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CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
2002 American Institute of Physics 0-7354-0068-7
CRYSTAL FAILURE AND CRACK FORMATION DURING PLASTIC
FLOW
C. S. Coffey1 and J. Sharma2
1 Indian Head Division, Naval Surface Warfare Center, Indian Head, MD 20640-5035
2
Carderock Division, Naval Surface Warfare Center, Bethesda, MD 20817-5700
Abstract. Atomic Force Microscopy (AFM) of molecular crystals subjected to impact or shock has shown
increasing lattice and molecular damage with increasing levels of plastic flow. While AFM measures only
surface distortions, evidence exists showing that the molecular and lattice distortions persist deep within the
crystal interior. This evidence of lattice damage has been used to provide a physical model of lattice softening
and eventual failure during large plastic deformation due to impact or mild shock. For a simple geometry, the
onset of crystal failure and crack trajectory during impact is determined and compared with experiment.
INTRODUCTION
pressures in excess of 10 GPa. The distortions persist
throughout much of the bulk of the crystals.
This is a continuation of an effort to understand at
the fundamental level the plastic deformation and
energy dissipation that occur in crystalline solids as
they undergo plastic flow due to shock or impact.
Retaining the conventional notion that plastic
deformation is due to the creation and motion of
dislocations, the previous efforts have dealt with the
energy dissipated in the crystal by moving
dislocations, the description of this motion in terms
of quantum tunneling and, most recently, describing
the damage and distortion that occur in the crystal
lattice during plastic deformation.1"6 Here, it is shown
that when the deformed and damaged lattice potential
is taken into account it is possible to describe the
softening and eventual failure and cracking of the
crystal that occur at high levels of plastic flow.
DISTORTED LATTICE POTENTIAL
It has been suggested that the damage and distortion
that appear in impacted or shocked crystals is due to
the dislocations created during plastic flow.5 Since
the length of the crystal along the slip plane does not
change during shear induced plastic flow, the
dislocations created on the active slip planes must
compress and distort the lattice and molecules lying
on these slip planes. The averaged potential of the
distorted lattice of a crystal that has undergone plastic
deformation can be approximated as5'6
dx
Lo
dx"
Lo
(1)
Atomic Force Microscopy studies of plastically
deformed molecular crystals have shown that
permanent lattice and molecular distortions occur at
almost all levels of deformation.7"10 These
observations extend over the range of deformation
produced by the lowest force level of a microindenter to the deformation produced by shock
where U0 is the potential of the undistorted lattice and
represents the lattice state of lowest energy. The
second term comes about when the crystal is
subjected to moderate damage. The quantity dtydx
is a measure of the lattice damage due to the
deformation, N is the number of dislocations on an
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tracked. To illustrate this consider crystal failure and
crack development due to an impact or mild shock
load applied to an aggregate of crystals that contains
a simple cavity about which the crack forms. Let the
aggregate be composed of similar crystals and let
them be bound together by a low strength bonding
agent that has no role in determining crystal failure.
Within the aggregate assume a cylindrical cavity of
initial height h and radius r0. Let the cavity be loaded
from the top by a pressure P due to the impact or
mild shock as shown in figure 1. This geometry, first
investigated by De Vost,11 is of particular interest
because the loading force on the cavity can be related
to the shear stress responsible for plastic deformation
and crack formation.
Pressure
active slip plane and 5 is the displacement introduced
on the slip plane by a single dislocation. L0 is the
length of the crystal, d is the molecular spacing of the
undeformed lattice so that Lo/d is the number of
molecules/atoms on a line in the slip direction in the
slip plane that must share the fixed crystal length
with the N dislocations. From stability arguments, the
damage must initially act to increase the lattice
potential so that the quantity diydx must be positive.
This approximation assumes that the lattice
deformation due to the dislocations is distributed
evenly over the length of the slip plane which is in
keeping with the AFM observations.
The third term becomes important during severe
plastic deformation. With increased plastic flow the
AFM studies show that the lattice and the molecules
increasingly distort, rotate and slide past one
another.7"10 This implies that for a highly deformed
crystal the deformed lattice potential must decrease
causing the crystal to soften and eventually fail. For
example, the Peierls-Nabarro approximation of the
undeformed lattice potential has the form U0 «
UoSin(2nx/d) insuring that d2iydx2 will be negative.
