Impact Calculations
A Review and Assessment
K. A. Holsapple
University of Washington, 352400
Seattle, WA 98195
holsapple@aa.washington.edu
Part 1: Introduction
Rocks:
A Modeling
Challenge
Part 1: Introduction
The modeling of material
behavior is the biggest
shortcoming in code
calculations, and the primary
reason for bad results..
Part 1: Introduction
What I won’t talk about, but are
important:
Eulerian v. Lagangian codes
Handling Mixtures in Eulerian codes
Boundaries in Eulerian codes
Grid distortion in Lagrangian
N-body codes: Limited material behavior:
simple coefficient of restitution
Large-scale macro-porosity: cannot smear
into a continuum
Part 1: Introduction
Energy Limit
Water
Wet sand and rocks
Dry Sand
But, there is lots of data, and it is
consistent if organized in appropriate
ways..
Part 1: Introduction
QuickTime™ and a DV/DVCPRO - NTSC decompressor are needed to see this picture.
And, while much of it is for explosions,
Impacts and Explosions are (almost) the same…
Part 1: Introduction
QuickTime™ and a GIF decompressor are needed to see this picture.
The Siren of CPU Power!
Part 1: Introduction
Understanding the processes:
Regions of Impact Processes
1. r ~0-> a: Coupling of the energy and momentum of the impactor into the
asteroid
2. r ~ a -> 2a: Transition into point source solution, shock breakaway.
3a. r ~ 2a -> : Shock decays with distance, strength (and gravity) become
important
3b. r ~ 5a -> 15a: Crater boundary, depending on problem
Part 1: Introduction
What do we need?

Balance Laws (easy: continuum
mechanics: balance of mass,
momentum, energy)

Material behavior (very hard: 100 Mbar
down to partial bars!)

Robust computer codes
Part 1: Introduction
Material Behavior: Three regimes
EOS
Single species EOS
Single species EOS
e(P, r)
e(P, r)
P>>r c2
Mixture Theory
(including porosity)
Solids
Flow,
fractur
e,
failure
P~r c2
Stress-Strain Equations
Yield
Flow & Failure
Fracture
P<<r c2
Yield surface,
Flow rule
Fracture Criteria
Part 2: Source region, EOS
Region 1: Source Coupling

Other Names:





Penetration
region
Contact and
Compression
region
Coupling region
Early-time region
Isobaric Core
region
 Characteristics
•High Pressure >> c^2
• => Hydrodynamic
• => Primarily determined by
EOS
In this regime, the energy and
momentum of the impactor are
transferred into the target.
This is the bailiwick of the EOS..
EOS
Single species EOS
Single species EOS
e(P, r)
e(P, r)
Mixture Theory
(including porosity)
P>>r c2
Part 2: Source region, EOS

It is the EOS that determines
the initial high-pressure
response, including:
 The
Maximum Initial Pressure
 The Transition into the Point-Source
field
 The exponent m of that point source
 The source effect on all subsequent
scaling
Part 2: Source region, EOS
Stress Waves: Pressure
Decay
PROPERTIES:
1. Max Pressure=r0[(c0
/2) U+(s/4)U2]
2. Transition to a
Point-Source
Solution
3. If P>>r0c2, The
Point-Source is SelfSimilar and Power-Law
4. But every impact
velocity case
approaches a PointSource Solution (but not
self-similar)
Part 2: Source region, EOS
And, pressure
decay is
problem
dependent:
Pressures decay
much faster in a
porous material,
the point source
has m~1/3, P~r-6
Part 2: Source region, EOS
Point-Source Impacts


Initially, the flow field
depends on all three of
the impactor measures:
 radius a, velocity U
and mass density r
However, soon all
signatures of those
disappear, and there
remains but a single
measure of the
impactor:
aUmdn
Then all aspects of
the process can
depend only on
aUmdn
and not separately
on the three
measures..
Part 2: Source region, EOS
Scaled Pressure Decay
Scaled Pressure Decay Assuming
a Point-Source
100
run 101 U=80 km/s
run 101b U=40
km/s
run 101a U=20
km/s
Self Similar
Scaled Pressure=P/(rc2)
10
Point-Source, Selfsimilar regime
All Velocity Cases in
a given material
approach the same
Point-Source
Solution after a few
impactor radii
1
Point-Source, NotSelf-Similar
0.1
0.01
0.1
1
10
Scaled Range=(r/a)(c/U) m
100
Thus, there is no
signature of the
impact velocity
except in the source
region r<2a
Part 2: Source region, EOS
EOS => EOS
Different sources have the same farfield results whenever the point-source
property holds!
The EOS determines the EOS: The
Equation Of State determines
Equivalence Of Sources
Part 2: Source region, EOS
“Equivalent” Sources: mass x
0.82
Q
U=5 km/s
U=30 km/s
Depends on:
U=50 km/s
Mass=1010 kg Mass=5 108 kg Mass=2.2 108 kg
TNT
Q=4.2 1010
Mass=2.4 1010
(24 Mtons)
Part 2: Source region, EOS
So, what do we need to define
the EOS?
¨
1. Equations of State for
solid
¨
2. + modified by mixtures

3. + modified by Porosity
Part 2: Source region, EOS
Analytical Single-phase EOS
Models

Murnaghan: Non-linear elastic, no
thermodynamics, limited uses.

