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Generating Penetration Resistance Functions With a Virtual Penetration Laboratory (Vpl) - Applications to Projectile Penetration and Structural Response Simulations
Citation preview
GENERATING PENETRATION RESISTANCE
FUNCTIONS WITH A VIRTUAL PENETRATION
LABORATORY (VPL): APPLICATIONS
TO PROJECTILE PENETRATION AND
STRUCTURAL RESPONSE SIMULATIONS
MARK D. ADLEY*, ANDREAS O. FRANK, KENT T. DANIELSON,STEPHEN A. AKERS and JAMES D. CARGILE
U.S. Army Engineer Research and Development Center
ATTN: CEERD-GM-I 3909
Halls Ferry Road Vicksburg, MS 39180-6199 USA*[email protected]
BRUCE C. PATTERSON
U.S. Air Force Research Laboratory Munitions DirectorateEglin AFB, FL
STEPHANIE TERMAATH
Applied Research Associates, Inc. 6320 Southwest Blvd.
Fort Worth, TX
Received 27 September 2008
Accepted 5 December 2010
A new software package called the Virtual Penetration Laboratory (VPL) has been developed
to automatically generate and optimize penetration resistance functions. We have used this
VPL code to generate highly \tuned" penetration resistance functions that can distinctly model
the penetration trajectory of steel projectiles into rate-independent, elastic-perfectly plasticaluminum targets. Projectiles with arbitrary nose geometry were considered in this example (i.e.
conical, ogival, and spherical nose shapes). The penetration resistance of the aluminum target
was determined by numerically solving a series of spherical and cylindrical cavity expansionproblems. The solution to these cavity expansion problems were obtained with an explicit,
dynamic ¯nite element code that accounts for material and geometric nonlinearities. The
resulting cavity expansion equations are then transformed to penetration resistance functions
using various transformation algorithms, in order to determine an appropriate method tospatially distribute the resisting stresses on the projectile nose. The resulting penetration
resistance functions were then used in a penetration trajectory code to predict the actual
trajectories observed from a set of similar experiments.
Keywords: Penetration mechanics; constitutive modeling; cavity expansion.
*Corresponding author.
International Journal of Structural Stability and DynamicsVol. 12, No. 4 (2012) 1250024 (25 pages)
#.c World Scienti¯c Publishing Company
DOI: 10.1142/S0219455412500241
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1. Background
1.1. Introduction
The penetration mechanics group at the U.S. Army Engineer Research and Devel-
opment Center (ERDC) conducts research investigating the process of projectile
penetration. Research results are used to improve the predictive capability of pen-
etration trajectory algorithms. The research e®orts include: (a) extensive material
property experiments that provide stress, strain, and strength data for ¯tting con-
stitutive models,1 (b) development / modi¯cation and validation of complex con-
stitutive models such as the Nonlinear-Inelastic-Fracture (NIF) model and the
Microplane model,2,3 (c) an active projectile penetration experimental program that
is conducted at the ERDC 83mm ballistic research facility,4 (d) and the develop-
ment of computational algorithms that transfer the knowledge gained to user
friendly software packages used for the prediction and analysis of projectile pen-
etration problems.5
Over the past few decades, many \hydrocodes" have been developed to simulate
various impact and penetration events. Typically these codes employ either an
Eulerian, Lagrangian, or \mixed" formulation. The mixed formulation might be an
Arbitrary Lagrange Euler (ALE) formulation, or a coupled formulation that links a
Lagrangian code with an Eulerian code, or a Lagrangian code with an analytical code
that models penetration resistance. Some of these codes include CTH6 an Eulerian
based code, PRONTO,7 DYNA3D,8 EPIC,9 and ParaAble10 which are Lagrangian
codes.
Eulerian based codes allow the material to °ow through the mesh and thereby can
handle large material deformations more easily than Lagrangian based codes. These
codes can be very useful for modeling the large deformations often observed in the
target material. Lagrangian based explicit ¯nite element (FE) codes use a mesh that
is attached to the material under consideration and are very popular for simulating
the structural response of the projectile to impact and penetration loads. These codes
behave reasonably well for simulating a penetration event when plastic deformations
of the projectile are not excessive and erosion of the projectile is not a primary
concern. However, they can have some di±culty when simulating the target behavior
during penetration. This is largely due to the excessive element distortion and
material damage that occurs in the target. In particular, target material break-up
and subsequent removal can be di±cult to predict and must typically be legislated in
advance using specialized algorithms. Alternatively, the regions of high deformation
within the target can be treated with Eulerian based codes, constant mesh rezoning
and re¯nement, or a variety of meshless methods. There are a number of sophisti-
cated meshless methods currently available,11,12 and some of these methods have
been used successfully to solve projectile penetration problems13 as well as problems
involving the high strain-rate fracturing/fragmentation behavior of concrete slabs.14
There are also numerical methods available that involve the coupling of meshless
methods with FEs.9,15 However, the computational cost of treating the target on a
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fully ¯rst principles basis in any of these available methods can be extremely high
and at times excessive, especially for fully three-dimensional complex geometries.
Several of the aforementioned approaches are so computationally intensive that it
is not practical to apply them to problems such as projectile design calculations,
which often require numerous iterations and trade studies on material properties and
projectile geometry. For those types of problems a faster running methodology is
required. Early attempts to solve that dilemma involved the use of penetration
resistance functions with a separate rigid-body trajectory analysis to uncouple the
projectile/target interaction and greatly reduce the size and complexity of the FE
analysis.16 This approach looked promising because penetration resistance functions
have been used successfully in rigid-body penetration codes to predict the trajectory
of projectiles impacting complex targets.5 In this approach, the time-history of the
penetration loads are determined by a rigid projectile penetration analysis that
models the target resistance with penetration resistance functions. These loads are
then used as input to a dynamic FE code. A shortcoming of this approach is that the
computed penetration loads are not a function of the projectile's structural response.
