Numerical simulation of ceramic composite armor subjected to ballistic impact

K. Krishnan a, S. Sockalingam a, S. Bansal a, S.D. Rajan b,⇑

aHawthorne & York, Intl., Phoenix, AZ 85040, USAbCivil, Environmental and Sustainable Engineering Program, Arizona State University, Tempe, AZ 85287, USA

a r t i c l e i n f o

Article history:

Received 19 March 2010

Received in revised form 24 September

2010

Accepted 3 October 2010

Available online 8 October 2010

Keywords:

A. Plates

B. Impact behavior

C. Finite element analysis

B. Delamination

Ceramic

a b s t r a c t

Armor systems made of ceramic and composite materials are widely used in ballistic applications to

defeat armor piercing (AP) projectiles. Both the designers and users of body armor face interesting

choices – how best to balance the competing requirements posed by weight, thickness and cost of the

armor package for a particular threat level. A finite element model with a well developed material model

is indispensible in understanding the various nuances of projectile–armor interaction and finding effec-

tive ways of developing lightweight solutions. In this research we use the explicit finite element analysis

and explain how the models are built and the results verified. The Johnson–Holmquist material model in

LS-DYNA is used to model the impact phenomenon in ceramic material. A user defined material model is

developed to characterize the ductile backing made of ultra high molecular weight polyethylene

(UHMWPE) material. An ad hoc design optimization is carried out to design a thin, light and cost-effective

armor package. Laboratory testing of the prototype package shows that the finite element predictions of

damage are excellent though the back face deformations are under predicted.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Both the designers and users of body armor face interesting

choices – how best to balance the competing requirements posed

by weight, thickness and cost of the armor package for a particular

threat level. An armor system made of a single material may be

good enough to resist the impact of small caliber ammunition.

However a multi-component armor system such as a hard faced

ceramic armor with composite backing is necessary and is widely

used to defeat armor piercing (AP) projectiles. These projectiles

have a hard core material such as hardened steel or tungsten car-

bide and the ceramic face helps blunt and erode the projectile tip

during impact. The composite backing absorbs the kinetic energy

of the decelerated projectile and also catches the ceramic and pro-

jectile fragments preventing them from doing further harm.

Alumina (Al2O3), Boron Carbide (B4C), Boron Silicon Carbide

(BSC) and Silicon Carbide (SiC) are some of the ceramics that are

commonly used. The range of composite materials used as backing

and spall minimizing material include UHMWPE materials, aramid

woven fabrics such as Kevlar and Twaron, fiber glass materials

such as S2-glass and E-glass and so on.

A number of different analytical models have been developed to

model ceramic and ceramic composite armors. Anderson and

Walker [1] develop an analytical model that describes the

dwell or interface defeat of the projectile during its impact against

a ceramic. Dwell or interface defeat is where the projectile erodes

with no penetration. This erosion is due to the pressure at the

interface between the ceramic and the projectile exceeding the

erosion strength of the projectile [9]. There is an inverse relation-

ship between hardness and fracture toughness of the armor ceram-

ics [18]. Hardness of the ceramic blunts the tip of the AP projectile

whereas high fracture toughness provides multi-hit capability to

the armor. Generally B4C ceramics have high hardness and low

fracture toughness whereas SiC ceramics have low hardness and

high fracture toughness compared to B4C. Boron Silicon Carbide

(BSC) a B4C–SiC blend ceramic has light weight, high strength

and high fracture toughness which provide ballistic resistance

against armor piercing threats. It combines the high hardness of

Boron Carbide and high fracture toughness of Silicon Carbide.

Benloulo and Sanchez-Galvez [3] develop a simple one dimen-

sional analytical model to simulate the ballistic impact of ceramic

composite armor. The penetration process is divided into three

phases. In the first phase the ceramic is intact, and the velocity

and mass erosion of the impacting projectile are described by

Tate’s equation. The fracture of ceramic and the damage to the

composite occurs in the second phase. In the last phase, failure of

the composite takes place.

Lee and Yoo [21] discuss the numerical modeling and experi-

mental study of ceramic metal armor systems with a metal back-

ing. Using smoothed particle hydrodynamics (SPH), the ceramic

(Alumina) is modeled using Mohr–Coulomb strength model and

linear equation of state (EOS) in AUTODYN. Lundberg [26] uses

Johnson–Holmquist (JH1 model for SiC and JH2 model for Alumina

1359-8368/$ - see front matter � 2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compositesb.2010.10.001

⇑ Corresponding author.

