47th Lunar and Planetary Science Conference (2016)
1492.pdf
STEPS TOWARD IMPLEMENTATING THE GRADY-KIPP FRAGMENTATION MODEL IN AN
EULERIAN HYDROCODE. B. C. Johnson1, T. J. Bowling2, and H. J. Melosh3, 1Department of Earth, Environmental and Planetary Sciences, Brown University, 324 Brook Street, Providence, RI 02912, USA.
(Brandon_Johnson@Brown.edu). 2 Department of the Geophysical Sciences, University of Chicago, 5734 S. Ellis
Avenue, Chicago, Illinois 60637. 3Department of Earth, Atmospheric, and Planetary Sciences, Purdue University,
550 Stadium Mall Drive, West Lafayette, IN 47907, USA
𝑌 = 𝑌# 1 +
&
&'
(
)
[6]. Where 𝑌# is the yield stress
without the added term, 𝜀 is the equivalent strain rate,
and 𝜀# is a material constant.
Results: The color scale on Figure 1 is truncated at
a fragment size of ~3 cm, comparable to the radius of
the target. Material with these low fractures areas represent fragments that are actually resolved in the mod-
el. As expected, the smallest fragments are located
under the point of impact and large spalls are located
along the entire free surface. With our relatively high
resolution (as compared to [5]) of 20 cells per projectile radius and a self-consistent definition for the
threshold strain, we are able to resolve the observed
‘intact’ inner core of material [11] whereas [5] was
unable to reproduce this feature.
log10(Fragment size (m))
0
-1.5
-1
-2
-2
-2.5
-3
cm
Introduction: The details of impact fragmentation
and ejection are critical for understanding secondary
impact cratering, the formation of planetary regolith,
and the ejection of meteorites from the terrestrial planets. Smooth Particle Hydrocodes (SPHs) give robust
estimates of fragment sizes especially in lower velocity
collision [eg. 1], but so far SPHs have not achieved the
resolution necessary to resolve fast ejecta produced by
hypervelocity impacts. Recent modeling [2] shows that
high velocity ejecta comes from very near the projectile where the point source approximation used in
Lagragian hydrocodes and analytical models may not
be valid [3,4]. Here we detail steps toward implementing the Grady-Kipp fragmentation model [5] into an
Eulerian hydrocode. In the future we may consider
other fragmentation models [6]. Although our end goal
is to model the formation of fast ejecta and gain valuable insight into the processes outlined above, here we
focus on reproducing results from laboratory scale
impact experiments to validate our numerical methods.
Methods: In this work we use the 2D version of
the iSALE shock physics code [7-9]. We model basalt
impactors striking spherical basalt targets at vertical
incidence while matching the target mass, projectile
mass, and impact velocity from laboratory experiments. Fracture area and damage are calculated in the
code following [5]. As clearly demonstrated by [10],
the constant threshold strain for failure used by [5] is
not hydrodynamically self-consistent. Thus, we use a
randomly seeded threshold strain similar to [10] but
still calculate fracture area according to [5] rather than
making cracks and their growth explicit as in [10]. To
fit the experimental data, we held the Weibull parameter m=9.5 constant while varying the k parameter.
Tensile strength and failure are determined by the
Grady-Kipp model [5] where ‘shear’ failure is determined by [8] however we include a strain rate hardening term such that the yield stress is
-3
-4
-5
-3.5
-6
-4
-7
0
1
2
3
4
cm
Figure 1: Tracers colored according the logarithm of
final peak fragment size (50 µs after impact) for
Weibull parameters k=1037 and m=9.5. This is a model
of a 1.00 x 10-2 kg basalt sphere striking a 0.410 kg
basalt sphere at 599 m/s similar to experiment #820520
of [11].
Although we are working on producing a reliable
algorithm to calculate the size of these resolved intact
fragments, without estimates of their size we can only
speculate on how they might change the large end of
47th Lunar and Planetary Science Conference (2016)
the calculated fragment Size Frequency Distribution
(SFD). Figure 2 shows some estimates of the fragment
SFD where the dashed curve simply uses the method
of [5] and effectively ignores these intact fragments,
which account for a large fraction of the target mass
(Fig. 1). The solid lines assume all tracers with peak
fragment sizes more than two times larger than the
largest fragment given by the previous method make
up a single large fragment. This assumption is also
clearly problematic as there are many resolved fragments (Fig 1.). Based on the average cell mass resolved fragments may contribute to the SFD at normalized fragment masses exceeding 1.3x10-4. Although
Weibull parameter k and m vary significantly with
material, our best-fit to experiments occurs close to
experimentally derived estimates of k=1.59x1038 and
m=9.5 for Dresser basalt [5, 13]. Another potential
problem with all current fragmentation models and an
area for future exploration is the fact that fracture area
is not accumulated during shear failure.
There is clearly much work to be done, but our initial models represent an important first step toward a
better understanding fast ejecta fragments, secondary
cratering, and the ejection of meteorites. Because we
are able to directly resolve large fragments, this model
may help determine the size of spall plates without
significant modifications.
Cumulative number N
10
1492.pdf
References: [1] Jutzi M. et al. (2009) Icarus, 201,
802-813. [2] Johnson B. C. and Melosh H. J. (2014)
Icarus, 228, 347-363. [3] Head J. N. et al. (2002) Science, 298, 1752-1756. [4] Melosh H. J. (1984) Icarus,
59, 234-260. [5] Melosh et al. (1992) JGR, 97, 735759. [6] Ramesh K. T. et al. (2015) Planet. Space Sci.,
107, 10-23. [7] Amsden, A. et al. (1980) LANL Report,
LA-8095. [8] Collins G. S. et al. (2004) MAPS, 38,
217-231. [9] Wünnemann K. et al. (2006) Icarus, 180,
514-527. [10] Benz W. and Asphaug E. (1995) Comput. Phys. Commun., 87, 253-265. [11] Nakamura A.
and Fujiwara A. (1991) Icarus, 92, 132-146. [12] Takagi Y. et al. (1984) Icarus, 59, 462-477. [13] Lindholm
U. S. et al. (1974) Int. J. Rock. Mech. Min. Sci. Geomech. Abstr., 11, 181-191.
Acknowledgements: H. J. M. acknowledges support from NASA grant NNX15AL61G. We gratefully
acknowledge the developers of iSALE-2D, including
Gareth Collins, Kai Wünnemann, Dirk Elbeshausen,
and Boris Ivanov.
Figure 2 (below): Comparison of cumulative fragment
size frequency distributions for the impact conditions
from Figure 1 to laboratory measurement (blue diamonds) and previous estimates from Lagrangian hydrocodes (red triangles). The black curve represents a
separate run with different Weibull parameters.
Laboratory impact
Takagi et al. (1984)
Exp #820520
3
Lagrangian Hydrocode
Melosh et al. (1992)
k=1E37 m=9.5
10
k=1E37 m=9.5
'normal' treatment
2
k=1E38 m=9.5
10 1
10 0 -6
10
10 -5
10 -4
10 -3
10 -2
Normalized Mass M/Mtarget
10 -1
10 0