Consequently, various e®orts have been made to couple the loads predicted by
penetration resistance functions with deformable projectile models using beam,17
shell,18,19 and solid20,21 FEs. These e®orts have led to the development of compu-
tational tools for the analysis of projectiles subjected to the intense loading histories
that occur during impact and penetration. The penetration resistance functions can
provide the response of the target during a penetration event using purely analytical
expressions and therefore do not require the target response to be solved on a ¯rst
principles basis. In many instances this approach reduces the computational
resources by orders of magnitude while preserving the ¯delity of the impact and
penetration loading histories. The paper under consideration contains a discussion of
a process that can be used to develop penetration resistance functions for use with
either rigid-body penetration trajectory codes or in the framework of a deformable
body FE code.
As discussed in the previous paragraph, using penetration resistance functions to
model the target in projectile penetration simulations has proven to be a very
e®ective method of solving penetration problems. However, one potential short-
coming of this approach is that the penetration resistance functions are usually based
on very limited material property data, e.g. the uncon¯ned compressive strength and
the mass density of the concrete. This restriction has not been a serious limitation in
the past because most of the target materials of interest have been very similar.
Therefore, it was possible to calibrate the parameters in the penetration resistance
function to penetration data and then use that equation for a number of similar
targets. However, due to the wide variety of target materials now under consider-
ation, the approach of basing target resistance on very limited material property
data is no longer adequate. The work presented in this paper is aimed at correcting
this de¯ciency by providing a methodology and a software package that allows
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analysts to quickly develop penetration resistance functions that are based on a more
complete set of material property data. Speci¯cally, the VPL code allows an analyst
to use a sophisticated constitutive model with parameters that have been determined
by reproducing stress�strain data from a number of material property experiments,
e.g. uniaxial strain, uncon¯ned compression, triaxial compression, hydrostatic
compression, etc. Therefore, penetration resistance functions developed with the
VPL code are a function of many of the factors that de¯ne the behavior of a material,
e.g. bulk moduli (initial, tangent, locking), shear modulus, uncon¯ned compressive
strength, Mises limit strength, characteristics of the pressure-volume response and
the undamaged and damaged failure surface, etc. This capability to quickly develop
high-¯delity penetration resistance functions based on sophisticated material models
and detailed material property data is the important contribution represented by the
VPL software package.
1.2. Overview of virtual penetration laboratory (VPL) methodology
The method used in this paper for developing penetration resistance equations are
based on an analogy between the penetration problem and the cavity expansion
problem. This analogy has been used successfully by a number of researchers to
model various penetration problems.22�26 However, the methodology adopted in the
vast majority of this research has involved the use of very simple constitutive models
and other simplifying assumptions that allow the derivation of a closed form solution
to the cavity expansion problem. One notable exception to that trend is the recent
work of Warren, Fossum and Frew.27
The methodology presented in this paper uses the FE method to solve the
equations governing the expansion of a cavity in a given target material. The ¯rst
step in the process involves the solution of a series of one-dimensional cavity
expansion problems where each problem is de¯ned as the opening of a spherical or
cylindrical cavity at a speci¯ed constant expansion velocity. Each cavity expansion
solution is run until the radial stress (normal to cavity wall) approaches a constant
value. Each of these solutions then represents a point in radial stress versus radial
velocity space. The next step in this process involves the determination of a quad-
ratic function that represents the best approximation to that family of points in a
least squares sense. This resulting quadratic equation then provides an expression
for determining the cavity expansion resistance of the target material as a function
of velocity.
The ¯nal step requires a transformation of the cavity expansion resistance
equation to a penetration resistance function. This is where the analogy between
cavity expansion and penetration resistance becomes the critical link in this process.
Speci¯cally, it is necessary to somehow correlate the velocity at the nose of the
projectile during penetration with the cavity expansion velocity. However, due to
various fundamental geometric di®erences between the cavity expansion problem
and the penetration that occurs along the nose of a projectile, there is no unique or
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distinct way to link these two problems. For example, the cavity expansion problem
necessarily has either a spherical or cylindrical geometry. However the penetration
problem has a nose shape associated with the projectile that is distinctly 3D, which
cannot be described as uniquely spherical or cylindrical (as shown in Fig. 1).
Thereby, some simpli¯cations must be made as to how we can transform the cavity
expansion velocity (which is always normal to the cavity surface), into the local
velocity along the projectile nose (which is seldom normal to the cavity surface).
One way to accomplish this is by replacing the radial cavity expansion velocity
with the component of the projectile's velocity that is normal to the surface of the
projectile at the point under consideration (i.e. the center of an element face that is
located in the outer surface of the mesh of the projectile). The resisting stress pre-
dicted by the transformed equation is then interpreted as the normal stress that is
acting on the surface of the projectile at the point under consideration (i.e. the
penetration resistance of the target as a function of velocity). This penetration
resistance function can then be implemented into a trajectory code to provide pen-
etration predictions. Due to various uncertainties in this process, a number of
numerical simulations based on penetration experiments should be conducted in
order to determine the ¯delity of a given penetration resistance function.
