BALLISTIC IMPACT SIMULATION USING

DISCRETE ELEMENT METHODS

R.KARTHIKAYEN

B.Tech (Mechanical Engineering)

Third Year,

National Institute of Technology, Thiruchirappalli

This Report is submitted for Summer Fellowship Program at

I.I.T. Madras. July 2011

Under the supervision of

Prof. C.Lakshmana Rao

Department of Applied Mechanics

Indian Institute of Technology Madras

Contents

Abstract

Nomenclature

1. Introduction

2. Discrete Element Methods(DEM)

2.1 Computational steps in DEM

2.2 Critical Time step determination

2.3 Stability in Explicit Integration

2.4 Assumptions in PFC-2D

3. Issues in PFC-2D Simulations

3.1 Material modelling in PFC-2D

3.1.1 Determination of stiffness parameters-kn & ks

3.1.2 Determining the failure criteria-bond strength

3.1.3 Friction co-efficient and damping constant

4. Simulation and Results

5. Conclusion

5.1 Scope of future work

Acknowledgements

References

Abstract:

The development of Armour and ammunition is important for any nation’s security. The

study of impact is integral in design of armour and ammunition as it involves the study of

destruction and disintegration of both the projectile and target. The study can be carried

out experimentally, but the costs, time and the difficulty to monitor every parameter in

ballistic experiments motivate ballisticians to numerically simulate the event for a better

understanding. This study of ballistic impact involves a two-dimensional numerical

simulation implemented using discrete element method. PFC-2D (Particle flow Code in 2

Dimensions), a commercial discrete element software is used for the simulation. The depth

of penetration for various speeds of projectile hitting a thick target is obtained and

compared with experimental results from literature. The study is carried out with a copper

projectile and an aluminium target.

Nomenclature

ν Poisson’s ratio

ξ Critical damping ratio

ω Angular velocity

E Young’s modulus

Kn Equivalent normal stiffness

Ks Equivalent shear stiffness

kn normal spring stiffness

ks shear spring stiffness

F(t) Resultant force on particle at time t

x(t) Position of particle at time t

Velocity of particle at time t

Acceleration of particle at time t

Relative normal displacement between two contact entities

Relative tangential displacement between two contact entities

m Mass of particle

unit vector in normal direction at a contact

unit vector in tangential direction at a contact

g Acceleration due to gravity

ktrans Translational stiffness

krot Rotational stiffness

I Moment of inertia of the particle

DEM calculation time step

T Total time of a simulated event

tcr Critical time step in DEM

ωmax maximum natural frequency of a DEM mesh

Kic Fracture toughness of the material

Chapter 1

Introduction

An impact is a large force applied in a short duration of time. The problem with impacts is

that it is very tough to determine the response of the material and the mechanics

developed for quasi-static or gradual loading cannot be applied.

Ballistics is the science of motion and effects of projectiles. It involves the study of missiles

and bullets which have been developed to destruct and destroy life and property. The field

of ballistics encompasses the design of guns, rifles, armour, and ammunition. The study of

ballistics involves various sciences like impact mechanics, dynamics and fracture mechanics.

The study of ballistic impact is necessary for both protective (armour design) as well as

destructive (ammo design) purposes. The ballistic impact is governed by many parameters

like Velocity and yaw of the projectile; Geometry of the projectile: conical, flat or

hemispherical nose shapes; Material properties of the projectile and target: Stiffness,

hardness, Toughness; Geometry of the target: thickness and face area; Location of the point

of impact. [9]

The parameters used to quantify impact in general are

Depth of penetration in case of incomplete penetration of the target

Residual velocity if the projectile penetrates the target completely

Simulation is replication of phenomena and events. A phenomenon is a behavioural pattern

of a system. The experimental study of phenomena is tedious as it involves continuous

monitoring of parameters which have been identified to affect the behaviour of a system.

This difficulty combined with cost and time factors associated with experiments have

stimulated the study of various events by simulating them. Simulations help us identify the

parameters and understand their effects on a certain event. These factors also motivate

ballisticians to study ballistic events using simulations.