E-mail address: s.rajan@asu.edu (S.D. Rajan).

Composites: Part B 41 (2010) 583–593

Contents lists available at ScienceDirect

Composites: Part B

journal homepage: www.elsevier .com/locate /composi tesb

and Boron Carbide) model in AUTODYN. The determination of tran-

sition velocity (transition from interface defeat to penetration) for

various combinations of projectile, target material and target con-

figuration is studied. Simha et al. [31] develop and use a constitu-

tive model for ceramic (Alumina) and implement into EPIC

Lagrangian finite element code. The model consists of strength

model based on Hugoniot elastic limit for compression, viscoelastic

flow rule, damage model for compression and tension and Mie-

Gruneisen EOS.

Nemat-Nasser et al. [29] discuss experimental techniques used

to study the performance of Alumina armor tiles wrapped with

thin layers of several different materials such as carbon–fiber/

epoxy, E-glass-/epoxy, etc. Details and results from a two-dimen-

sional finite element model using DYNA2D are presented. They

show that release waves emanating from the projectile edges re-

duce the pressure and increase the shear stress at a distance equal

to the projectile diameter, ahead of the projectile. Grujicic et al.

[11] analyze the performance of ceramic/composite armor sub-

jected to AP projectile impact. They model the Alumina ceramic

in AUTODYN using a polynomial equation of state, Johnson–Holm-

quist 2 (JH-2) strength model [16] and JH-2 failure model along

with an erosion model. The composite material, S2-glass, is mod-

eled using an orthotropic material model [6].

UHMWPE materials are widely used in ballistic applications be-

cause of their low weight, high tenacity and high specific modulus.

These materials have a unidirectional construction in which the fi-

bers lie parallel to each unlike fabrics that are woven. A thermo-

plastic resin is used as the binding agent. Typically, the material

used for armor applications is made up of several 0–90� layers

(or plies). The two most popular examples of UHMWPE material

are Spectra� manufactured by Honeywell [4] and Dyneema� man-

ufactured by DSM (DSM [8]). The UHMWPE fibers have a modulus

in the range of 90–140 GPa and a failure strain of 2.9–3.8% [14].

These fibers have a very high energy absorption capability and high

sonic velocity compared to aramid, S2-glass, polyamide and similar

materials.

As stated earlier, some armor packages include ceramics and a

backing material. There are different ways of bonding the two

materials including use of spray on adhesive, adhesive tape, auto-

claving/vacuum bagging, etc. Zaera et al. [34] study the effect of the

adhesive layer thickness on the performance of the ceramic/metal

armor. They show that the adhesives – a soft adhesive (polyure-

thane) and a hard adhesive (rubber–modified epoxy) show strain

rate dependent behavior. In the follow up publication [23], the

adhesive is modeled in AUTODYN using Steinberg–Guinan model

and the Mie-Gruneisen EOS. By analyzing the depth of penetration

and the projectile residual velocity, the authors conclude that the

thickness of the epoxy resin adhesive significantly affects the per-

formance of the system.

A finite element model with a well developed material model is

indispensible in understanding the various nuances of projectile–

armor interaction and finding effective ways of developing light-

weight solutions. In this research we use the Lagrangian solver

and explain how the models are built and the results verified using

LS-DYNA [24]. The Johnson–Holmquist material model [7] in LS-

DYNA is used to model the impact phenomenon in ceramic mate-

rial. A user defined material model is developed to characterize the

ductile backing (UHMWPE) material. The modeling and calibration

of the ceramic material model are presented in Section 2. In Sec-

tion 3 we review the development of constitutive model for the

UHMWPE composite. In Section 4 we present the ballistic simula-

tion of ceramic composite armor and compare the simulation re-

sults with tested samples. Finally, we present some thoughts on

the current work and potential for future improvements.

2. Material modeling and simulation

In this section we discuss the details of the finite element mod-

els for the three different components used – the bullet, the cera-

mic plate and the backing material, UHMWPE.

