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
NEW DIRECTIONS AND NEW CHALLENGES IN
ANALYTICAL MODELING OF PENETRATION MECHANICS
James D. Walker
Southwest Research Institute, P.O. Drawer 28510, San Antonio, TX, 78228
Abstract. With the development of plasticity theory in the 1940s and 1950s, the modeling of penetration has become increasingly analytical and accurate. Currently, analytic penetration models are able to
accurately predict depths of penetration for simple penetration geometries where the target and projectile
are metals. The most accurate models use sophisticated plasticity analysis to obtain target resistances,
but they usually end up expressible as relatively simple explicit differential equations. The most promising development in recent years is the centerline momentum balance, and this technique will be reviewed with some discussion of the meaning of terms by way of first principle physics. A recent model
that successfully predicts ballistic limits of fabrics will also be discussed. In addition to addressing what
is known, the most pressing questions that need to be answered and what is currently known on those
problems will be discussed. Questions include: the transition from eroding to rigid penetration; the
stress state transition for eroding projectiles; calculation of impact crater diameter; fracture time for ceramic targets; modeling targets comprised of anisotropic composites and fabrics; analytical approaches
to projectile yaw; and modeling back surface bulging, failure and perforation.
One of the results of the last 50 years of penetration modeling is that, in a generic sense, the target
resistance can be written in the form
INTRODUCTION
Analytic penetration models are models that predict the detailed penetration history of a target based
upon first principles assumptions about mechanics.
Penetration of a target essentially divides into two
parts: how the target resists penetration, and what
happens to the projectile as penetration occurs. In
cases where these parts can be dealt with separately,
good models currently exist for penetration when the
target is a metal. Where the specifics of the target
and projectile together matter a great deal (as it can
with crater radius, the transition of eroding to rigid
penetration, and yawed penetration), there is still
work to be done towards the goal of an explicit, accurate, analytical model of penetration.
As with any overview paper of this sort, personal
biases appear. In particular, this paper relies heavily
on the centerline momentum balance, which in the
previous decade yielded excellent penetration models as well as insight into the penetration process [1].
ipy+*,<«>
2
CD
where u is the penetration velocity, p, is the target
density, and Rt is a term with units of strength (the Rt
notation is due to Tate [2]). This resistance explicitly develops in both the centerline momentum balance [1] and cavity expansion techniques [3]. The
first term in Eq. (1) reflects the inertia of the target
and the second term reflects the strength of the target. Rt(u) shows a weak dependence on penetration
velocity. The calculation of Rt(u) depends on plasticity theory, and major advances towards its calculation occurred in the 1980s and 1990s based on
plasticity theory as developed in the 1940s and
1950s. In an energy context, the first term of Eq. (1)
represents the energy that is temporarily being
stored in the target as kinetic and elastic compres1273
sive energy, and the second term represents energy
that is immediately being dissipated through plastic
flow. This understanding of the various terms is
recent, and represents a large step forward in the
understanding of the penetration process [4]. Before, there was considerable discussion on the topic
of "energy vs. momentum," but now it is fairly clear
how the two approaches in modeling relate to each
other. One result of this understanding is the recognition that in order to model from an energy perspective, it is necessary to include the intermediate
energy transfer mechanisms. Including such terms
makes the modeling very tedious and therefore this
paper will address the topic of penetration modeling
from the momentum perspective.
The understanding of penetration described
above is in large part due to analytic models and
large scale numerical simulations (hydrocodes).
One of the tools available to today's analytic modeler is large scale numerical simulation. Simulations
allow the stresses and velocities within the target to
be examined. This numerical ability to look within
the targets and projectiles during the penetration
event provides ideas for better models as well as
provides direct comparisons for the analytic model
results. Modern analytic models provide not only
final depth of penetration results, but provide depth
vs. time information and velocity and stress states
within the target and projectile during the penetration event. The model development [1] was greatly
aided by results from hydrocode calculations.
With the hydrocode, verification of the analytic
model can occur, meaning that when the same constitutive models are used in the large scale numerical
simulations as are used in the analytic models, then
nearly identical agreement between the simulations
and analytic model confirms that the mechanics in
the model have been implemented as intended. Today's analytic models can use quite sophisticated
constitutive models for materials, and so such verification can be extremely useful. For example, such a
verification of a penetration model using a pressure
depended flow stress for the target with a cutoff
(thus producing an interior boundary in the target
flow region that had to be determined by the model)
was performed in [5].
There is, of course, still the additional step of
validating the model against experimental results.
