CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
FRAGMENTATION OF EXPANDING CYLINDERS AND THE
STATISTICAL THEORY OF N. F. MOTT
Dennis Grady
Applied Research Associates, 4300 San Mateo Blvd., A-220, Albuquerque, New Mexico 87110
Abstract. The seminal investigation of the explosive fragmentation of steel cylindrical shells by N. F.
Mott in the early 1940's led to an elegant statistics-based theory of dynamic fragmentation (Mott,
1947). Experiments in which rapidly expanded metal rings undergo dynamic fragmentation provide
ideal data for testing the fragment size and statistical distribution predictions of Mott's theory. In this
work the theoretical development of Mott is examined and compared with expanding metal ring
experimental data.
INTRODUCTION
The fragmentation of hollow metal shells subjected
to rapid expansion by impulsive internal pressure
loading continues to be a problem of both practical
and intellectual interest. The seminal modeling of
this fragmentation process by Mott (1947) continues
to be a fascinating framework for investigating the
consequences of differing fracture physics. The
model is illustrated in Figure 1 in which a uniformly
expanding metal cylinder (the Mott cylinder) has
exhausted
deformation hardening and the
circumferential stress versus strain curve is
horizontal tangent at a nominally constant dynamic
tension Y . Fracture is assumed to occur within a
fairly short time through a statistical fracture
activation and growth process. Fractures activate at
random points on the Mott cylinder. Growth is the
process of stress wave propagation by which release
of the tensile stress is communicated to regions away
from fractures points. Density of the material is p
while the nominal stretching rate £ is the ratio
VIR of the expansion velocity and cylinder radius.
In the present effort we will investigate two
alternative theories underlying the fracture activation
and growth process, and then offer possible
explanations for reconciling them. First is the
energy-based approach (Grady et al., 1984; Kipp and
Grady, 1985). Second is the statistical strain-tofracture theory of Mott (1947).
Radial
Expansion
Velocity - V
Regions Stretching
At Constant Strain Rate
Regions Stress-Relieved
After Mott-Wave Passage
FIGURE 1. The Mott Cylinder. A model for the statistical
fragmentation of an expanding cylindrical metal shell.
799
ENERGY-BASED
FRAGMENTATION THEORY
At the heart of the energy-based fragmentation
theory are details of the fracture activation process.
In the 1947 paper Mott offered a solution for the
propagation of stress release away from the fracture
point (the Mott wave) by considering instantaneous
fracture and tensile stress drop at a point and
subsequent motion of a rigid region behind the
propagating Mott wave through momentum
considerations (Mott, 1947; Kipp and Grady, 1985).
The solution,
x = J(2Y/pey2,
STATISTICAL STRAIN-TOFRACTURE THEORY
Mott proceeded by arguing that energy dissipated in
the fracture process was unimportant. Rather, he
chose to assume that fracture occurred
instantaneously at points on the Mott cylinder
according to a probabilistic expression which
increased rapidly with the strain at that point. In
particular he proposed a probabilistic hazard
function of the exponential form,
A, (£) = Ae™ ,
that provided the chance of fracture within unit
length of the cylinder at the strain 8 . It is readily
shown that Equation 4 leads to an extreme value
distribution of the Gumbel type where 1.28/G is the
standard deviation of the distribution (Hahn and
Shapiro, 1967). Other extreme value distributions
could of course be considered.
Mott (1947) then used Equation 4 for the fracture
activation law and Equation 1 for the propagation of
Mott waves to determine the probabilistic number of
fracture sites and the probabilistic distribution in
position of these sites over the fracture activation
and growth duration. Through a graphical method
he determined the characteristic fragment length and
the distribution in lengths about this mean.
Grady (198la, 1981b) applied statistical methods of
Johnson and Mehl (1938) to provide an analytic
solution to the theoretical approach proposed by
Mott. It can be shown that the average fragment
length is
(1)
was obtained for the position of the Mott wave as a
function of time t .
The solution of Mott was extended (Grady et al.,
1984; Kipp and Grady, 1985) to account for
dissipation at the fracture point and the timedependent release of tensile stress. A similar
solution for the position of the Mott wave, again
assuming rigid-plastic properties, yields,
1 Y2 2
= ———t
12 py
(4)
(2)
over the activation time T from initial perturbation
at the fracture site until fracture separation. The
energy dissipated in the fracture is 2y during the
time T .
It is readily recognized that if two fractures begin to
activate within a sufficiently small linear region of
the Mott cylinder than their respective Mott waves
will interact before activation is complete and one or
the other will arrest and not go to completion. To
achieve theoretical closure it was reasoned that twice
the distance traveled by the Mott wave over the time
T ,
r YIp82Y
l
2
a
while the probability density distribution in fragment
lengths is,
/w-fff]'>4'W)*>JV*'
(3)
L
. \L» )
o
(6)
should correspond to an average fragment length. A
coupled statistical theory leading to prediction of the
probably distribution in fragment lengths was not
pursued.
although a simpler strain-to-fracture hazard function
than Equation 4 was assumed to carry through the
analytic solution (Grady, 198la, 1981b) for the
fragment distribution.