The aforementioned process has been automated in a software package Virtual
Penetration Laboratory (VPL) recently developed by ERDC. One of the motivating
factors behind this e®ort is the extensive library of advanced constitutive models and
Fig. 1. Schematic showing the geometrical di®erences between the projectile penetration problem, andeither the cylindrical or spherical cavity expansion problem.
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the associated material ¯ts available at ERDC, e.g. the Advanced Fundamental
Concrete (AFC) model28,29 and the M4 microplane model.3,30�32 The ERDC library
of constitutive models contains a number of sophisticated models that simulate
complex material behaviors such as the brittle-to-ductile transition, post-peak soft-
ening, strain-rate e®ects, pore collapse, etc. Combining the ERDC library of con-
stitutive models/¯ts with existing ERDC trajectory code algorithms, in a fast
running automated software package (VPL), provides ERDC researchers with a
powerful tool to develop high-¯delity penetration resistance functions. The VPL
code can be used for new target materials of interest, as well as improving pen-
etration resistance functions for commonly used target materials such as concrete.
The VPL code will be also useful as a research tool to study the e®ect of the level of
sophistication of a material model, as well as the parameter values used in the model,
on the resulting penetration resistance equation.
1.3. Overview of results
We have provided a detailed discussion of the VPL methodology, along with a
demonstration of its use in simulating actual penetration experiments. We have
demonstrated the development of various penetration resistance functions for arbi-
trary nose shaped projectiles penetrating aluminum targets. We have considered
various methods to link the cavity expansion geometry with the projectile nose shape
geometry and discuss their di®erences. We also have compared calculations from the
derived penetration resistance functions with available experimental data.
2. Example of Virtual Penetration Laboratory (VPL) Experiments
2.1. Motivation and material models
Most analytically determined penetration resistance functions have been developed
from relatively simple constitutive models.22�26 However, there is a set of more
complex nonlinear constitutive models available in most wave propagation codes.6,9
These models can simulate, with varying degrees of success, the response of materials
under extreme loading environments. Some of these models can capture the funda-
mental and often complex mechanical behaviors required to accurately simulate
penetration problems. In order to provide a high-¯delity solution to the cavity
expansion problem, researchers at the ERDC have recently implemented several of
these more complex constitutive models into the large-strain Lagrangian cavity
expansion code Virtual Cavity Expansion (VCE). The models available in VCE
include, Johnson�Cook (JC) model for metals,33 Hull model for geomaterials,9
Holmquist�Johnson�Cook model for geomaterials,34 and Microplane model M3 for
concrete.35,36 These models can be used in both spherical and cylindrical ¯nite-
element cavity expansion simulations, in order to extract the penetration resistance
of various target materials. For the example we present in this paper, we have used
the JC constitutive model to simulate penetration into aluminum targets.
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2.2. Nomenclature
G Shear modulus
P Pressure
e Speci¯c internal energy
q Arti¯cial viscosity pressure
u Displacement
"� Volume strain
� Poisson's ratio
"ij Strain tensor
!ij Spin tensor
�ij Total stress tensor
sij Deviatoric stress tensor
�ij Kronecker delta
2.3. Virtual cavity expansion (VCE) ¯nite element code
The large-strain Lagrangian cavity expansion code VCE utilizes the FE method to
solve the equations governing the behavior of solids subjected to large magnitude
short-duration load histories. The governing equations consist of the conservation
equations, the strain-displacement equations, and the constitutive equations.37,38
The conservation of mass equation is written as
�: þ � _ui;i ¼ 0: ð1Þ
Equation (1) is actually embodied in the determinant of the deformation gradient
since VCE is a Lagrangian code. The conservation of linear momentum equation is
expressed as
�€ui ¼ �ji;j þ �fi: ð2ÞThe conservation of angular momentum (assuming nonpolar media) results in the
following equation:
�ij ¼ �ji: ð3ÞFinally, the conservation of energy equation (assuming adiabatic conditions) is
written as
_e ¼ 1
��ijDij: ð4Þ
The strain-displacement relations are satis¯ed by solving the velocity strain
equations:
":ij ¼
1
2ð _ui;j þ _uj;iÞ
!ij ¼1
2ð _ui;j � _uj;iÞ
: ð5Þ
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The stress tensor is split into deviatoric and hydrostatic components. The deviatoric
components are computed as follows:
_sij ¼ 2G ":ij �
1
3�ij _uk;k
� �: ð6Þ
The hydrostatic component of stress (pressure) is given by
P ¼ fðe; "vÞ; ð7Þwhere the pressure P is de¯ned as a function of internal energy and volumetric strain
by the equation of state. The total stress tensor is computed as follows:
�ij ¼ sij � �ijðP þ qÞ: ð8ÞFinally, the Jaumann stress rate equation is given by
�̂ij ¼ �:ij þ �im!mj � !im�mj ð9Þ
VCE utilizes the FE method to accomplish the spatial integration of the governing
partial di®erential equations; the temporal integration is performed with an
explicit ¯nite di®erence scheme. VCE uses an updated Lagrangian Jaumann (ULJ)
kinematic formulation to handle the large displacements, large rotations, and large
strains present in the types of problems under consideration.39 The ULJ formulation
utilizes the true Cauchy stress as the measure of stress, the velocity strains (":ij) or
rate of deformation tensor (Dij) as the strain measure, and the Jaumann stress rate
tensor as the objective stress rate. The frame of reference used in the ULJ formu-
lation is the current con¯guration of the body. Implementing a new material model
in VCE involves replacing Eqs. (6) and (7) with the equations that represent the
new model.