In the case of large numbers of edge dislocations
created during severe deformation of the lattice the
molecules on or near a slip plane are distorted and
move slightly away from the slip plane and into the
bulk of the crystal. If the displacement is shared
equally by the L</d molecules that lie along lines in
each of these directions, the expansion coefficient is
dx2/dN2«5(Nd/L0)2.
Shear on crystal at crack tip
Crack Trajectory
Crack Tip
Crack
Cylindrical Cavity
Figure 1. Schematic of shear crack development in a crystal
aggregate containing a cylindrical cavity of height h and radius j.
The crystal aggregate is loaded from the top by an impact or mild
shock of pressure P.
LATTICE SOFTENING, CRYSTAL FAILURE
AND CRACK FORMATION
The material above the cavity is supported against
the applied pressure by the crystals adjacent to the
side walls of the cavity. These opposing forces create
a shear stress on the crystals at the upper corner of
the cylindrical walls of the cavity as shown in the
insert of Fig. 1. With increasing shear stress these
crystals experience severe deformation and
eventually fail and initiate a crack. The crystals
immediately ahead of the failed crystals on the crack
trajectory next experience the shear stress and these
in turn eventually fail and further extend the crack. In
this way the failure-crack region propagates through
the sample on the surface of maximum plastic
The molecules of severely deformed crystals are
usually less securely bound to their neighbors
suggesting crystal softening and eventual failure.
When approximating the severely deformed lattice
potential the third term of the lattice potential U,
Equation (1), must be taken into account since this
term serves to decrease the lattice potential and
accounts for crystal softening and failure.
Because it is difficult to describe in detail the
softening term of the lattice potential the shear stress
responsible for severely deforming the crystal will be
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rr
deformation.
For simplicity, assume that to first order the cavity
retains its shape during loading. Choose a cylindrical
coordinate system for which the origin of the z axis is
located at the top center of the cavity as shown in
Fig. 1. Further, assume that the plastic deformation is
greatest at the tip of the crack but can be neglected in
the region beyond the crack tip. The average
direction of propagation of the crack is determined
by the condition that the vertical component of the
shear stress, iz, be a maximum at the crack tip. At the
tip of the crack the z component of the shear stress is
,r-z tan0
R
•[-Z
Ur
, (4)
that the crack is mainly vertical and the added
complications of crack closure and load sharing
when e > 0° are not a concern. The crystals at the
crack tip experience a shear stress, iz, arising from
the force loading the top of the cavity and, at a
slightly greater radius, an equal but opposite force
originating from the material in the side walls. Let i f
be the average shear stress at which the crystals in the
aggregate fail and for simplicity let if be isotropic so
that for this analysis there is no need to take into
account crystal orientation. At failure t z « if-p'-(total
volume of crack tip), where p' is the crystal number
density. Evaluating the total volume of the crack tip
gives T Z « 2nifp'r(_r_z), where _r and _z are the
radial and z dimensions of the failure region at the
crack tip. The component of the shear stress due to
the external load, Fz, can be estimated since F2 «
iz-(l/2 area of crack tip in the radial direction) so that
Fz « 2n2plTfr2Q")2_z. In the case of a cylindrical
cavity loaded by a pressure P, Fz = nr2? so that the
pressure necessary to cause cracking and failure is
cos
(2)
,r-z tan© .