Currently, analytic penetration models are able to
accurately predict depths of penetration for simple
penetration geometries where the target and projec-
tile are metals. Part of the reason for producing
analytic models in the days of relatively successful
large scale numerical simulations is that the analytic
models are fast. Analytic models allow the parameter studies necessary for design and optimization. Also, when an analytic model agrees with experiment, there is confidence that the event being
studied is understood from a fundamental physics
viewpoint. This paper primarily discusses analytic
models where a central axis of symmetry is assumed, but for those interested in 3D problems
lacking traditional symmetries, the potential time
savings through use of analytic models is enormous.
In order to clarify some of the discussion, it will
be helpful to explicitly write down a penetration
model so that the various terms can be identified. In
particular, the method of the centerline momentum
balance involves taking the momentum balance,
du. d o (2)
P— = ——
dt
dXj
and then integrating along the centerline to obtain
(3)
The u terms appear because we are in an Eulerian
framework. For the specific model described in [1],
three primary assumptions are made to produce the
equations of motion. First, a velocity profile is assumed along the centerline in both the target and the
projectile. Second, the rear of the projectile is assumed to decelerate by elastic waves traveling up
and down the length of the projectile. Third, a
hemispherical velocity field is assumed within the
target that, combined with rigid plastic assumptions,
provides a stress field making it possible to compute
the derivative of the shear stress along the centerline, as required by the last term of Eq. (3). The
velocity field is derived from a potential and describes the plastically flowing target material. In
addition to a velocity field describing behavior deep
within a target, using a multiplicative blending of
potentials a velocity field has been developed that
describes target material motion near the back surface as the target bulges [6]. This flow field produces back surface bulges that agree very well with
computer simulations and experimental data. With
the above three assumptions, the centerline momentum balance equation becomes
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L
a
d (v-
- 2 -
(4)
(cc+1)
~dt 1
2
where
where v is the velocity of rear of the projectile and u
is the penetration velocity, L is the length of the
projectile, s is the plastically flowing zone in the
projectile, a(u) is the extent of the plastic zone
within the target, R is the crater radius and Yt is the
flow stress of the target. The term in brackets on the
right hand side is the target resistance. The deceleration of the rear of the projectile is
, v —u s
(5)
v —•
1 +——— + where op is the projectile flow stress and c is the bar
wave speed in the projectile. Projectile erosion is
L = -(V-M)
(6)
These three equations are the central part of what
has become known as the Walker-Anderson model,
and the full development can be found in [1]. There
are also additional assumptions required, one to
determine the extent of the plastic zone within the
projectile, one to determine the extent of the plastic
zone within the target and one to determine the
crater radius. The last two are important topics and
will be discussed further below. This model agrees
very well with experiment for larger L/Ds (D is
projectile diameter) and models using the same ideas
have been produced that do well predicting
penetration into thick ceramics and glasses [5].
Recently a model that predicts well the ballistic
limit of fabrics has been produced [1]. This model
begins by assuming the fabric is comprised of elastic
springs connected where the fabric yarns cross.
Next the static deflection problem is solved. The
static solution allows an explicit calculation of the
strains. The strains give a force vs. deflection curve.
Next, the longitudinal wave speed is calculated.
These pieces combine to form a momentum balance
where the deceleration of the impacting projectile
and inertially involved fabric is balanced against the
force versus deflection curve. When the state of the
projectile just coming to rest is set to occur when the
strain in the fabric equals the fiber breakage strain,
the following equation for ballistic limit results [6]:
RblhJR p = — — +
V 8 (X
X is the areal density of the fabric divided by the
areal density of the projectile, p=(1.6)2 a constant
determining how much fabric material is inertially
involved in the impact, cf is the fiber wave speed and
er is the fiber breaking strain. This model predicts
ballistic limit extremely well.
Though not a
centerline momentum balance but rather a Lagranian
model, the model indicates a successful approach
that may be applied to composites and other
nonflowing materials.
Though analytic models have become very accurate (predictions to within 5% on depth of penetration for simple metal projectiles into monolithic
metal targets), there are still problems. For example,
no one model predicts the full L/D effect, that is, the
observation that low L/D projectiles penetrate
deeper in terms of L than larger L/D projectiles.
Experimentally, the effect is surprisingly large [8].
PROJECTILE MODELING PROBLEMS
Three problems at present arise in the modeling of
projectiles.
1. Transition in stress state. When the projectile
nearly completely erodes during the penetration
event, as it approaches L/D=1 the projectile stress
state changes from uniaxal stress to uniaxial strain.