2.1. Damage mechanics in ceramic armor

The impact of a projectile on the surface of the ceramic material

generates compressive shock waves that propagate through the

ceramic plate [19]. These stress waves are reflected back as tensile

waves once they reach the free surface. The ceramic material frac-

tures if the magnitude of the reflected tensile wave exceeds the dy-

namic tensile strength of the material. Radial cracks are formed at

the bottom of the ceramic material due to the initial impact and

travel from the bottom to the top of the ceramic plate. Meanwhile

a fracture cone (conoid) is formed at the impact zone on the top of

the ceramic tile and grows towards the back face of the ceramic. In

the case of a composite armor system where the ceramic tile is

backed by a ductile material, part of the compressive waves is

transmitted into the ductile backing. The rest of the waves are re-

flected back into the ceramic plate. The amount of stress waves

that are transmitted depends on the mechanical impedance of

the ductile backing. The thickness of the adhesive layer, used to

bond the ceramic tile to the ductile backing, also determines the

percentage of reflected and transmitted stress waves.

Nomenclature

Cijkl material (stiffness) matrixA, B, C JH-2 parametersa, b, c strain rate constants in the composite material modelM; N JH-2 parametersK1, K2, K3

JH-2 parametersD1, D2 JH-2 parametersd4 constant used in the failure strain modelG shear modulusJ2 second invariant of deviatoric stress tensorP pressurePHEL pressure at Hugoniot elastic limitsij deviatoric stress tensorT maximum tensile hydrostatic pressure the material can

withstand

x1; x2; x3 material coordinate directionsb Hardening parametert Poisson’s ratioepeff

effective plastic strainefp plastic strain to fracture in JH-2 material model_� strain rateDekl incremental strain/ yield functionr�i normalized intact strength

r�f fractured material strength

ry yield stressro initial yield stressrij stress tensorHEL Hugoniot elastic limit

584 K. Krishnan et al. / Composites: Part B 41 (2010) 583–593

2.2. Bullet and aluminum models

The depth-of-penetration (DOP) test methodology [27] is used

to evaluate the performance of the various ceramic materials

and has been successfully used to characterize and rank armor

ceramics for vehicle protection [33]. We first use the DOP tests

to calibrate the bullet model. The properties of the bullet (0.30

caliber M2 AP) are given in Tables 1 and 2. The calibration is

performed by considering the impact of the bullet into Al

6061-T6 block of size 600 � 600 � 600. Tetrahedral elements are used

to mesh the projectile. The finite element model and the mesh

details are shown in Fig. 1 and Table 4. Erosion strains for the

components of the bullet and aluminum are calibrated simulta-

neously using several metrics such as the depth of penetration,

exit velocity (for uncontained bullets), damage to the bullet

and aluminum, and the final state/orientation of the bullet. Ero-

sion values of 1.0 and 3.0 are used for the bullet jacket and the

lead filler materials respectively. �MAT_ADD_EROSION with prin-

cipal strain at failure criterion is used to model the erosion of

the bullet core with a value of 0.12. The response of the target

plate (Al 6061-T6) is simulated using Johnson–Cook [15] mate-

rial model and the properties are given in Tables 2 and 3 (Kauff-

man et al., 2004).

A typical DOP test sample is shown in Fig. 2 (X-ray measure-

ment shows bullet captured in the monolithic aluminum DOP test

sample). In this study, the average depth of penetration taken from

five tests for a monolithic block is 50.8 mm (200). Using the fine

mesh for the bullet, these tests are simulated using varying mesh

sizes and erosion strain values for the aluminum block. Four mesh

sizes for the aluminum block are considered keeping the aspect ra-

tio close to 1:1 [35] – eight noded solid elements with one point

integration are used in the model.

(a) Very fine – 0.5 mm � 0.5 mm � 0.5 mm.

(b) Fine – 1.0 mm � 1.0 mm � 1.0 mm.

(c) Coarse – 1.5 mm � 1.5 mm � 1.5 mm

(d) Very coarse – 2.0 mm � 2.0 mm � 2.0 mm

The finite element analysis results for the different mesh sizes

and the appropriate erosion strain values that match the DOP value

as closely as possible are shown in Fig. 3a–d. Even though all the

models are able to predict the DOP, the finite element model with

a very fine mesh is able to model the material damage in the

aluminum block more accurately than other models.