2.4. Material property data and the model ¯t (Aluminum)
For the example presented in this paper we have used the VPL code to simulate
previously conducted penetration experiments into aluminum targets.40 The JC
model is ¯t to the material property values that are appropriate for 6061-T651
aluminum bars. The target is modeled as a rate-independent, elastic-perfectly plastic
material. The material values, which are provided in the previous reference, are:
poisson's ratio of 0.33, a yield stress of 400MPa, a bulk modulus of 69GPa, and a
density of 2,707Kg/m3. Since the aforementioned material constants de¯ne an
elastic-plastic material with a constant °ow stress, the JC material constants are
simpli¯ed signi¯cantly. Speci¯cally, all of the parameters de¯ning the JC9 strength
are zero with the exception of cohesive strength parameter (C1 ¼ 400Mpa), and the
equivalent plastic strain exponent (N ¼ 1:0). The shear modulus is 27.5GPa. Since
damage accumulation and element failure were not considered in the cavity expan-
sion simulations, the only remaining nonzero constants are the bulk modulus
(K1 ¼ 69GPa), and the Gruneisen coe±cient (� ¼ 2:0).
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2.5. Virtual cavity expansion (VCE) simulations
The VCE cavity expansion code was used to conduct a series of spherical and
cylindrical cavity expansion simulations. The simulations generated a family of
points in radial stress versus radial velocity space, which represents the stress
required to open a cavity in the given target material. Since the VCE code is a
one-dimensional code the mesh simply consists of 200 equal length bar elements.
Constant velocity boundary conditions are applied to the surface of the cavity, and a
soaker element is employed at the far end of the mesh to create a nonre°ective
boundary.
In order to run the cavity expansion simulations, the VPL input ¯le requires the
minimum and maximum desired cavity expansion velocities and the total number
(n) of velocities to compute. This allows the user to determine the velocity regime
of interest and the resolution between velocity increments (i.e. constant velocity
increments with n total calculations). The VPL code will then automatically run
the VCE code, twice for each cavity expansion velocity under consideration (i.e. one
spherical and one cylindrical cavity expansion simulation). Each cavity expansion
simulation is carried out until the radial stress at the surface of the cavity has
achieved a near constant value. From these simulations, a function representing the
¯nal radial stress versus velocity can be determined. This function is determined
using the least squares method by ¯tting a quadratic equation to the points in
radial stress versus velocity space. Two quadratic ¯ts are determined, one for the
spherical and another for the cylindrical cavity simulations. It should be noted that
the VPL code also o®ers other least squares ¯tting options if desired (i.e. the
analyst can choose to set the linear term equal to zero, or the quadratic term equal
to zero).
The example shown in this paper considered 11 cavity expansion velocities that
were equally distributed between 15m/s and 1,500m/s. We have shown the resulting
value of radial stress as a function of time for the cylindrical and spherical cavity
expansions at a velocity of 610m/s (Fig. 2). The entire family of points generated by
all of the cylindrical cavity expansion simulations is shown in Fig. 3, along with the
quadratic function that best ¯ts those points in a least squares sense. Likewise, the
family of points generated by the spherical cavity expansion simulations is shown in
Fig. 4, along with their ¯tted quadratic function. It can be seen that these quadratic
equations provide a good ¯t to the data. It should be noted that the equations shown
in Figs. 3 and 4 provide a means of interpolating between the constant velocity
cavity expansion simulations, i.e. the velocity of a projectile is not constant during a
penetration event, so this interpolation is required in order to compute the resistance
stress for any velocity that occurs during the penetration event.
It is interesting to note that the normal stress values predicted by the cylindrical
cavity expansion model for the lowest expansion velocity is �20% lower than the
value predicted by the spherical equation, whereas at the highest expansion velocity
the trend is reversed and the normal stresses predicted by the cylindrical model
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are �30% larger than the spherical model. A careful study of the cavity expansion
results also reveals that the character of the cylindrical ¯t is more linear than the
spherical ¯t as shown by the plots and the relative magnitudes of the coe±cients of
the linear terms (B parameters). These characteristics can have a signi¯cant e®ect on
the time history of the projectile loads when these equations are used as the basis for
penetration resistance functions.
0
2000
4000
6000
8000
10000
12000
0 200 400 600 800 1000 1200 1400 1600Velocity (m/s)
No
rmal
Str
ess
(MP
a)
quadratic fitcavity expansionsolutions
Fig. 3. Cylindrical cavity expansion solutions with quadratic ¯t.
0
1
2
3
4
5
6
7
8
9
10
-0.01 0 0.01 0.02 0.03 0.04 0.05time (msec)
Rad
ialS
tres
s / Y
ield
Str
eng
th
cylindricalspherical
Fig. 2. Radial stress versus time generated by the VCE code for cylindrical and spherical cavity expansion
at a velocity of 610m/s.
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2.6. Transforming cavity expansion equations
into penetration resistance functions
The least squares ¯ts to the cavity expansion solutions presented in the previous
section must somehow be transformed into penetration resistance functions, in order
to facilitate penetration trajectory simulations for our target material (i.e. alumi-
num). This can be done by ¯nding or substituting a suitable component of the local
velocity at the nose of the projectile for the given cavity expansion velocity. However,
due to the geometric mis-match between the projectile nose and the cavity expansion
simulations (Fig. 1), this is not necessarily unique and can be done in several ways.
Nonetheless, once the cavity expansion equations have been transformed, the pen-
etration resistance functions provide the normal stress distribution acting on the
nose of the projectile.