Comparison of Mott's
graphical distribution and the analytic distribution in
800
a of a few percent to a few tens of percent for
existing data are certainly reasonable, however, and
the predicted inverse linear dependence on strain
rate of Equation 5 is seen in at least some of the data
(Grady and Bensen, 1983).
Further, several
comparisons of the Mott statistical size distribution
(Figure 2) with experimental data have been made
(Grady et al., 1984; Grady and Benson, 1983) and
results are quite satisfying. Comparison with the
data of Weisenberg and Sagartz (1977) shown in
Figure 2 is representative.
2.0
Weisenberg &
Sagartz Data
Analytic
Solution
1.5
Mott Graphic
1-0
0.5
DISCUSSION
Two theories of fragmentation following the
fundamental fracture activation and growth
framework of Mott have been pursued. That of
Grady and Kipp is based on energy dissipated within
the fracture activation process and the length scale
governing the predicted fragment size contains that
fracture energy. Mott's statistical strain-to-fracture
theory, on the other hand, ignores fracture energy
and predicted that the average fragment length is
proportional to a length scale which is a unique
dimensional combination of the flow stress Y,
density p and strain rate 8 . Magnitude of the
average fragment length is determined by the
temporal standard deviation in fracture activation
(8 =£t). As I / O goes to zero so does the average
fragment size provided from Equation 5.
Both theories have attractive features and
experimental data to date do not strongly favor one
over the other. Several issues should be considered
which would effectively merge the two. First, a
statistical variation in the fracture energy y could be
considered in the energy-based theory of Grady and
Kipp. This in turn would lead to a statistical
variation in the activation time T and hence the
temporal variation in fracture assumed by Mott.
This statistical variation in y was tacitly assumed in
the energy-based approach of Grady and Kipp
anyway. Otherwise the predicted average fragment
length in Equation 3 would more appropriately be a
minimum fragment length. Secondly, even if each
fracture energy were nominally the same, temporal
variations in initial fracture perturbations should be
expected.
0.0
0.05
0.1
0.15
0.2
Fractional Length of Circumference
FIGURE 2. Comparisons of analytic and graphical
solutions for the Mott statistical fragment size distribution and
expanding ring data of Weisenberg and Sagartz (1977).
Figure 2 suggests a lack of sensitivity to the
assumed hazard function but this has not been
proved. Comparisons with expanding ring data of
Weisenberg and Sagartz (1977) are also shown.
COMPARISON WITH EXPERIMENTAL
DATA
An extensive series of fragmenting aluminum and
copper ring experiments using magnetic loading
(Grady and Benson, 1983) has been compared with
the energy-based prediction of fragment length in
Equation 3. The data are reasonably consistent with
fracture energies measured by independent methods
although the average fragment length dependence on
strain rate is closer to an inverse first power rather
than the predicted inverse two-thirds power. More
recent unpublished expanding ring data on other
metals have shown inverse two-thirds power
dependence, however. Further, exploding cylinder
fragment size data are reasonably predicted with the
energy-based fragment size relation using published
fracture toughness properties (e.g., Reedal et al.,
1999).
Mott's theoretical prediction of fragment length
based on statistical strain-to-fracture concepts is less
readily compared with data due to the unavailability
of the parameter a (1.28/a is the standard
deviation in strain-to-fracture). Calculated values of
801
Thus there is every reason to expect that a merging
of the energy-based concepts of Grady and Kipp
with the temporal fracture statistics concepts of Mott
would lead to an improved predictive theory of
dynamic fragmentation.
REFERENCES
1. Mott, N. F. (1947), Proc. Royal Soc., A189, 300-308, January.
2. Grady, D. E., M. E. Kipp, and D. A. Benson (1984), Inst.
Phys. Conf. Ser. 70, 315-384.
3. Kipp, M. E. and D. E. Grady (1985), J. Mech. Phys. Solids,
33,399-415.
4. Hahn and Shapiro (1967)
5. Grady, D. E. (1981a), J. Geophys. Res. 86, 1047-1054.
6. Grady, D. E. (1981b), Shock Waves and High-Strain-Rate
Phenomena in Metals, M. A. Meyers and L. E. Murr, Eds.,
Plenum, New York, 181-191.
7. Johnson and Mehl (1938)
8. Reedal, D., L. Wilson, D. Grady, L. Chhabildas, and W.
Reinhart (1999), Proceedings of the 15th U. S. Army
Symposium on Solid Mechanics, 569-585, Myrtle Beach, SC,
12-14 April, 1999, Battelle Press, Columbus, OH.
9. Grady, D. E. and D. A. Benson (1983), Exp. Mech., 23, 393400.
10. Wesenberg, D. L. and M. J. Sagartz (1977), J. Appl. Mech.,
44, 643-646.
802