Tillotson (1962): Powers in density + thermal
component~E + vapor interpolation

Mie-Gruneisen: Linear Us-up + thermal
component~E + vapor interpolation

Puff (1966): vapor
Simple algebraic descriptions, no phase cha
Part 2: Source region, EOS
Analytical, Explicit Three-Phase

Gray (1971), ESA, Philco-Ford (1969),
Barnes et al. (1967 and on)

Aneos (1970) solid, liquid, vapor


No molecules, no mixture theory, limited solid phase
changes
Panda (1981+) Various combinations of cold,
thermal, electron and multi-phase models

Allows mixtures of molecular species, multiple phases
Part 2: Source region, EOS
Complete
Equations
of State
E(r,P)
(Cold+Thermal
+Electron
components)
Part 2: Source region, EOS
What properties are sometimes
available for calibration??
1.Loading along the Hugoniot from
shock measurements, but usually not
energy or temperature directly.
U<10km/s.
2. Maybe some data on adiabats
from shock unloading.
3. Static melt and vapor points at
atmospheric pressure.
4. Other data at one bar such as
specific heat, thermal expansion.
5. The critical point.
Part 2: Source region, EOS
Porosity Addition..solid + voids




Herrmann’s P-alpha
(1969)
Carroll-Holt (1972)
Seaman and Linde
POREQST (1969)
Holt et al. (1971)
Part 2: Source region, EOS
Herrmann’s P-alpha

Distension ratio
a=rsolid/rtotal
Ps
 ranges from r0 (initial) to
(fully crushed)

A crush curve defines
crushing: decrease of the
void volume:


a=f(P,Pe,Ps)
Instantaneous state
variable
P(r, T,a)= Psolid(rsolid,T)/a
e(r,T,a)= esolid(rsolid,T)
Pe
Part 2: Source region, EOS
4
Pure #75 sand 36%
70%
3.5
Loose #75 sand
Dense #75 sand
45%
Distension, alpha
Crush
curves for
porous
Sands:
Crush
begins at
Pe,
complete at
Ps
57%
3
45%
Coarse Sand
2.5
2
1.5
1
1.00E+06
1.00E+07
1.00E+08
1.00E+09
1.00E+10
Pressure
(From Kevin Housen)
Part 2: Source region, EOS
Pure #75 sand 36%
4
Crush is
limited to one
decade!
57%
Loose #75 sand
3.5
Dense #75 sand
45%
Distension, alpha
The P-Alpha
model
Pe=1e7
Ps=2e9:
70%
45%
3
Coarse Sand
P-Alpha Model
2.5
2
1.5
1
1.00E+06
1.00E+07
1.00E+08
1.00E+09
1.00E+10
Pressure
You don’t always get what you think!!
Part 2: Source region, EOS
Furthermor
e:
The actual
path
followed in
CTH with
Ps=20 mpa
You don’t always get even what
you think you didn’t get!!
QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture.
P=85
mpa
Part 2: Source region, EOS
Finally, supposing we have a
reasonable EOS, how do we
use it in Wave Codes?

Direct analytical evaluation, or..

Tabular Data Tables (Sesame)
 We have Sesame tables from Panda,
Aneos, Seslan (Los Alamos) and
others
Part 2: Source region, EOS
Summary of EOS

The model choice will fix the scaling and other
important features of a solution

The tools are there for complex models, but it is very
hard to get the data to calibrate the models, even for
“simple” ones like Aneos (24 constants)

I think many users are unaware of the major
uncertainties: the very existence of a pre-existing
model gives it unwarranted credibility