2.7. Penetration resistance functions and the spatial
stress distribution along the projectile nose
Now we shall discuss the transformation of the cavity expansion equations into
penetration resistance functions in more detail. Speci¯cally, we will describe some
possible ways to accomplish this and how that a®ects the resulting spatial normal
stress distribution acting on the nose of the projectile.
Experimentally measuring the spatial (and temporal) distribution of normal
stress that occurs on the nose of a projectile during a penetration event has been an
elusive thing. Hence, this spatial distribution is not exactly known and has been an
ongoing e®ort for many researchers in the penetration mechanics ¯eld. However, this
quadratic fitcavity expansionsolutions
0
2000
4000
6000
8000
10000
0 200 400 600 800 1000 1200 1400 1600Velocity (m/s)
No
rmal
Str
ess
(MP
a)
Fig. 4. Spherical cavity expansion solutions with quadratic ¯t.
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distribution can be assumed. For example, several di®erent spatial distributions of
the normal stress have been documented in previous studies.23,25,41
In order to provide maximum °exibility in evaluating various possibilities for the
spatial distribution of normal stress acting on the nose of the projectile, the VPL code
o®ers six di®erent algorithms for distributing this stress. These six di®erent algor-
ithms can be separated into two distinct groups, as follows: (1) direct substitution
algorithms and (2) nose performance algorithms. Speci¯cally, these six options are as
follows: (1) cylindrical cavity expansion by direct substitution, (2) spherical cavity
expansion by direct substitution, (3) combined cylindrical and spherical cavity
expansion by direct substitution, (4) cylindrical cavity expansion with nose per-
formance factor, (5) spherical cavity expansion with nose performance factor, and (6)
combined cylindrical and spherical cavity expansion with nose performance factor.
These options are discussed in detail below.
2.7.1. Direct substitution algorithms
The ¯rst group of transformations (direct substitution algorithms) assumes a direct
correlation can be drawn between the cavity expansion velocity and the local
velocity along the projectile nose. This can be done by assuming that the normal
stress distribution can be represented by replacing the radial cavity expansion
velocity (V ) with the particle velocity (Vn) at the projectile-target interface. The
particle velocity Vn is computed by taking the scalar product between the local
velocity vector and the unit outer normal vector to the projectile nose at the point
under consideration (Fig. 1). Note that employing this assumption for the cylind-
rical cavity expansion equation implies that the side of the cylinder is tangent to
the surface of the projectile at the point under consideration. This interpretation
was employed in order to avoid using the anomalous equations that result from
assuming the cylinder is aligned with the axis of the projectile (i.e. as the nose of
the projectile becomes blunt the magnitude of the normal stress becomes
unbounded).
Transformation option one: The penetration resistance function based on the
cylindrical cavity expansion equation is obtained by simply substituting the normal
velocity (Vn) for the radial cavity expansion velocity, as follows:
�c ¼ Ac þ BcVn þ CcV2n : ð10Þ
It should be noted that the coe±cients A, B, and C are not dimensionless in any of
the penetration resistance functions described in this section. This allows for a
simpler form of these equations.
Transformation option two: As with option one, the penetration resistance function
based on the spherical cavity expansion equation is obtained by simply substituting
the normal velocity (Vn) for the radial cavity expansion velocity, as follows:
�s ¼ As þ BsVn þ CsV2n : ð11Þ
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Neither option one or two described above, make any considerations for matching
the local geometry of the projectile nose with the geometry considered in the cavity
expansion problem (i.e. cylindrical or spherical).
Transformation option three: The penetration resistance function is obtained by
substituting the normal velocity (Vn) for the radial expansion velocity into an
equation that is computed as a weighted average of the cylindrical and spherical
equations. This combined equation then becomes a function of the local shape of the
projectile nose at the point of interest. Thereby this provides a geometrically more
consistent method for transforming the cavity expansion equations to penetration
resistance functions. For example (Fig. 1), the cylindrical cavity expansion equation
becomes dominant near the shoulder of the projectile nose (i.e. where the cylindrical
stress distribution is more geometrically appropriate), and the spherical cavity
expansion equation becomes more dominant near the nose tip (i.e. where the
spherical stress distribution is more geometrically appropriate).
This option allows for greater °exibility in penetration problems involving pro-
jectiles with arbitrary nose shapes. The following method of combining the cylind-
rical and spherical cavity expansion equations has been considered;
� ¼ �ccos2�þ �ssin
2�; ð12Þwhere � is the penetration resistance stress normal to the surface of the projectile
node, �c is the resisting stress computed from cylindrical cavity expansion, �s is the
resisting stress computed from spherical cavity expansion, and � is the angle between
the surface of the projectile and a tangent that is parallel with the axis of the
projectile (Fig. 5). Notice that � is zero at the shoulder where the projectile nose
meets the aft-body.
One method to combine the cylindrical and spherical resisting stresses assumes
that the surface of the spherical cavity and the side surface of the cylindrical cavity
Projectile
θ
Nose
Point ofInterest
Projectile
θ
Nose
Point ofInterest
Fig. 5. Schematic showing the angle theta (�) used to combine the cylindrical and spherical cavity
expansion equations.
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are each tangent to the surface of the projectile at the point under consideration.
This assumption necessarily dictates that the velocity used in both cavity expansion
equations becomes the component of velocity that is normal to the surface of the
projectile nose (Vn), as follows:
Vn ¼ V sin �; ð13Þwhere V is the velocity of the projectile.