The change in stress state allows larger decelerations
of the back of the projectile and results in larger recovered residual projectiles than the model predicts.
When and how the transition occurs is not well
known. In [9] a relatively smooth transition based
upon remaining L/D was used.
2. Modeling complex projectiles with centerline
approximations. Choosing to go the route of the
centerline momentum balance, the problem arises of
modeling projectiles that are not simple rods with
hemispherical noses but have complex 3D structure,
such as jackets. Work recently performed in this
area modeled a 0.30" APM2 projectile in the context
of the centerline momentum balance [10]. Various
approaches of allocating the projectile material were
examined. The conclusion for that work was to
model the projectile as a length of lead followed by
a length of steel. However, a straightforward, con1275
sistent algorithm for defining complex projectiles in
the context of a centerline momentum balance
would be useful.
3. Projectile side loading. The models discussed
above all assume the projectile load occurs on the
nose of the projectile. However, if the projectile
impacts with obliquity or yaw, or the target plate is
in motion, the side of the projectile could be impacted by the crater wall. Such a collision leads to
questions of how the projectile will deform and
break. Accurately modeling these behaviors probably will require discretizing the projectile.
problems, but in fact the issue of projectile nose
shape is a target modeling issue, since it assumes the
projectile is penetrating in such a fashion as to
maintain its nose shape (eroding penetration always
tends towards a hemispherical eroding nose on the
projectile, regardless of the initial nose shape of the
projectile [13]. Also, topics such as "self sharpening" effectively fall under the crater-radius problem,
discussed later). It should be straightforward, assuming that the approach of producing a flow field
leads to terms that can be inserted in the momentum
balance, to model the target material flow around a
different nose shape. However, this has proven to
be difficult for the following reason. Hemispherical
flow has only one length scale, based on the radius
of the crater. Thus, given the crater radius and an
extent of the flow field, the flow pattern is defined.
For flow fields that lack this spherical symmetry,
there are at least two length scales. Thus, producing
a good flow field for a pointed or ogival projectile
also requires calculating flow field extents in terms
of at least two variables. Perhaps the flow field extents can be related in terms of a constant, but an
argument must be put forward for doing so.
6. Breakout models. When a target of finite
thickness is impacted, the projectile may perforate
the target. When the thickness of the target is relatively large (at least a couple of projectile diameters)
and the target is a ductile metal, then a flow field has
been developed for the target that reproduces the
shape of the back of the plate [6], but the next step is
the target failure. Breakout has been addressed
through assumptions about internal failure and geometric failure modes in [14] based on experimentally-based observations and empiricism, but a first
principles approach is needed. Most failure modes
involve fracture of the material or the localization of
shear, two notoriously difficult subjects in applied
mechanics, particularly in a predictive context.
The next problems address different target materials. In each case the material does not engage in
ductile flow, as does a metal. Because the target
response is so "nice" for a metal, it can be accurately
addressed through analytical modeling. However,
these next target materials do not behave in the same
fashion, and in many ways are not "nice."
7. Failure time for thin ceramic tiles. As a
simplification, ceramics in analytic penetration
models have two modes: breaking and broken (or
perhaps better wording is failing and failed). For
thick ceramics, the penetration by large L/D projec-
TARGET MODELING PROBLEMS
There are still a number of open problems with
respect to target modeling. Here are six of them.
4. Extent of plastic flow in the target a(u). All
penetration models that rely on an assumed flow
field require the calculation of the extent of the flow
field. This topic is dealt with in the centerline momentum balance models through the cavity expansion technique. In penetration models that use the
cavity expansion technique directly the production
of an extent of the flow field within the target is implicit. To describe what is currently done in [1], a
cavity expansion is calculated where the penetration
velocity u is the assumed driving velocity for the
inner surface of the cavity. The subsequent velocity
of the interface between the elastic and elastic/plastic response region c(u) is then used to obtain
a scaling factor CL(U)=C(U)/U that is then used to calculate the flow field extent cuR (R is the crater radius). This approach seems to work fairly well, but
success in part could be due to the fact that in the
model the flow field extent appears as a logarithmic
term in the target resistance (Eq. 4). However, there
is still work to be done here. For example, at higher
velocities a fix is required to keep a greater than 1,
since the equation of state used for the compressible
metal is linear and does not have the higher order
terms that become important at higher compressions.