It is Extremely complex to construct a multiplespecies model using Panda.
Part 2: Source region, EOS
But wait, there’s more…
Porosity adds yet another large
uncertainty
Phase changes
Kinetic Effects
Multiple Species
And all of that assumes we know the
actual material, which we don’t in
most or our applications!
Part 3, Stress-Strain
Stress-Strain
EOS
Single species EOS
Single species EOS
e(P, r)
e(P, r)
Mixture Theory
P>>r c2
(including porosity)
Solids
Stress-Strain Equations
P~r c2
Part 3, Stress-Strain
Stress-Strain behavior
• When P≈rc2
the material no longer
behaves as a fluid.
• Then we need a constitutive equation for
the stress-strain behavior
• Almost always, in wave codes that is
simply an isotropic linear elastic relation
(which is undoubtedly extremely crude).
Part 4, Strength
Which brings us to the strength
parts..
EOS
Single species EOS
Single species EOS
e(P, r)
e(P, r)
P>>r c2
Mixture Theory
(including porosity)
Solids
Flow,
fractur
e,
failure
P~r c2
Stress-Strain Equations
Yield
Flow & Failure
Fracture
P<<r c2
Yield surface,
Flow rule
Fracture Criteria
Part 4, Strength
The “F” words:
Flow, Fracture and Failure
Models for these fall into three groups:
• “Degraded Stiffness”, no explicit flow or fracture.
• “Flow” including plasticity and damage, used to model
microscopic voids and cracks leading to an inability to
resist stress.
• “Fracture”, involving actual macroscopic cracks and
voids which are tracked, leading to an inability to resist
stress.
Part 4, Strength
In a continuum theory, the first
two can be included directly, the
latter is difficult, unless some
statistical approach is used to
smear them out.
Part 4, Strength
Some Real
Data
Part 4, Strength
Yield
depends
on
pressure
Cohesion
Tensile
strength
ANGLE OF FRICTION
Part 4, Strength
Damage and
degradation
leading to
ultimate failure
occur at some
limiting strain
Part 4, Strength
PressureDependent
Ductility:
Failure Strain
depends on
pressure..
Part 4, Strength
Pressure
Dependent
Yield and
Ductility
Part 4, Strength
Bulking:
increase
in volume
at failure
Part 4, Strength
Damaged
material:
Cohesionless,
but not Fluid.
Grady Kipp, failed
Part 4, Strength
Tensile fracture depends strongly on strain rate
Strength v. Strain Rate
from Various Studies
Concrete (plai n)
Concrete (polyes ter)
1000
Lim esto ne (Oakhal l)
Oil Sh ale (80ml /kg )
Dynamic Strength (Mpa)
Codes
Arkansa s Novaculite
Wes terly Gran ite
(Lip kin )
100
H&H Gra nite (Crack
Distri buti on)
Ful ly Cracked, Larg e
(Vari ous Ma teria ls)
Melos h et al. (Basa lt)
Dresse r Basa lt
10
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
Strain Rate (1/sec)
1.E+05
1.E+06
1.E+07
Benz a nd Aspha ug,
199 4
Part 4, Strength
Flow and Fracture Models should
include:
Strength v. Strain Rate
from Various Studies
Concrete (plai n)
Concrete (polyes ter)
1000
Lim esto ne (Oakhal l)
Dynamic Strength (Mpa)
Oil Sh ale (80ml /kg )
Arkansa s Novaculite
Wes terly Gran ite
(Lip kin )
100
H&H Gra nite (Crack
Distri buti on)
Ful ly Cracked, Larg e
(Vari ous Ma teria ls)
Melos h et al. (Basa lt)
Dresse r Basa lt
10
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
Benz a nd Aspha ug,
199 4
Strain Rate (1/sec)
••Degradation
•Pressure
Pressure
dependent
with especially
strain
failure
Yield
over
envelope
envelope
(damage)
and
even
Permanent
when
fullystrains
failed
•Rate
dependent,
ductility:
infailure
Failure
tension
strain
increases
with pressure
Part 4, Strength
We can do all of that using:
•An Explicit Yield Envelope
•Envelope is pressure dependent
•A damage measure
•Envelope changes with damage
•Damage accumulation depends on pressure
Part 4, Strength
Flow and Fracture: Yielding and Cracking
Lets put it all together; Start with the yield
envelope:
Initial Yield=F(stresses) or G(strains)
•Isotropic=> s1, s2, s3
(Or three stress invariants)
•Commonly only 2, e.g.
J2=F(P)
Or max shear=f(pressure)
Part 4, Strength
Special Case: VonMises (metals)
Shear Stress
Pressure
Part 4, Strength
Special Case: Mohr-Coloumb &
Drucker-Prager
Shear Stress
Tan( )
Uniaxial Compressive Strength
Cohesion
Uniaxial Tensile Strength
Pressure
Part 4, Strength
Mohr-Coloumb with Mises Cutoff:
Part 4, Strength
Yield Functions in ShearPressure Space
Mises Cutoff
Crush
Friction Angle
Tensile Strength
Compressive Strength
Shear Stress
cohesion
Pressure
Part 4, Strength
Now add Improvements:
•A Damage measure. e.g. accumulated
plastic strain or plastic work
•Then the envelope degrades with damage
•Damage accumulates slower at higher pressure
leading to more ductility at higher pressure
•The envelopes also depend on rate, mostly the
tension parts
•And they depend on temperature
Part 4, Strength
An example that has many of these is
that of Johnson & Holmquist, 1990
depends on rate
Yield Envelope
=0
ge
a
m
da
static
interpolate with damage
depends on rate
d
e=
g
a
am
1
static
Pressure
Part 4, Strength
But, Improvements Needed
Temperature dependence
Tensile point depending most strongly
on rate
General EOS, not just non-thermo
algebraic
Porosity
Interdependence between porosity,
strength
Size Dependence
Part 4, Strength
Stress
Another approach is to use Stiffness Degradation
rather than an explicit yield envelope
(e.g. Grady-Kipp Damage Model)
Modulus E(strain)
Strain
Part 4, Strength
The Grady-Kipp Model
Special nature





