generating penetration resistance functions with a virtual penetration laboratory (vpl) -...

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

1250024-1

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

http://dx.doi.org/10.1142/S0219455412500241

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

M. D. Adley et al.

1250024-2

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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

Generating Penetration Resistance Functions with a Virtual Penetration Laboratory

1250024-3

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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

M. D. Adley et al.

1250024-4

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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.


1250024-5

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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.

M. D. Adley et al.

1250024-6

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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


1250024-7

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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).

M. D. Adley et al.

1250024-8

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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


1250024-9

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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.

M. D. Adley et al.

1250024-10

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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.


1250024-11

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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

M. D. Adley et al.

1250024-12

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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.


1250024-13

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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

M. D. Adley et al.

1250024-14

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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).


1250024-15

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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

M. D. Adley et al.

1250024-16

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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

)


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


1250024-17

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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

)



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

)



Fig. 9. Simulations of ogival nose penetration experiments as follows: (a) using direct substitution


M. D. Adley et al.

1250024-18

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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

)


250 500 750 1,000 1,250 1,500

Velocity (m/s)


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

)



Fig. 10. Simulations of conical nose penetration experiments as follows: (a) using direct substitution



1250024-19

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

(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.

M. D. Adley et al.

1250024-20

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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.


1250024-21

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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

M. D. Adley et al.

1250024-22

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

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

1. S. A. Akers, M. D. Adley and J. D. Cargile, Comparison of constitutive models forgeologic materials used in penetration and ground shock calculations, in Proc. of the 7thInt. Symposium on the Interaction of Conventional Munitions with Protective Structures(Mannheim FRG, Germany, 1995).

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).

3. Z. P. Bazant, M. D. Adley, I. Carol, M. Jirasek, S. A. Akers, B. Rohani, J. D. Cargile andF. C. Caner, Large-strain generalization of the microplane model for concrete andapplication, J. Eng. Mech. 126(9) (2000) 971�980.

4. D. J. Frew, J. D. Cargile and J. Q. Ehrgott, WES geodynamics and projectile penetrationresearch facilities, in Proc. of the Symposium on Advances in Numerical SimulationTechniques for Penetration and Perforation of Solids (ASME Winter Annual Meeting,New Orleans, LA, 1993).

5. R. P. Berger, M. D. Adley and J. D. Cargile, A three-dimensional rigid-body trajectorycode for predicting projectile motion under general impact conditions, in Proc. of TheSixty Sixth Shock and Vibrations Symposium (Biloxi, MS, 1995).

6. R. L. Bell, M. R. Baer, R. M. Brannon, R. A. Cole, D. A. Crawford, M. G. Elrick, E. S.Hertel, Jr., S. A. Silling and P. A. Taylor, CTH User's Manual and Input Instructions,Version 6.01 (Sandia National Laboratories, Albuquerque, NM, 2003).

7. L. M. Taylor and D. P. Flanagan, PRONTO 3D: A three-dimensional transient soliddynamics program, Report SAND 87-1912, Sandia National Laboratories, Albuquerque,NM (1989).

8. R. G. Whirley and B. E. Engelmann, DYNA3D: A nonlinear explicit, three-dimensional¯nite element code for solid and structural mechanics-user manual, Lawrence LivermoreNational Laboratory Report, UCRL-MA-107254 (1993).

9. G. R. Johnson, R. A. Stryk and S. R. Beissel, User Instructions for the 2001 Version of theEPIC Code (Alliant Techsystems Inc., Hopkins, MN, 2001).


1250024-23

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

10. K. T. Danielson and M. D. Adley, A meshless treatment of three-dimensional penetratortargets for parallel computation, Comput. Mech. 25(2/3) (2000) 267�273.

11. T. Belytschko, Y. Krongauz, D. Organ, M. Fleming and P. Krysl, Meshless methods:An overview and recent developments, Comput. Meth. Appl. Mech. Eng. 139 (1996)3�47.

12. V. P. Nguyen, T. Rabcsuk, S. Bordas and M. Du°ot, Meshless methods: A review andcomputer implementation aspects, Math. Comput. Sim. 79 (2008) 763�813.

13. T. Rabczuk and T. Belytschko, A three dimensional large deformation meshfree methodfor arbitrary evolving cracks, Comput. Meth. Appl. Mech. Eng. 196(29�30) (2007)2777�2799.

14. T. Rabczuk and J. Eibl, Simulation of high velocity concrete fragmentation using SPH/MLSPH, Int. J. Num. Meth. Eng. 56 (2003) 1421�1444.

15. T. Rabcsuk, S. P. Xiao and M. Sauer, Coupling of meshfree methods with ¯nite elements:Basic concepts and test results, Comm. Num. Meth. Eng. 22(10) (2006) 1031�1065.

16. W. J. Stronge and J. C. Schulz, Finite element analysis of projectile impact response, inFinite Element Handbook, ed. H. Kardestuncer (McGraw-Hill, New York, 1987).

17. D. B. Longcope, Coupled bending/lateral load modeling of earth penetrators, SAND90-0789, Sandia National Laboratories, Albuquerque, NM (1990).