Also, the cavity expansion solution gives a sharp
decrease in a as the velocity increases. Though it
has been shown that a does indeed decrease with
increasing velocity [11], such a large change for
small velocities seems surprising. There are other
methods that have been proposed based on theorems
from plasticity, and for a careful discussion see [12].
5. Different projectile nose shapes. It may be
thought that this topic should be under projectile
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tiles is dominated by the response of the failed ceramic material. This failed material is usually modeled, in both analytic models and large scale hydrocode simulations, as a pressure dependent yield with
a cut-off, also referred to as a Drucker-Prager yield
surface with cutoff. Such a constitutive model
seems to model well failed ceramic material, and an
analytic model has been developed that uses this
constitutive model [5]. Once the ceramic is ground
up (comminuted), it essentially flows, thus allowing
the modeling of the failed target material with the
same flow fields as seen in flowing metals. Because
of the success in using a pressure dependent constitutive model for failed ceramic, the behavior of the
failed ceramic is not considered an open problem.
What is an open problem is modeling the time it
takes the ceramic to break. The kinematics of fracture is an important question for light armors,
meaning armors where the ceramic thickness is on
the order of the projectile diameter. In these armors,
the ceramic exerts significant stopping power to the
projectile while it is in the process of breaking producing dwell, where the projectile erodes against
an essentially anvil-like ceramic face - and the
amount of time the ceramic takes to break (or, more
accurately, grind up into small pieces) is a large determining factor in the performance of the ceramic
target. Currently, analytical models taking into account ceramic fracture usually just include a fracture
time, defined as the time that it takes for the ceramic
to break and begin behaving like a failed (pressuredependent) material. There is to date no explicit
calculation of the fracture time based upon the properties of the ceramic. Ref. [9] used hydrocode calculations to calibrate a curve fit to the fracture time
for different ceramic thicknesses for a given impact
velocity. Since every ceramic is different, and the
fracture time depends on impact velocity as well as
tile thickness, this approach is not reasonable for
analytic modeling.
A first principles model to arrive at this fracture
time is needed, but where to begin? It is still not
known what material parameters are important in
ceramic fracture, and the fact that for some ceramics
(e.g. boron carbide) the microstructure properties
seem to have little influence on the final ballistic
performance leaves considerable room for concern
as to the relevant properties.
8. Nonflowing target resistance calculations.
Composite materials, such as carbon and glass fiber
reinforced composites and fabrics with resin, intro-
duce new complexities because they do not nicely
flow. Modeling them in the context of the centerline
momentum balance requires the ability to compute a
stress gradient term for use in the centerline momentum balance, or it may require the use of a Lagrangian framework for the target linked with the
projectile treated in an Eulerian framework. This
approach would allow for the computation of elastic
in-plane stresses within the target. (This approach
may also be best for very thin metal plates.) Dealing
with these materials is proving difficult, not just for
analytic models but also for large-scale numerical
simulations. Once successful constitutive models
are developed, it should be possible to apply them in
the analytic modeling framework. Damage is also
more complicated, since, in addition to deformation,
fiber breakage can occur.
9. h/R ratio for fabrics during impact. When a
fabric is impacted with a projectile, a pyramid is
formed in the fabric as it deflects and deforms. The
edges of the pyramid run in the direction of the fibers. An outstanding problem in modeling fabrics is
determining, from first principles, the ratio of the
height to the radius of the base of the pyramid (h/R).
It appears from experiments that this ratio is constant during the penetration, but that fact has yet to
be shown through modeling. The model described
in [7] assumes a constant h/R value. The model is
very successful at predicting the ballistic limit of a
fabric based on its fiber density, Youngs modulus,
and failure strain. However, the model predicts
smaller h/R than are seen in experiments; experimentally, h/R ~ 2/3, while the model predicts -1/3 or
less. The reason for the difference is most likely due
to looseness and crimping in the fabric, but this has
yet to be demonstrated.
COUPLED PROJECTILE/TARGET
MODELING PROBLEMS
Finally, the last two problems require the simultaneous consideration of the projectile and the target.
10. Crater diameter. An explicit equation for
crater diameter has proven elusive. In [1] an experimental curve fit was assumed universal for all
materials to provide a crater diameter based on impact velocity. However, experimentally it is observed that the crater diameter decreases with penetration depth. It is straightforward to include in the
analytic model a time (or depth) dependent crater
diameter. The equations for crater diameter clearly
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optimization of today's light armors comprised of
ceramic tiles attached to composite plates backed by
loose fabric.