In the numerical simulations, the Johnson–Cook damage model

for 6061-T6 is not used. Rather, the erosion strain at which the

6061-T6 elements erode is calibrated for different mesh sizes. Ta-

ble 4 summarizes the mesh size versus erosion strain for all the dif-

ferent mesh sizes. As expected, the erosion strain value that gives

the accurate DOP increases as the mesh is refined. The mesh size

versus erosion strain data (from Table 4), gives the flexibility to

chose the failure strain value for any given mesh size ranging be-

tween very fine and very coarse.

Table 1

Bullet construction and properties.

Bullet construction Properties Material model used in LS-DYNA

Bullet jacket Gilding copper �MAT_PLASTIC_KINEMATIC

Bullet core Steel [5] �MAT_PIECEWISE_LINEAR_PLASTICITY

Filler Lead [2] �MAT_PLASTIC_KINEMATIC

Bullet weight

(g/grains)

10.8/166

Bullet velocity

(ft/s)/(m/s)

2880/878

Table 2

Bullet material properties.

Parameters Bullet jacket Bullet core Lead filler Al 6061-T6

Density (kg/mm3) 8.858 � 10�6 7.85 � 10�6 1.127 � 10�5 2.7 � 10�6

Elastic modulus (GPa) 117.20 210.0 17.0 69.0

Poisson’s ratio 0.40 0.33 0.40 0.29

Yield strength (GPa) 0.3447 1.40 0.008 –

Tangent modulus (GPa) 0.0 15.0 0.015 –

Fig. 1. Finite element models (coarse, medium and fine mesh) of 0.30 caliber M2

AP.

Table 3

Al 6061-T6 JC parameters [22].

A (GPa) B (GPa) n C m

Al 6061-T6 0.324 0.114 0.42 0.002 1.34

Fig. 2. Experimental DOP test on monolithic aluminum.

K. Krishnan et al. / Composites: Part B 41 (2010) 583–593 585

2.3. Ceramic material model

The Johnson–Holmquist [16,17] ceramic constitutive model

was proposed to describe the response of brittle materials to

large deformations. The constitutive model comprises of three

parts – strength, pressure and damage. The model includes a

(a) Very fine mesh (b) Fine mesh

(c) Coarse mesh (d) Very coarse mesh

Fig. 3. Cross-section through FE mesh showing depth of penetration.

Table 4

Mesh size versus erosion strain study for aluminum.

Very fine Fine Coarse Very coarse

Erosion strain DOP (in.) Erosion strain DOP (in.) Erosion strain DOP (in.) Erosion strain DOP (in.)

1.4 2.02 1 1.566 1 1.306 1 1.01

1.2 2.25 0.75 1.91 0.6 1.88 0.4 1.67

1 2.57 0.7 2.06 0.5 2 0.35 1.81

0.6 2.29 0.45 1.979 0.3 2.04

0.5 2.58 0.4 2.23 0.1 3.92

Table 5

BSC properties.

Density (kg/mm3) Youngs modulus

(GPa)

Poisson’s

ratio

Fracture toughness

(MPa m1/2)

2.75 (10�6) 400.0 0.20 4.5

Table 6

BSC JH-2 parameters.

JH-2 parameter Calibrated value

A 0.94

B 0.65

C 0.005

M 0.85

N 0.67

D1 0.001

D2 0.50

586 K. Krishnan et al. / Composites: Part B 41 (2010) 583–593

representation of the intact and fractured strength, a pressure–

volume relationship that can include bulking and a damage model

that transition from an intact state to a fractured state. The strength

and damage relationships are given by the following set of equa-

tions [24].

The normalized intact strength is given by

r�i ¼ AðP� þ T�ÞNð1þ C ln e�Þ ð1Þ

The fractured material strength is defined as

r�f ¼ BðP�ÞMð1þ C ln e�Þ ð2Þ

The hydrostatic pressure, before the onset of damage is

P ¼ K1lþ K2l2 þ K3l3 ð3Þ

The plastic strain to fracture under a constant pressure P, is

efp ¼ D1ðP� þ T�ÞD2 ð4Þ

where P� ¼ P=PHEL, T� ¼ T=PHEL, _e� ¼ _e= _e0, where _e is the actual strain

rate and _e0 ¼ 1:0 is the reference strain rate, l ¼ q=q0 � 1 for cur-

rent density q and initial density q0.

2.4. Depth-of-penetration test with ceramic strike face

The DOP model used in the bullet study is used here to calibrate

the Johnson–Holmquist material constants. A fine mesh resolution

is required to capture failure involving spall and crack propagation.