By substituting the appropriate cavity expansion equation (Eqs. 10 and 11)
into the assumed normal stress distribution Eq. (12) we can get the following
penetration resistance equation for combined cylindrical and spherical cavity
expansion;
� ¼ ðAccos2�þ Assin
2�Þ þ ðBccos2�þ Bssin
2�ÞVn þ ðCccos2�þ Cssin
2�ÞV 2n ; ð14Þ
where the parameters subscripted with a c refer to the cylindrical cavity expansion
equation, and parameters subscripted with an s refer to the spherical cavity
expansion equation.
2.7.2. Nose performance algorithms
The second group of transformations (nose performance algorithms) is similar to the
¯rst group but includes the use of a nose performance factor. Speci¯cally this method
assumes that the normal stress distribution along the nose of the projectile can be
represented by replacing the radial cavity expansion velocity (V ) with the resultant
local velocity (Vres) at the projectile-target interface. The nose performance factor (a
function of the local nose shape at the point of interest) is then multiplied by the two
dynamic terms in the cavity expansion equation. The nose performance factor cur-
rently used in the VPL code has been predetermined as sin(�).
Transformation option four: This penetration resistance function is based on the
cylindrical cavity expansion equation and includes the nose performance factor. It is
obtained by substituting the resultant projectile velocity (Vres) for the radial
expansion velocity into the cylindrical equation, as follows:
�c ¼ Ac þ ðBcVres þ CcV2resÞ sin �: ð15Þ
Transformation option ¯ve: This penetration resistance function is based on the
spherical cavity expansion equation and includes the nose performance factor. It is
obtained by substituting the resultant projectile velocity (Vres) for the radial
expansion velocity into the spherical equation, as follows:
�s ¼ As þ ðBsVres þ CsV2resÞ sin �: ð16Þ
Transformation option six: This penetration resistance function is based on the
combined cylindrical and spherical cavity expansion equation Eq. (12) and includes
the nose performance factor. It is obtained similar to transformation option three
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described above, except that it considers the resultant projectile velocity (Vres) and
includes the nose performance factor, as follows:
� ¼ðAccos2�þ Assin
2�Þ þ ððBccos2�þ Bssin
2�ÞVres
þ ðCccos2�þ Cssin
2�ÞV 2resÞ sin �: ð17Þ
2.8. Implementation of penetration resistance functions
into a penetration trajectory code
In order to use one of the derived penetration resistance functions (described above)
in a penetration trajectory code, the computed stress is interpreted as a boundary
value acting on the projectile (i.e. the normal stress acting on the surface of the
projectile). Thereby we have linked the cavity expansion equation (which describes
the target resistance) to the projectile motion or structural response via the pen-
etration resistance function.
For example, the projectile is discretized with a surface mesh that distinctly
describes the projectile geometry, as shown in Fig. 6. From the projectile surface
mesh, the normal component of velocity acting at the center of each element face can
then be computed. The given penetration resistance function is then evaluated to
determine the normal stress acting on each element face of the projectile. The force
and moment contributions can then be computed by integrating the stresses
spatially over the surface of the projectile. Once the total forces and moments have
been determined, they can be substituted into the six equations of motion. The
equations of motion are then integrated temporally with a ¯nite di®erence scheme, in
order to determine the trajectory of the projectile. This is done automatically within
Fig. 6. Rigid-body projectile model with surface mesh for applying normal stress (i.e. penetration resist-
ance function).
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the VPL software package. Thereby a series of projectile penetration problems can be
solved in order to evaluate various details of the penetration behavior of a given
projectile. For example, the penetration depth as a function of impact velocity can be
quanti¯ed, among other things.
3. Penetration Experiments
For the example presented in this paper we have used the VPL code to simulate
previously conducted penetration experiments into aluminum targets.40 This has
allowed us to compare our simulation results with a thorough set of penetration data.
We shall only brie°y describe this set of experimental data herein.
Projectile experiments for three di®erent projectile nose shapes were conducted,
as follows: (1) Conical nose shape, (2) Ogival nose shape, and (3) Spherical nose
shape. The projectiles were made from high-strength steel (T-200 and C-300,
maraging steel) and had a diameter of 7.10mm (as shown in Fig. 7). The targets
were made form aluminum bars (6061-T651) and were 152mm in diameter. This
provided a target width-to-projectile diameter ratio of more than 21. The projectiles
were shot at normal incidence into the targets with impact velocities ranging from
0.4 to 1.4 km/sec. The projectiles were not seen to signi¯cantly erode during the
penetration event and had a post-test nose geometry that was similar to the pre-test
geometry. The results from these penetration experiments (i.e. penetration depth
versus impact velocity) will be used as a measure to examine the penetration
simulations from the VPL code.
Spherical Nose
Ogival Nose
Conical Nose
Fig. 7. Projectiles used for penetration simulations (conical, ogival, and spherical nose shapes).23
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4. Penetration Simulations using the VPL Code
We have provided penetration simulations of the experimental data brie°y described
above in Sec. 3.40 We have focused our e®orts in order to highlight the use of the
VPL code as a research tool. Speci¯cally, we have intended to examine the form of
the penetration resistance function or resulting normal stress distribution on the
projectile nose. This was accomplished by providing simulations using both the
cylindrical and spherical cavity expansion data ¯ts according to Eqs. (10), (11), (15)
and (16) described above. We have then compared the computed penetration depth
versus impact velocity to the available experimental data.
4.1. Spherical nose penetration simulations
First, we shall discuss the spherical nose penetration simulations, since this pro-
jectile geometry provides the best geometric link between the cavity expansion
calculations and the penetration calculations (see Figs. 1 and 7). For example, we
can hypothesize that the direct substitution spherical cavity ¯t Eq. (11) should
provide a good result for our penetration predictions and this is in fact the case. We
have plotted the results from the spherical nose penetration simulations for the
direct substitution algorithms (i.e. Eqs. (10) and (11)) in Fig. 8(a) and the results
for the nose performance algorithms (Eqs. (15) and (16)) in Fig. 8(b). It can be seen
that the spherical ¯t provides a better comparison to the experimental data.