In conclusion, analytic penetration models can be
expected to become more capable and address more
complicated targets, projectiles, and impact geometries in the future. Solution of the problems presented in this paper will allow more detailed examination and optimization of armors.
will depend on both the strengths and densities of
the target and projectile. Failure strain in the projectile also comes into play in the diameter of the
formed crater. There is an additional level of need
in crater diameter modeling. In order to analytically
model certain projectile/target interactions, it is important to have a dynamic crater growth model that is, the model should include the transient motion of the crater wall as it moves outwards to its
final diameter.
11. Projectile rigid/eroding transition. A holy
grail of penetration modeling: When does a projectile penetrate as a rigid body, and when does it
erode, and what is the velocity that demarcates the
two regimes? To be able to determine this analytically would be a major achievement. Current models do a transition from eroding to rigid penetration
based on their calculations of front and tail velocities, and when the equations produce a larger value
for the front velocity u than for the back velocity v,
then it is assumed the penetration is rigid. However,
such an approach does not accurately predict
whether a striking projectile will initially penetrate
in a rigid fashion or will erode. Currently, a different model is used from the outset if it is known the
projectile does not erode throughout the whole
penetration event.
ACKNOWLEDGMENTS The author thanks those
he has worked with in terminal ballistics over the
years, particularly Charles Anderson of Southwest
Research Institute.
REFERENCES
1. Walker, J.D. and Anderson, Jr., C.E., Int. J. Impact
Engng 16, pp. 19-48(1995).
2. Tate, A., /. Mech. Phys. Solids 15, pp. 387-399 (1967).
3. Forrestal, M., Okajima, K. and Luk, V.K., J. Appl.
Mech. 55(4), pp. 755-760 (1988).
4. Walker, J.D., "Hypervelocity Penetration Modeling:
Momentum vs. Energy and Energy Transfer Mechanisms," Int. J. Impact Engng to appear (2001).
5. Walker, J.D., and Anderson, Jr., C.E., "An Analytic
Penetration Model for a Drucker-Prager Yield Surface
with Cutoff," in Shock Compression in Condensed
Matter-1997, AIP Conference Proc. 429, New York,
pp. 897-900(1998).
6. Walker, J.D., "An Analytic Velocity Field for Back
Surface Bulging," Proc., 18th Int. Ballistic Symp., pp.
1239-1246(1999).
7. Walker, J.D., "Constitutive Model for Fabrics with
Explicit Static Solution and Ballistic Limit," Proc.,
I8h Int. Ballistic Symp., pp. 1231-1238 (1999).
8. Anderson, Jr., C.E., Walker, J.D., Bless, S.J., and
Partom, Y., Int. J. Impact Engng 18, pp. 247-264
(1996).
9. Walker, J.D., and Anderson, Jr., C.E., "An Analytical
Model for Ceramic Faced Light Armors," Proc., 16th
Int. Ballistic Symp. 3, pp. 289-298 (1996).
10. Chocron, S., Grosch, D.J., and Anderson, Jr., C.E.,
"DOP and V50 Predictions for the 0.30-Cal APM2
Projectile," Proc., l$h Int. Ballistic Symp., pp. 769-776
(1999).
11. Anderson, Jr., C.E., Littlefield, D.L., and Walker, J.D.,
Int. J. Impact Engng 14, pp. 1-12 (1993).
12. Walker, J.D., "On Maximum Dissipation for Dynamic
Plastic Flow," Proc., 15th Int. Ballistic Symp.l, pp. 6774(1995).
13. Walker, J.D. and Anderson, Jr., C.E., Int. J. Impact
Engng 15(2), pp. 139-148 (1994).
14. Ravid, M., and Bodner, S.R., Int. J. Engng Sci. 21(6),
pp. 577-591 (1983).
CONCLUSIONS
There are of course more problems, but these ones
are central. For example, it is likely that a solution
to the crater diameter problem (#10) and the transition in stress state within the projectile problem (#1)
will solve the L/D effect problem within the current
model. Also, solution of the transient crater diameter problem (#10) and the projectile side loading
problem (#3) will make it possible to solve complicated, 3D projectile/target interaction problem. For
example, with that additional information, it should
be possible to model oblique and yawed impacts.
Adding a good breakout model (#6) will provide all
the pieces required for the interaction of rods with
armors comprised of dynamically moving plates.
Such modeling will allow examination and optimization of modern armors employing plates at angle
and in motion. Finally, solution of the ceramic failure time problem (#7), the nonflowing-target resistance problem (#8) and the fabric problem (#9) coupled with modeling complex projectiles on the centerline (#2) would allow a detailed examination and
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