There is no significant difference in the results using a uniform

mesh throughout the armor panel instead of a locally refined mesh

in the impact region. A uniform mesh is suitable when modeling

multiple hits on an armor panel [32]. The penetration into the

backing material is measured and compared with the penetration

of the projectile into a monolithic block of the backing material

as shown in Fig. 4.

As before, the DOP is measured from X-rays of the impacted cyl-

inders. The ballistic testing for measuring DOP requires firing the

0.30 caliber M2 AP rounds at a velocity of 2880 fps. All impacts

are at normal incidence (00 obliquity). The ceramic-faced targets

are prepared by adhering a ceramic tile to an aluminum block

using 24-h-cure epoxy. The tile is pressed into the face of the alu-

minum plate forcing the epoxy to flow between the ceramic and

aluminum leaving a minimal layer of epoxy.

The element erosion option in LS-DYNA is used to remove the

highly distorted elements which otherwise would significantly re-

duce the time step and the stability of the simulation. Since the ele-

ment erosion is dependent on the mesh size, a study of mesh size

versus erosion criteria is done. For each mesh size, the erosion

strain values are modified to match the experimental DOP test va-

lue. It should be noted that erosion is a non-physical parameter

and the optimal erosion strain values (Table 7) may not have phys-

ical significance.

2.4.1. Calibration of JH-2 parameters using DOP test

The ceramic and aluminum materials are modeled using 8-

noded hexahedron elements. A one point reduced integration

scheme is used along with hourglass control using Flanagan–Bely-

tschko stiffness formulation. The ceramic materials are modeled

using the Johnson–Holmquist (JH-2) material model available in

LS-DYNA. The material model parameters for the ceramic materials

were obtained from the Cronin et al. [7].

As a first step in the calibration process, the DOP test of

8.76 mm (0.34500) BSC ceramic attached to a 152.4 mm (600) alumi-

num block is used in this study. The ceramic strike face blunts and

thereby stops the projectile with no penetration into the aluminum

block. The depth of indentation on the ceramic tile is taken as the

metric to compare the effect of mesh size and erosion strain. The

indentation on the ceramic tile was measured to be approximately

4 mm (0.15700). Using a fixed set of JH-2 parameters, the erosion

values for different mesh sizes were determined to match this

depth of indentation. The corresponding finite element analysis re-

sults for the different mesh sizes are shown in Fig. 5a–d and sum-

marized in Table 7.

Next, using the erosion strains for the aluminum block (Table 4)

and the bullet materials from the previous study, and the very fine

mesh, the JH-2 strength and damage parameters A, B, C, M, N, D1

and D2 are calibrated such that the DOP and the extent of damage

incurred by the ceramic matches closely with the experimental re-

sults. Figs. 6 and 7 show the comparison of experimental and sim-

ulation results after the optimal values are obtained. The BSC

material properties from the manufacturer and the calibrated JH-

2 parameters are given in Tables 5 and 6.

As we will see later the bullet–ceramic interaction is perhaps

the most important component in the overall behavior of the ar-

mor package.

Fig. 4. Depth-of-penetration (DOP) measurement technique.

Table 7

Mesh size vs. erosion strain study for BSC ceramic.

Very fine Fine Coarse Very coarse

Erosion strain DOP (mm) Erosion strain DOP (mm) Erosion strain DOP (mm) Erosion strain DOP (mm)

1 33.3 0.5 33.31 1 7.51 1 5.4

2 18.51 1 27.2 2 4.51 1.5 3.6

4 6.01 2 8.01 2.5 4 2 1.8

6 4.5 3 6.01 3 3 3 1.8

7 4.5 4 4 4 1.5 4 0

8 4 5 4 5 1.5 5 0

K. Krishnan et al. / Composites: Part B 41 (2010) 583–593 587

2.5. UHMWPE modeling

The failure mechanisms of a typical UHMWPE composite panel

when impacted with a bullet consisting of a lead core can be briefly

explained as follows [13,14]:

1. During the initial stage of impact where the strain rate is very

high, the failure of the material is due to fiber breakage. In this

stage, the bullet behaves as a rigid body with very little

deformation.

2. In the second stage the bullet deforms (mushrooms) losing its

kinetic energy. Very little penetration of panel takes place.