However, it should be noticed that the cylindrical ¯t is still reasonable. Also it
0
5
10
15
20
25
30
250 500 750 1,000 1,250 1,500
Velocity (m/s)
Pen
etra
tio
n D
epth
(cm
)
cylindrical fitspherical fitexperimental data
(a) Direct substitution algorithms
0
5
10
15
20
25
30
Pen
etra
tio
n D
epth
(cm
)
cylindrical fitspherical fitexperimental data
250 500 750 1,000 1,250 1,500
Velocity (m/s)
(b) Nose performance algorithms
Fig. 8. Simulations of spherical nose penetration experiments as follows: (a) using direct substitution
algorithms and (b) using nose performance algorithms.23
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should be noted that the direct substitution algorithms provide a slightly greater
depth of penetration than the nose performance algorithms.
4.2. Ogival nose penetration simulations
Now we shall examine the ogival nose penetration simulations. These simulations
clearly provide a geometric mis-match between the cavity expansion calculations and
the penetration calculations (see Figs. 1 and 7). The results from these simulations
for the direct substitution algorithms (i.e. Eqs. (10) and (11)) are shown in Fig. 9(a)
and the results for the nose performance algorithms (Eqs. (15) and (16)) are shown in
Fig. 9(b). As with the spherical nose penetration simulations, it can be seen that
there are only minor di®erences between the cylindrical and spherical data ¯ts.
Again, however the spherical ¯t seems to provide the best comparison to the
experimental data. Also, the nose performance algorithms now seem to more clearly
provide a better comparison to the experimental data, whereas the direct substi-
tution algorithms slightly over-predict the penetration depth.
4.3. Conical nose penetration simulations
Finally, we shall examine the conical nose penetration simulations. These simu-
lations provide the biggest geometric mis-match between the cavity expansion cal-
culations and the penetration calculations (see Figs. 1 and 7). The results from these
250 500 750 1,000 1,250 1,500
Velocity (m/s)
0
5
10
15
20
25
30
Pen
etra
tio
n D
epth
(cm
)
cylindrical fitspherical fitexperimental data
(a) Direct substitution algorithms
250 500 750 1,000 1,250 1,500
Velocity (m/s)
0
5
10
15
20
25
30
Pen
etra
tio
n D
epth
(cm
)
cylindrical fitspherical fitexperimental data
(b) Nose performance algorithms
Fig. 9. Simulations of ogival nose penetration experiments as follows: (a) using direct substitution
algorithms and (b) using nose performance algorithms.23
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simulations for the direct substitution algorithms (i.e. Eqs. (10) and (11)) are shown
in Fig. 10(a) and the results for the nose performance algorithms (Eqs. (15) and
(16)) are shown in Fig. 10(b). Again, it can be seen that there are only minor di®er-
ences between the cylindrical and spherical data ¯ts, with the spherical ¯t providing
the best comparison to the experimental data. Also, the di®erences between the nose
performance and direct substitution algorithms are seen to increase, with the nose
performance algorithms providing the better comparison to the experimental data.
4.4. Summary of penetration simulations
We have provided penetration simulations using the VPL code for arbitrary nose
shapes, including a spherical, ogival, and conical nose. Generally, the VPL code
provides good agreement with the experimental penetration data. Also only minor
di®erences were seen between using either the spherical or cylindrical cavity
expansion data ¯ts, with the spherical data ¯t providing slightly better results. This
may suggest that the form of the penetration resistance function (i.e. stress distri-
bution acting on the nose of the projectile) is not critical in evaluating the depth of
penetration. This is most likely due to the fact that the spatial stress distribution on
the nose is integrated during the penetration calculations and thereby the total force
provides the dominant in°uence on penetration resistance in the equations of motion
(see Sec. 2.8 above). This will e®ectively de-emphasize the importance of the spatial
stress distribution on the nose if the only experimental data available is the depth of
0
5
10
15
20
25
30
Pen
etra
tio
n D
epth
(cm
)
cylindrical fitspherical fitexperimental data
250 500 750 1,000 1,250 1,500
Velocity (m/s)
(a) Direct substitution algorithms
0
5
10
15
20
25
30
250 500 750 1,000 1,250 1,500
Velocity (m/s)
Pen
etra
tio
n D
epth
(cm
)
cylindrical fitspherical fitexperimental data
(b) Nose performance algorithms
Fig. 10. Simulations of conical nose penetration experiments as follows: (a) using direct substitution
algorithms and (b) using nose performance algorithms.23
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(a) (b)
(c)
Fig. 11. Example of projectile structural response calculations using the ParaAble FE code and various
forms of the penetration resistance function. Shown are the e®ective plastic strains: (a) for Eq. (18) and (b)
for Eq. (19) along with a comparison of the resulting projectile deformation, and (c) for a given exper-
imental penetration test.
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penetration. Therefore, it is very important to investigate the e®ect of nose stress
distribution on other aspects of the penetration problem such as the deceleration
time history, and the structural response of the projectile. The VPL software
package, in conjunction with a deformable-body trajectory code, is a very powerful
tool for conducting those type of investigations.
