will be presented in
detail in a forthcoming paper. We give the results in Table 1.
We add some comments with regard to the concept of local
wave speeds near the crack tip, which play a critically important role in explaining our simulation results. Conceptually, it
is clear that the fracture process in brittle solids involves
breaking of atomic bonds and is intrinsically a highly nonlinear
process. The anharmonic material properties of solids near the
cohesive strength of atomic bonds would in general be quite
Fig. 3. Snapshot pictures of a crack traveling in the harmonic slab, where the progression in time is from the top to the bottom. The boxed snapshot
pictures represent a progression in time from the top to the bottom.
Abraham et al.
PNAS 兩 April 30, 2002 兩 vol. 99 兩 no. 9 兩 5779
SCIENCES
Longitudinal wave
Shear wave
Bulk harmonic
Fig. 4. Snapshot pictures of a crack traveling in the anharmonic slab, where the progression in time is from the top to the bottom. The boxed snapshot
pictures represent a progression in time from the top to the bottom.
energy is transported near the region of bond breaking in front
of a crack tip. For example, for a mode I crack in a homogeneous isotropic elastic, the local wave speed is calculated to be
公␴max兾␳ (12, 13), where ␴max is the cohesive strength and ␳ is
the density of the undisturbed materials. The cohesive strength
is typically around 1兾10 of the shear modulus, suggesting that
the local wave speed for a mode I crack is ⬇1兾3 of the shear
wave speed. Interestingly, experiments (14, 15) and MD
simulations (16, 17) show that mode I cracks exhibit a dynamic
instability at 30% of the shear wave speed, which suggests a
possible dependence on the local wave speed (12, 13). We note
that the local wave speeds differ from the conventional wave
speeds both qualitatively and quantitatively.
In the present study, the focus is shifted to study the effect of
stiffening anharmonic behaviors of materials (as in many polymers)
on a mode II crack propagating along a weak interface described by
the LJ potential. The solid itself is described by the tethered LJ
potential. We follow the same approach used in refs. 12 and 13 and
define the local wave speed as the wave speed of the solid at the
location adjacent to the interface where the shear stress is exactly
equal to the cohesive strength of the interface. Note that in this case
the local wave speed depends on both the interface cohesive
strength and the nonlinear elastic properties of the solid.
The Computer Experiments Results and Discussion. Fig. 2 presents
distance-time histories of a crack moving in the three different
slab configurations: two weakly bonded harmonic crystals
(designated harmonic), two weakly bonded anharmonic crystals (designated anharmonic), and a harmonic crystal weakly
bonded to an anharmonic crystal (designated mixed). The
cracks begin their motion when the Griffith criterion is
satisfied. The respective dip-spike regions for each history
represent the birth of a new crack. For the harmonic and
anharmonic simulations (see Figs. 3 and 4), we observe one
such region representing the birth of a daughter crack, the
former traveling at the longitudinal sound speed for the
harmonic solid and the latter achieving a supersonic sound
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speed of Mach 1.6. For the mixed simulation (see Fig. 5), we
see two such dip-spike regions where a daughter crack is born
from the mother crack, which then gives birth to a granddaughter crack at a later time.
In Figs. 3 and 4, the harmonic and anharmonic simulations
are shown, respectively. The boxed snapshot pictures in each
figure represent a progression in time from the top to bottom.
In the top images, the mode II daughter cracks are born. The
early-time occurrence of the Mach cone attached to the crack
tip is evident in the anharmonic slab; in the middle image of
Fig. 3, the crack in the harmonic slab has a single Mach cone
and a circular stress-wave halo, which indicates a crack speed
equal to the longitudinal sound speed of the linear solid. This
is in contrast to the middle image of Fig. 4 where the two Mach
cones for the crack in the anharmonic slab signify that the
crack is traveling supersonically. For the bottom images of the
two figures, we conclude that the crack in the anharmonic slab
wins.
In Fig. 5, snapshot pictures of the crack traveling in the
mixed slab are presented, where the progression in time is from
the top to bottom. The material properties for the harmonic
and anharmonic regions are labeled, and the different sound
waves associated with the crack’s dynamics are denoted. The
sequence shows the following progression of events: (i) the
daughter crack is born; (ii) the daughter crack is traveling at
the longitudinal sound speed of the harmonic slab; (iii) the
granddaughter crack is born; and (iv) the granddaughter crack
speeds ahead to Mach 1.6, matching the crack speed in the
anharmonic slab (movies of the brittle fraction simulations
may be viewed and downloaded at http:兾兾www.almaden.ibm.com兾st兾Simulate).
We have shown that the behavior of the crack in the
harmonic crystal is controlled by the conventional elastic wave
speeds (6). In contrast, the crack behavior in the anharmonic
crystal is controlled by the local wave speeds, which play an
important role in the dynamic behavior of crack propagation.