The thermopressed material starts delaminating resulting in a

bulge in the non-strike (rear) face of the panel.

3. Finally the bullet, behaving like a rigid body, starts penetrating

the panel and is eventually stopped because it has lost its

kinetic energy.

However, in the case of an AP projectile, the core remains rigid

and does not undergo a mushrooming deformation. The two pri-

mary modes of failure under this scenario are fiber breakage and

delamination. It has been reported that UHMWPE fibers undergo

brittle failure at high strain rates and ductile failure at low strain

rates [10]. Koh et al. [20] show that the mechanical properties of

Spectra Shield laminated composite roll are highly rate sensitive.

The failure stress and the stiffness increase up to a certain critical

strain rate during which the failure strain decreases. Interestingly,

when the strain rate is higher than the critical strain rate, the prop-

erties showa reverse trend. Theyattribute this to thermal effects –as

the strain rate increases beyond the critical rate, frictional and visco-

elastic hysteresis effects cause the temperature in localized areas of

the specimen failure surface to increase significantly such that the

failureagainbecomes increasinglyductile. Theyalsoput forwardan-

other explanation – as the strain rate increases, there is less time for

filaments to align themselves in the direction of the applied stress.

This implies that filaments that are already aligned fail by brittle

fracture, while those that are not aligned fail by intermolecular slip-

page or shearing for strain rate less than 400 s�1.

A brief review of the constitutive model for the UHMWPE mate-

rial [32] is presented here. The developed constitutive model is

implemented as a user defined material model in LS-DYNA [12].

The constitutive behavior of the composite panel is modeled using

a nonlinear orthotropic model. The essential behavior is assumed

as follows where changes in the strains as supplied by LS-DYNA

are used in computing the changes in the stress.

Since sheets of UHMWPE are laid parallel to the 1–2 plane and

thermopressed, we assume that the material is isotropic in the 1–2

plane (see Fig. 8). The through thickness direction of the plate is

aligned with the x3-axis and the behavior in the 1–2 plane is

decoupled from the behavior in the 3-direction. The material

(a) Very fine mesh (b) Fine mesh

(c) Coarse mesh (d) Very coarse mesh

Fig. 5. Cross-section through FE models (BSC ceramic plus aluminum).

(a) Experimental DOP = 39 mm (b) FEA DOP = 46 mm

Fig. 6. Comparison of experimental and FEA DOP for 0.100 thick BSC ceramic strike face with aluminum backing.

588 K. Krishnan et al. / Composites: Part B 41 (2010) 583–593

behavior is characterized from experimental results and include

in-plane and out-of-plane stress–strain curves in tension and com-

pression, out-of-plane unloading and reloading curves, strain-rate

effects, delamination characterization and finally, strain-based fail-

ure criteria. Experimental results show that the behavior in the 1–2

plane is essentially isotropic. The quasi-static properties of the

UHMWPE composite panel obtained from laboratory testing are gi-

ven in Table 8.

(a) Experimental DOP = 8 mm (b) FEA DOP = 6 mm

Fig. 7. Comparison of experimental and FEA DOP for 0.200 thick BSC ceramic strike face with aluminum backing.

x1

x2

x3

Fig. 8. Material coordinate system associated with composite panel.

Table 8

UHMWPE material properties.

Parameters UHMWPE

Density (kg/mm3) 9.7 (10�7)

Elastic modulus (GPa) 6.0

Poisson’s ratio 0.15

Yield strength (GPa) 0.413

Tangent modulus (GPa) 3.6

Fig. 9. Body armor plate (a) FE model top view and (b) cross-sectional through point of impact.

K. Krishnan et al. / Composites: Part B 41 (2010) 583–593 589

Dr11

Dr22

Dr33

Dr23

Dr31

Dr12

0

B

B

B

B

B

B

B

B

@

1

C

C

C

C

C

C

C

C

A

¼

C11 C12 0 0 0 0

C12 C11 0 0 0 0

0 0 C33 0 0 0

0 0 0 C44 0 0

0 0 0 0 C44 0

0 0 0 0 0 C12

0

B

B

B

B

B

B

B

B

@

1

C

C

C

C

C

C

C

C

A

De11De22De33De23De31De12

0

B

B

B

B

B

B

B

B

@

1

C

C

C

C

C

C

C

C

A

ð5Þ

2.6. LS-DYNA solver parameters

The parameters IBULK, bulk modulus and IG, shear modulus

are required for user defined materials for transmitting bound-

aries, contact interfaces, rigid body constraints and time step cal-

culations. Since these parameters significantly affect the

simulation results for high strain rates they are conservatively

set to the highest possible stiffness expected in the simulation.