18. M. D. Adley, A simply coupled penetration trajectory/structural dynamics model, inProc. of The Sixth International Symposium on the Interaction of Nonnuclear Munitionswith Structures (U.S. Army Engineer Waterways Experiment Station, Panama CityBeach, FL, 1993).

19. R. E. Moxley, M. D. Adley and B. Rohani, Impact of thin-walled projectiles with concretetargets, J. Shock Vib. 2(5) (1995) 355�364.

20. M. D. Adley, K. T. Danielson and R. E. Moxley, Coupling penetration resistance func-tions and ¯nite elements for deformable projectile impact analysis, University of Min-nesota AHPCRC Preprint, No. 98-061 (1998).

21. T. L. Warren and M. R. Tabbara, Spherical cavity-expansion forcing function inPRONTO 3D for applications to penetration problems, SAND97-1174, Sandia NationalLaboratories, Albuquerque, NM (1997).

22. M. J. Forrestal, F. R. Norwood and D. B. Longcope, Penetration into targets described bylocked hydrostats and shear strength, Int. J. Sol. Struct. 17 (1981) 915�924.

23. M. J. Forrestal and V. K. Luk, Dynamic spherical cavity-expansion in a compressibleelastic-plastic solid, J. Appl. Mech. 55 (1988) 275�279.

24. M. J. Forrestal, B. S. Altman, J. D. Cargile and S. J. Hanchak, An empirical equation forpenetration depth of ogive-nose projectiles into concrete targets, Int. J. Imp. Eng. 15(4)(1994) 395�405.

25. S. Hanagud and B. Ross, Large deformation, deep penetration theory for a compressiblestrain-hardening target material, AIAA J. 9(5) (1971) 905�911.

26. T. L. Warren and M. J. Forrestal, E®ects of strain hardening and strain-rate sensitivityon the penetration of aluminum targets with spherical-nosed rods, Int. J. Sol. Struct.35(28/29) (1998) 3737�3753.

27. T. L. Warren, A. F. Fossum and D. J. Frew, Penetration into low-strength(23 MPa) concrete: Target characterization and simulations, Int. J. Imp. Eng. 30 (2004)477�503.

28. M. D. Adley, A. O. Frank, K. T. Danielson, S. A. Akers and J. L. O'Daniel, The virtualpenetration laboratory: New developments for projectile penetration in concrete, Com-put. Concr. 7(2) (2010) 87�102.

29. G. Voyiadjis, I. R. Ionescu and E. Buzaud, Materials under Extreme Loadings: Appli-cation to Penetration and Impact (Wiley, Hoboken, NJ, 2010), pp. 267�290.

M. D. Adley et al.

1250024-24

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

30. Z. P. Bazant, F. C. Caner, I. Carol, M. D. Adley and S. A. Akers, Microplane model M4for concrete. I: Formulation with work-conjugate deviatoric stress, J. Eng. Mech. 126(9)(2000) 944�953.

31. F. C. Caner and Z. P. Bazant, Microplane model M4 for concrete. II: Algorithm andcalibration, J. Eng. Mech. 126(9) (2000) 954�961.

32. Z. P. Bazant, F. C. Caner, M. D. Adley and S. A. Akers, Fracturing rate e®ect and creepin microplane model for dynamics, J. Eng. Mech. 126(9) (2000) 962�970.

33. G. R. Johnson and W. H. Cook, A constitutive model and data for metals subject to largestrains, high strain rates and high temperatures, in Proc. of Seventh Int. Symposium onBallistics (The Hague, The Netherlands, 1983).

34. T. J. Holmquist, G. R. Johnson and W. H. Cook, A computational constitutive model forconcrete subjected to large strains, high strain rates, and high pressures, in FourteenthInt. Symposium on Ballistics (Quebec City, Canada, 1993).

35. Z. P. Bazant, Y. Xiang and P. C. Prat, Microplane model for concrete. I: Stress�strainboundaries and ¯nite strain, J. Eng. Mech. 122(3) (1996) 245�254.

36. Z. P. Bazant, Y. Xiang, M. D. Adley, P. C. Prat and S. A. Akers, Microplane model forconcrete. II: Data delocalization and veri¯cation, J. Eng. Mech. 122(3) (1996) 255�262.

37. J. A. Zukas, High Velocity Impact Dynamics (Wiley-Interscience, New York, 1990).38. L. E. Malvern, Introduction to the Mechanics of a Continuous Medium (Prentice-Hall,

New Jersey, 1969).39. K. J. Bathe, Finite Element Procedures in Engineering Analysis (Prentice-Hall,

New Jersey, 1982).40. M. J. Forrestal, K. Okajima and V. K. Luk, Penetration of 6061-T651 aluminum targets

with rigid long rods, J. Appl. Mech. 55 (1988) 755�760.41. F. E. Heuze, An overview of projectile penetration into geological materials, with

emphasis on rocks, Int. J. Rock Mech. Min. Sci. & Geomech. 27(1) (1990) 1�14.


1250024-25

Int.

J. S

tr. S

tab.

Dyn

. 201

2.12

. Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

UN

IVE

RSI

TY

OF

FLO

RID

A o

n 05

/22/

13. F

or p

erso

nal u

se o

nly.

generating penetration resistance functions with a virtual penetration laboratory (vpl) -...

Documents