A similar dependence has been identified in a related phenomenon where hyperelasticity plays an important role in
Abraham et al.
Abraham et al.
velocity barrier. The limiting speed of the daughter crack is
more than 50% higher than the longitudinal wave speed and
cannot be explained by the linear theory of intersonic fracture.
In comparison, the calculated local wave speed is approximately equal to (only 10% lower than) the observed limiting
speed. In calculating the local wave speeds, we have ignored
the large gradient of deformation field near the crack tip. In
view of this simplification, we conclude that the local wave
speed provides a reasonable explanation of the observed
limited crack speed.
In the mixed case, the nucleation of the daughter crack still
occurs at the Rayleigh wave speed for reasons discussed above.
The daughter crack breaks the velocity barrier at the Rayleigh
speed and propagates near the longitudinal wave speed. This
behavior is similar to the harmonic case. However, at a velocity
of 10.35, we observe a granddaughter crack forming ahead of
the daughter crack. The critical speed at which this transition
occurs is very close to the local shear wave speed in the
anharmonic crystal. The granddaughter crack rapidly accelerates toward the local longitudinal wave speed of the
stretched nonlinear solid. It is tempting to conclude that
the nucleation of the granddaughter crack is controlled by the
local Rayleigh wave speed magnified by the local stress
c oncentration, although this issue is worth further
investigation.
In summary, we conclude that the local wave speeds play a
dominant role in the behavior of cracks in the anharmonic crystals.
Table 2. Observed speeds at which a daughter crack or a
granddaughter crack nucleates and the limiting speeds for the
harmonic, anharmonic, and mixed simulation cases
Daughter crack
Granddaughter
Limiting speed
Harmonic
Anharmonic
Mixed
3.5
—
9.4
4.1
—
15
4.1
10.35
15
PNAS 兩 April 30, 2002 兩 vol. 99 兩 no. 9 兩 5781
SCIENCES
whether a solid will undergo brittle fracture or ductile failure
at the crack tip (18, 19). The local wave speed represents the
nonlinear, hyperelastic, material properties near cohesive failure of atomic bonds while the conventional elastic (shear and
longitudinal) wave speeds represent the material properties
under infinitesimal deformation and can differ significantly
from the harmonic wave speeds. The crack propagation speeds
observed in MD simulations are tabulated in Table 2 for
comparison.
Comparing the MD simulation results with the calculated
wave speeds for harmonic and anharmonic crystals, we reach
the following conclusions.
The harmonic case is consistent with the linear elastic theory
of intersonic crack propagation. We have previously discussed
this case in detail for two-dimensional harmonic MD simulation (8). The same theory is found to apply for the present
three-dimensional case. Essentially, the initial crack starts to
propagate when the Griffith criterion is satisfied. Near the
Rayleigh wave speed, the crack encounters a velocity barrier
and a vanishing of stress singularity at the crack tip; i.e., both
stress intensity factor and energy release rate vanish at the
Rayleigh wave speed. This velocity barrier is overcome by the
nucleation of a daughter crack at a distance ahead of the
mother crack. This distance corresponds to the shear wave
front at which a peak of shear stress increases to a critical
magnitude to cause cohesive failure of the interface. The
daughter crack’s speed is only limited by the longitudinal wave
speed. Comparing Tables 1 and 2, we see that the daughter
crack indeed nucleates at the Rayleigh wave speed and the
limiting speed agrees very well with the longitudinal wave
speed for the harmonic crystal. The mother-daughter mechanism described above is consistent with the Burridge-Andrew
model of intersonic crack propagation (20, 21).
In the anharmonic case, the nucleation of the daughter crack
is consistent with linear elastic theory. The mother crack
initiates according to the Griffith criterion and achieves a
limiting velocity equal to the Rayleigh wave speed. At this
point, it is necessary to nucleate a daughter crack to break the
APPLIED PHYSICAL
Fig. 5. Snapshot pictures of a crack traveling in the mixed slab, where the progression in time is from the top to the bottom. The sound waves associated
with the crack’s dynamics and the material properties of the harmonic and anharmonic regions of the mixed slab are labeled.
interpreted by elasticity theories. In particular, calculations
based on linear elasticity were able to predict the time and
location of the daughter crack as well as the initiation time of
the mother crack. Our atomistic simulation of the motherdaughter crack mechanism for intersonic shear crack is consistent with the continuum mechanics-based discovery made
earlier by Burridge (20) and Andrews (21). An important
message of the present study is that the nonlinear continuum
theory of local wave speeds is capable of predicting crack
velocities in strongly nonlinear solids. Indeed, it is quite
remarkable that the dynamic behavior of cracks may retain its
basic nature over such a wide range of length scales, from
atomistic calculations by using interatomic potentials all of the
way up to macroscopic laboratory experiments and continuum
elasticity treatments. This bridging of length scales in dynamic
materials failure should be of great interest to the general
scientific audience because it points out the fantastic power of
continuum mechanics.