The solver calculates the time step using a stability scale factor

which is 0.90 by default. Since the constitutive model has a

number of strain rate dependent properties, for stability reasons

we set the scale factor to 0.50 reducing the required time step.

The second order stress updates and invariant node numbering

in the �CONTROL_ACCURACY control card are turned on. Various

energy checks [25] are made to ensure that the results are stable

and makes physical sense.

Table 9

Simulation results for various configurations of equal areal density.

Configuration

(10 in. � 12 in.)

Ceramic

(%)

Composite

(%)

Undamaged

UHMWPE layers (%)

BFS

(mm)

0 70 30 40 18

1 40 60 30 22

2 30 70 13 39

Fig. 10. State of bullet at different times: (a) All models at time 0 ms. (b) Configuration 0 at time 0.049 ms. (c) Configuration 1 at time 0.049 ms. (d) Configuration 2 at time

0.049 ms. (e) Configuration 0 at time 0.15 ms. (f) Configuration 1 at time 0.15 ms. (g) Configuration 2 at time 0.15.

590 K. Krishnan et al. / Composites: Part B 41 (2010) 583–593

2.7. Delamination

The bonding between the sheets is represented numerically

using zero thickness cohesive elements available in LS-DYNA

[12]. The delamination can also be simulated several different

ways and the present work is based on the material model�MAT_COHESIVE_GENERAL [24]. Since density per unit area is

needed as input for the material model with zero thick cohesive

elements, the value is arrived at using trial-and-error. The proper-

ties such as peak traction and the relative displacement at which

the peak traction occur are required as input for the cohesive mate-

rial model which are obtained using a regression analysis as pre-

sented in our earlier work [32].

2.8. Ballistic simulation of ceramic composite armor

The calibrated ceramic model and the developed constitutive

model of the composite material are used to help design and man-

ufacture a certified body armor plate [28]. The plate is a doubly-

curved plate with approximate size as 10 in:� 12 in:� 1 in:. The

armor plate contains several components – the ceramic plate, the

UHMWPE composite panel that is the backing material, an epoxy

resin that is used to bond the ceramic and the composite material,

and a thin polyurethane shell that provides environmental protec-

tion for the entire package. During the armor plate qualification,

the plate is strapped to a clay block and is subjected to one shot

with a .30 caliber M2 AP projectile at 2880 ft/s. The basic motiva-

tion in this design study is to see how these two basic constituents

(ceramic and UHMWPE) can be mixed to yield a cost-effective ar-

mor plate.

To gauge the accuracy of the various models, two response mea-

sures are used.

(a) Response R1: Percentage of original thickness that is undam-

aged when the bullet is stopped.

(b) Response R2: Back face signature (BFS).

The finite element model of the plate is shown in Fig. 9a and the

cross-section showing the bullet and the plate just before impact is

shown in Fig. 9b. The composite part of the plate is divided into a

number of layers with each layer representing a few layers or

sheets of thermopressed material. Zero thickness cohesive ele-

ments are modeled between two adjacent FE layers. No boundary

conditions are used since our numerical experimentation showed

no perceptible difference between models with nodes restrained

at the straps versus the model with no nodes restrained. The finite

element mesh is refined in the projectile–armor contact region.

ERODING_SINGLE_SURFACE contact definition is used for the inter-

acting the projectile and armor.

Various configurations of ceramic and the composite were sim-

ulated keeping the areal density the same as the Configuration 0.

From amongst those configurations, two interesting ones are se-

lected where the percentage of ceramic is progressively less com-

pared to Configuration 0. Table 9 shows the simulation results

for all three configurations.

The results show that the performance of the armor plate

degrades as the percentage of ceramic is decreased even though

the overall areal density is constant. The analysis of the simulation

shows that the ceramic hard face blunts and erodes the tip of

the 0.30 caliber M2 AP projectile which is consistent with the

Fig. 11. Final state of the armor panel: (a) Configuration 0. (b) Configuration 1. (c) Configuration 2.