It is a Tensile Brittle Fracture Mechanism
 For fragmentation in mining
One-Dimensional Model
Synthesized for constant strain rate histories only
Governed by Crack Distributions (Weibull) and
growth
Implies rate and size-dependent strength
But Attractive Physics
Part 4, Strength
A Grady Kipp Implementation in
3D
•Damage is isotropic, so that when a crack is formed in
one directions, all directions lose stiffness
•As damage accumulates, the stiffness in both tension
and in shear decrease, eventually to zero.
•Therefore, material failed by the outgoing shock behaves
as water.
•Calibrated to disruption test, by adjusting the strength parameters
•But, for cratering, craters are way too large!
Part 4, Strength
The Grady-Kipp Approach
Shear
Da
m
ag
e
Af
fe
ct
Shear stress at a
given shear strain
Tensile stress at a
given tensile strain
Fully damaged material
Pressure
Part 4, Strength
So how can we improve the
models?
Compare, Compare, Compare
to real experiments
Large explosive field tests
Carefully controlled lab tests
to impact craters
(but what was the impactor?)
Test, Test, Test
real materials
Crushability
Strength in different states
Part 5, Concluding Remarks
Current Shortcomings:
• Damage is a scalar: not directional
• Behavior is isotropic, even after damage
• Most strength models do not address all types of “strength
• Codes often have “hidden features”
• Even the limited models available usually greatly surpass
• We do not often enough make comparisons to any experi
Part 5, Concluding Remarks
We are too anxious to “get results and
publish”
•
for the crater “X” on the moon “Y” of
planet
•
“Z”,
•
and do not make enough effort of
developing and
•
checking the codes and models
•
Part 5, Concluding Remarks
etc
We cannot model well enough to distinguish details for a
particular crater
We cannot handle mixtures well
We don’t have a good handle on even the mechanisms that
give very significant features: e.g. late-stage adjustments.
Mixing rocks and atmospheres, and porosity makes for
very difficult code calculations
We do primarily continuum mechanics, and cannot distinguish
individual particles, size distributions, ejecta particle
characteristics and fate
We don’t do chemistry
Part 5, Concluding Remarks
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Impact into a porous target
Part 5, Concluding Remarks
QuickTime™ and a Animation decompressor are needed to see this picture.
Porous cratering: Marker particles
Part 5, Concluding Remarks
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Impact into water
Part 5, Concluding Remarks
Summary
We have a long way to go
Calculations should be considered
order of magnitude at best:
Computer power will not save us and
might even be detrimental.
Part 5, Concluding Remarks
“On top of this, we observed that the current
exponential increase in computer power is
already leading to grave difficulties in
assessing the content of simulations of
complex phenomena, as well as comparing
them with high quality experimental data.
Among other challenges, this situation
leads to the erroneous possibility of thinking
that because output is complex, we must be
successfully
modeling
complex
phenomena.”
T. G. Trucano, “Prediction and uncertainty in computational modeling of complex
Phenomena: A whitepaper. SAND98-2776, 1998
Part 5, Concluding Remarks
However:
We have learned a lot, especially qualitative, of
cratering mechanics from codes.
Cratering regimes: strength, gravity, porous
Ejecta transport and emplacement
Effects of oblique impacts
Thermodynamic histories of material
Late-stage readjustments: Scope, although not
mechanism.
Calibration to real results, followed by minor
extrapolations are probably effective.
Part 5, Concluding Remarks
And, some recent advances are
promising:
Beginnings of all appropriate rock strength
features:
Damage, bulking, thermal, strain softening
E.g. O’Keefe et al, 2001, Holsapple and Housen, 2002
Mixed materials
Assuming non-equilibrium thermal
(O’Keefe et al. 2001 and others)
Assuming phase equilibrium
(Ivanov, 2003)
Atmospheric transport
e.g. Artemieva & Pierazzo (2003)
Part 5, Concluding Remarks
I believe it is the
correct modeling of the
down-side of the
Stress-strain curves
that we have not
done well, and that is
the key to important
aspects such as latestage readjustments.
Part 6, The End
THE END, Thank
You