r(——————)cos<
^
where e is the angle between the direction of the
crack and the vertical axis, R = (r2 +Z2)'72 and e + cp +
\|/ = n/2. Rather than deal with the above
transcendental equation, consider its asymptotic
behavior in the limits z -> 0, r ->• r0 and z » h and r
» r0. First, as z —» 0 and r —> r0 then 9 —> 0 and i z
-» i as they must. When z and r are large then
(5)
and is independent of the radius of the cavity when r
» jr. The quantity (_r)2_z *s a volume element in
the failure region at the crack tip. In most cases of
interest failure occurs in just single crystals at the
crack tip so that p'(jr)2_z ~ 1- F°r many brittle
molecular crystals the shear stress at failure and at
yielding are nearly equal, i f « I Y « 10 to 50 MPa, so
that from Equation (5) the pressure required to
initiate cracking and failure in these materials must
be in the range P * 60 to 300 MPa. The pressure-time
loading history of such an experiment shows the
increase in applied pressure until i z « Tf at which
point failure occurs and with it a rapid decrease of
the pressure in the sample. For cavities in aggregates
of explosive crystals the energy dissipated in the
crystals during rapid deformation often initiates
chemical reaction so that the observed rapid pressure
decrease due to failure is often followed immediately
by a rapid pressure increase due to reaction.11
(3)
The equation of the averaged crack surface is
determined by maximizing the vertical shear stress
due to the applied pressure, di z = 0. But diz =
dR-viz where R and v are vectors so that in the
asymptotic limit of large r and z
di z = 0 and i z is a maximum in the limit of large r
and z when r « z. This defines the average crack
surface for large r and z and gives the expected
asymptotic behavior that 8 -> n/4.11
The pressure required to initiate a De Vost crack
can be determined by examining the vertical force at
the tip of the crack needed to support the applied
pressure load. For simplicity, assume that r0 is much
greater than the failure zone at the crack tip, r 0 » jr.
Consider only the region near the cavity, r « r0, so
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8. Sharma, J. and Coffey, C. S. in Decomposition,
Combustion and Detonation Chemistry of
Energetic Materials, edited by T. B. Brill, T. P.
Russell, W. C. Tao and R. B. Wardle, MRS
Symposia Proceedings No. 418 (Materials
Research Society, Pittsburgh, 1995), p. 257.
9. Sharma, J., Hoover, S. M., Coffey, C. S., Tompa,
A. S., Sandusky, H. W., Armstrong, R. W. and
Elban, W. L. in Shock Compression of
Condensed Matter-1997, AIP Conf. Proc. 429,
563 (1997).
10. Sharma, J., Armstrong, R. W., Elban, W. L.,
and Coffey, C. S.in Appl Phys. Lett., Feb.
(2001).
11. DeVost, V. F. and Coffey, C. S., (unpublished
inNSWCTR81-249).
SUMMARY AND DISCUSSION
Plastic flow, energy dissipation and lattice failure in
crystalline solids involve the relative motion of the
atomic or molecular constituents of the crystal lattice
and are necessarily described by quantum
mechanical processes. The response of crystalline
solids to shock or impact has been determined in
terms of a deformed lattice potential and the notion
that plastic flow and energy dissipation occur due to
dislocation motion in which the dislocations move by
tunneling through the deformed lattice potential
barrier. This has been briefly summarized here
including the prediction that the crystal failure that
occurs during severe plastic deformation is
responsible for cracking. This prediction is not
available from classical physics.
ACKNOWLEDGEMENTS
The authors want to acknowledge the support that
they have received from the Indian Head and the
Carderock Divisions of the Naval Surface Warfare
Center. They especially want to thank Drs. C. W.
Anderson and J. M. Goldwasser of the Office of
Naval Research for their support. They also want to
thank their colleagues who helped with this work,
especially R. W. Armstrong, W. L. Elban, T. P.
Russell, S. E. Mitchell and J. P. Martin.
REFERENCES
1. Coffey, C. S., Phys. Rev. B 24, 6984 (1981).
2. Coffey, C. S., Phys. Rev. B 32, 5335 (1984).
3. Coffey, C. S., Phys. Rev. B 49,208 (1994).
4. Coffey, C. S., in Mechanics of Deformation at
High Rates, edited by R. Graham (SpringerVerlag, Berlin, 1996), Vol. 3.
5. Coffey, C. S. and Sharma, J., Phys. Rev. B 60,
9365 (1999).
6. Coffey, C. S. and Sharma, J., J. of Appl. Phys., 89,
4797(2001).
7. Sharma, J. and Coffey, C. S., in Shock
Compression of Condensed Matter, edited by S.
C. Schmidt and W. C. Tao (AIP Press,
Woodbury, NY, 1995), p. 811.
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