It is also interesting to note that the direct substitution algorithms seem to over-
predict the depth of penetration, where as the nose performance algorithms were seen
to provide a better comparison with the experimental data. This may suggest that
there is perhaps a more complex link between the cavity expansion calculations and
the penetration calculations than just a pure geometric link. For example, using a
direct geometric substitution between the cavity expansion velocity and particle
velocity on the projectile nose may not provide adequate resolution.
5. Projectile Structural Response Simulations
As an example, we have also included two structural response calculations using
the ParaAble FE code10 with various forms of the penetration forcing function, as
follows:
� ¼Aþ BV 2n ; ð18Þ
� ¼ðAþ BVnÞN: ð19ÞThe results from these calculations can be seen in Fig. 11, which shows a comparison
between using these two di®erent forms of the penetration resistance forcing function
(i.e. Eqs. 18 and 19). Since these simulations were conducted before the VPL code
was developed, the values of the parameters (A and B) were computed with sim-
pli¯ed analytical equations.41 This does not in°uence the present discussion because
it is the form of the equations that is currently under consideration, rather than the
speci¯c values of the constants. The e®ective plastic strain from the calculations are
compared with the structural deformation observed during a given experimental
penetration event. This clearly shows that the form of the penetration resistance
equation can have a signi¯cant in°uence on projectile structural response calcu-
lations. In this speci¯c example, using the form of the penetration resistance forcing
function described in Eq. (18) can preclude local buckling modes from taking place.
Speci¯cally, since the static term in Eq. (18) (A) is not a function of the normal
velocity it predicts that the entire outer surface of the projectile is subjected to a
signi¯cant level of normal stress. This issue is often addressed in practice by setting
the normal stress to zero for elements that have a normal component of velocity that
approaches zero. Although that algorithm is adequate in many cases it can lead to
extremely poor results in others. The example problem under consideration is
representative of the latter case, i.e. at the onset of buckling the elements just aft of
the nose have ¯nite values of Vn which means the aforementioned check for zero
normal velocity leads to the application of a large normal stress, and that stress
prevents the deformations that would have brought the buckling event to fruition.
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Since the true failure mode was prohibited by the form of Eq. (18), the simulation
predicted that the projectile would survive the penetration event unscathed. How-
ever, in the actual experiment a catastrophic failure was observed.
As seen in Fig. 11, the form of Eq. (19) completely avoids that problem due to the
fact that the nose performance coe±cient (N) is applied to both the static and
dynamic terms in the penetration resistance equation. Since the examples shown in
the previous section revealed that nose stress distributions that employ nose per-
formance coe±cients can provide excellent predictions of penetration depth, and the
present example shows these algorithms can be employed in a form that provides
more accurate predictions of structural response, it is clear that these algorithms are
worthy of further study. That fact is one of the motivations behind the development
of the VPL software package as the VPL provides researchers with the capability to
conduct detailed investigations which hold the promise of leading to the development
of very high-¯delity penetration resistance equations.
This example also brings to light another modeling issue that should be con-
sidered when using penetration resistance functions to model the target and FE
methods to model the projectile: Complete failure and break-up of the projectile
may make subsequent estimates of penetration depth inaccurate. Speci¯cally, the
normal stresses representing the target resistance are applied to the outside surface
of the projectile, but upon breakup there are additional surfaces that must be
considered. Large deformations of the projectile are accounted for in the simu-
lation, but in order to continue the simulation after breakup of the projectile the
de¯nition of the new surfaces that may be subject to penetration resistance
stresses must be updated. Although it is certainly possible to incorporate that type
of algorithm in projectile penetration simulations, it is usually not used because
the simulation is no longer of any practical interest if the projectile is failing
catastrophically. This type of surface updating algorithm was not used in the
projectile perforation simulations presented in this section, therefore the perfor-
ation velocities of the projectile (or projectile debris) as the projectile exited the
concrete slab are not reported herein.
6. Summary
We have provided a research tool (VPL) that can be used for investigating the
details of various projectile penetration events. Speci¯cally, the VPL code can be
used to quickly generate virtual (numerical) experiments and carefully examine the
data provided. The data generated by the VPL code has greater ¯delity and resol-
ution than can be measured from actual experiments. This can lead to a better
understanding of the fundamental phenomena associated with projectile pen-
etration. The ultimate goal for this tool (VPL) is to identify the best procedure for
developing penetration resistance functions that: (a) are based on a solid continuum
mechanics foundation associated with the cavity expansion problem, (b) provide
satisfactory predictions of the trajectory of the projectile, (c) exhibit the correct
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spatial normal stress distribution on the projectile nose, (d) provide the correct
acceleration-time history data (i.e. predict the correct time variation of the loads and
not just the ¯nal state or depth of penetration), (e) provide correct estimates of the
structural response of the projectile when used with FE codes.
Acknowledgments
The research reported herein was conducted as part of the U.S. Army Corps of
Engineers Survivability and Protective Structures Technical Area, Hardened Com-
bined E®ects Penetrator Warheads Work Package, Work Unit \HPC Prediction of
Weapon Penetration, Blast and Secondary E®ects". The third author's contri-
butions were performed in connection with contract DAAD19-03-D-0001 with the
U.S. Army Research Laboratory. The authors gratefully acknowledge M. Forrestal
and T. Warren for sharing their extensive notes on cavity expansion solution tech-
niques and for numerous discussions over the years. Permission to publish was
granted by Director, Geotechnical and Structures Laboratory.
References
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2. J. D. Cargile, Development of a constitutive model for numerical simulation of projectilepenetration into brittle geomaterials, Technical Report SL-99-11, U.S. Army EngineerWaterways Experiment Station, Vicksburg, MS (1999).
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