The mixed case behaves somewhat like an interface crack between
a soft material and a stiff material. Although the harmonic and
anharmonic crystals have identical material properties under infinitesimal deformation, the local material properties near the crack
tip are resembled by a bimaterial. Rosakis et al. (22) have previously
studied crack propagation along an interface between polymethyl兾
methacrylate (PMMA) (soft) and Al (hard) and found that the
crack speed can significantly exceed the longitudinal wave speed of
PMMA. Our present study shows that the crack behavior is
dominated by the local (nonlinear) wave speeds. This is not only of
theoretical interest, but also of practical importance. It is known
that many polymeric materials, especially rubbers, increase their
modulus significantly when stretched. The underlying physical
mechanism is that initial elasticity in rubbers is caused by entropic
effects. When stretched to large deformation, the polymeric chains
are straightened and covalent atomic bonds eventually dominate
their hyperelastic response. In such solids, the elastic modulus
increases with strain and the local wave speeds near a crack tip
would be larger than the linear elastic wave speeds. The dynamic
behavior of cracks in such solids should propagate at a speed
exceeding the conventional wave speeds.
Another point worth commenting on here is the remarkable
success of continuum theories in predicting the behaviors
obtained by atomistic simulations. We have found previously
(10) that the MD simulation results for shear crack propagation along a weak interface in harmonic solids are well
F.F.A. acknowledges the essential contributions of Brian Wirth and the
Lawrence Livermore National Laboratory ASCI White team under the
guidance of Steve Louis and Terry Heidelberg. F.F.A. is indebted to
Charles Rettner at the IBM Almaden Research Center for creating the
web pages related to this paper. The work of H.G. is supported by
National Science Foundation Grant CMS-9820988 and the Max Planck
Society.
1. Gibbs, W. (1999) Sci. Am. Extreme Engineering 10, 56 – 61.
2. Allen, M. P. & Tildesley, D. J. (1987) Computer Simulation of Liquids
(Clarendon, Oxford).
3. Feynman, R., Leighton, R. & Sands, M. (1963) The Feynman Lectures on
Physics (Addison–Wesley, Redwood City, CA).
4. Gordon, J. E. (1988) The New Science Of Strong Materials or Why You Don’t
Fall Though the Floor (Princeton Univ. Press, Princeton).
5. Freund, L. B. (1990) Dynamical Fracture Mechanics (Cambridge Univ. Press,
New York).
6. Broberg, B. (1999) Cracks and Fracture (Academic, San Diego).
7. Rosakis, A. J., Samudrala, O. & Coker, D. (1999) Science 284, 1337–1340.
8. Abraham, F. F. & Gao, H. (2000) Phys. Rev. Lett. 84, 3113–3116.
9. Abraham, F. F. (2001) J. Mech. Phys. Solids 49, 2095–2111.
10. Gao, H., Huang, Y. & Abraham, F. F. (2001) J. Mech. Phys. Solids 49, 2113–2132.
11. Broberg, B. J. (1964) J. Appl. Mech. 31, 546 –547.
12. Gao, H. (1996) J. Mech. Phys. Solids 4, 1453–1474.
13. Gao, H. (1997) Philos. Mag. Lett. 76, 307–314.
14. Fineberg, J., Gross, S. P., Marder, M. & Swinney, H. L. (1991) Phys. Rev. Lett.
67, 457– 460.
15. Fineberg, J., Gross, S. P., Marder, M. & Swinney, H. L. (1992) Phys. Rev. B
45, 5146 –5154.
16. Abraham, F. F., Brodbeck, D., Rafey, R. & Rudge, W. E. (1994) Phys. Rev.
Lett. 73, 272–275.
17. Abraham, F. F., Brodbeck, D., Rafey, R. & Rudge, W. E. (1997) J. Mech.
Phys. Solids 5, 1595–1605.
18. Abraham, F. F. (1996) Phys. Rev. Lett. 7, 869 – 872.
19. Abraham, F. F., Schneider, D., Land, B., Lifka, D., Skovira, J., Gerner, J.
& Rosenkrantz, M. (1997) J. Mech. Phys. Solids, 45, 1461–1471.
20. Burridge, R. (1973) Geophys. J. R. Astron. Soc. 35, 439 – 455.
21. Andrews, D. J. (1976) J. Geophys. Res. 81, 5679 –5687.
22. Rosakis, A. J., Samudrala, O., Singh, R. P. & Shukla, A. (1998) J. Mech. Phys.
Solids 6, 1789 –1813.
23. Abraham, F. F., Walkup, R., Gao, H., Duchaineau, M., Diaz De La Rubia,
T. & Seager, M. (2002) Proc. Natl. Acad. Sci. USA 99, 5783–5787.
5782 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.062012699
Abraham et al.
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