K. Krishnan et al. / Composites: Part B 41 (2010) 583–593 591

experiment. The very high compressive pressure generated causes

rupture of the ceramic material. The pressure wave generated in

the ceramic plate during the impact is initially positive and it

progresses radially outward in the planar (strike) face. At the inter-

face of the ceramic composite, part of the pressure gets reflected

and it becomes negative. The negative pressure which is tensile

causes the ceramic to fail in tension.

Fig. 10 shows how the bullet is damaged as it goes through

the ceramic plate first and then through the UHMWPE backing

for all the three configurations. Fig. 11 shows the final time

snapshot of the panel. Fig. 12 shows the evolution of the kinetic

energy of the bullet core. In the Configuration 0 model, approx-

imately 75% of the kinetic energy of the bullet core is dissi-

pated as it goes through the ceramic. Much less energy

dissipation of the bullet takes place through the other two

models in the same amount of time. The composite backing

helps to absorb the residual kinetic energy of the projectile

and eventually stops it.

Configuration 0 is chosen as the best design for a couple of rea-

sons. First, the configuration reduces the kinetic energy of the bul-

let very rapidly. This leads to minimal damage to the composite

backing and a much smaller BFS. Second, this ceramic-UHMWPE

combination leads to the most cost economical armor plate. A pro-

totype of Configuration 0 is made using the material and geometric

details from the FE model. Table 10 shows the comparison between

the tested plate and the FE predictions. The armor plate, damage on

the ceramic plate and the damage predicted by the simulation on

the ceramic material are shown in Fig. 13.

The FE model predicts that the bullet is contained in the

UHMWPE in agreement with the experiment. The complete ero-

sion of the bullet jacket and the filler material is well captured

along with the blunting of the core material. The BFS is under pre-

dicted by the model and a study is under way to understand the

reasons why.

Fig. 12. Kinetic energy of the bullet core versus time (FE simulation).

Table 10

Response metrics for configuration 0 armor panel.

Response metric FEA Experiment

Bullet contained Yes Yes

Percentage of undamaged UHMWPE layers (R1) (%) 40 40

Back face signature (R2) (mm) 18 35

Fig. 13. Using Configuration 0: (a) Tested armor plate. (b) FE prediction. (c) Close up of shot area showing ceramic damage. (d) Close up of FE prediction.

592 K. Krishnan et al. / Composites: Part B 41 (2010) 583–593

3. Concluding remarks

In this paper, we discuss the development of a finite element

model that is used as a predictive tool in the design of a body ar-

mor system involving ceramic and high-performance polyethyl-

ene. The ceramic material is modeled using JH-2 material model

and the JH-2 strength and damage parameters are calibrated using

depth-of-penetration tests. A similar procedure is used to build the

bullet finite element model. The UHMWPE model is developed and

discussed in a separate paper [32]. The numerical results for the

ceramic composite armor were found to be in good agreement

with the experimental data.

The following remarks and observations can be made:

(a) Ceramic model: The damage evolution mechanism in the

ceramic material during high velocity impact is a complex

phenomenon and the JH-2 material model with calibrated

parameters appears to provide a reasonable avenue for cap-

turing damage.

(b) UHMWPE model: One of the challenges is in capturing the

delamination phenomenon and the energy dissipation that

takes place in the process. Cohesive zone elements capture

the damage that takes place in the UHMWPE material simi-

lar to what is observed in the experimental specimens. The

BFS is under predicted by the simulation. Some of this can

be attributed to the temperature rise which occurs due to

the friction between the projectile and the armor plate and

also the friction between the armor plate layers. Further

research needs to be done in this area to improve the predic-

tive capability of the model.

(c) Bond line effects: In the current research, the bond line mate-

rial that is used to bond the ceramic to the UHMWPE mate-

rial is not modeled. There is experimental evidence that

there is an optimum thickness of the bond line material

for which the ballistic performance of the armor plate can

be maximized [23]. The authors are currently investigating

the tools required to model this component.

(d) Formal design optimization: A trial-and-error design process

may be useful in searching for an optimal solution when a

few design parameters are involved. To better understand

the armor design process and find better solutions, it is nec-

essary to carry out formal design optimization [30].

Acknowledgements

The authors would like to thank C.T. Wu (LSTC) for providing

valuable input on LS-DYNA capabilities and modeling issues deal-

ing with Lagrangian and SPH formulations.

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