The field of ballistics may be classified as interior ballistics dealing with the interaction of the

gun, projectile and propelling charge within the barrel of the gun, exterior ballistics covering

the projectile motion after its exits from the muzzle and terminal ballistics. This simulation

to be precise is in the domain of terminal ballistics which deals with the interaction of the

target with the projectile- the deformation and disintegration of the target and projectile.[9]

Impact simulations have been carried out using Finite element methods (FEM) in the past

[12]. But there are many issues to be addressed with finite element simulation of material

impacts, in case of disintegration and finite element modelling of particulate media. The

modelling and analysis of the debris which disintegrates from the bulk of the material is

difficult with FEM.

Cundall proposed the discrete element methods as a numerical model in 1971 and applied it

to solve problems in rock mechanics. The term discrete element methods (DEM) refer to a

family of numerical methods for computing the motion of a large number of particles like

molecules or grains of sand. In discrete element methods, the medium is considered to be

discontinuous. Its inherent ease to monitor individual particle motion encourages the use of

DEM in this simulation of ballistic impact. A commercial DEM software-PFC-2D (Particle Flow

Code) is being used for this purpose.

R.L. Woodward [8] had performed experiments to study the velocity dependence of

penetration of semi-infinite targets by cylindrical projectiles with a range of materials. He

studied the penetration of copper and tungsten projectiles with aluminium and copper

targets.

The objective of this study is to simulate one of the experiments conducted by Woodward

with a flat nose copper projectile on Aluminium target using discrete element methods in

two dimensions. The depth of penetration for various velocities of the projectile is obtained

and compared with the experimental results [8].

Chapter 2

Discrete Element Methods

The discrete element method (DEM) is a numerical model capable of describing the

mechanical behaviour of discontinuous media. It can be used in modelling particulate media

like soil and solids which disintegrate into particles upon impact like rock and concrete.

The discrete element and finite element model of a cantilever is shown in fig 1(a,b)

Fig 1 (a) Finite element mesh, (b) discrete element mesh

The discrete element method (DEM) considers any material to be composed of single or

multi-sized particles (referred as discrete elements) capable of independent motion and

held together by bonds. Two particles in contact interact with each other such that two

springs are placed between them as shown in the fig 1(b). The normal spring (with stiffness

kn) takes up normal forces (in case of a head-on collision) and the shear or tangential spring

(with stiffness ks) absorbs tangential forces. These stiffness and bond strengths are inputs

which must be given according to the material being modelled. In fig 1(b) the elements are

circular. Generally these discrete elements can take any shape and are rigid.

2.1 Computational Steps in Discrete Element Method:

1. Select the shape of the discrete element

2. Select the time step for simulation

3. Discretise the domain with discrete elements

4. Apply the initial and boundary conditions and loading.

5. Apply the laws of motion to compute the position of the particle

6. The displacement of the next step is calculated by the integration of equations of

motions.

The motion of particles in PFC-2D or any discrete element formulation is computed using

Newton’s laws of motion and Force displacement law in an explicit time scheme. In an

explicit time scheme, the position of a particle is computed using position, velocity and

acceleration of previous time steps. In an implicit time scheme, the position is calculated

using the previous time step data as well as present time step’s velocity.

For a particle, its neighbours are first detected using a contact detection algorithm. Its

displacement is used to find the resultant force F(t) using the force displacement law given

below.

(2.1)

Here Kn and Ks are the equivalent normal and shear stiffness which is calculated based on

the neighbours in contact of a particle. F(t) is the resultant force acting on a particle.

and are the relative normal and tangential displacement respectively between the

two contact entities. are the unit vectors in the normal and tangential directions

respectively.

This force F(t) is substituted in Newton’s law to find the resultant acceleration from

(Translational motion) (2.2)

Where m is the mass of the particle is the acceleration of the particle and g is the

acceleration due to gravity.

Velocities are calculated at mid-intervals. The acceleration is calculated as shown below.

(2.3)

The acceleration from equation (2.3) is substituted in (2.2) to calculate the velocity at time

as shown in (2.4)

(2.4)

The velocity is obtained by integrating the acceleration with respect to time. Then the

displacement is computed from previous position and the velocity of the particle calculated

in (2.4)

(2.5)

The displacement at time is found from equation (2.5). This displacement is put

into the force displacement law and the cycle continues. The cycle is shown in fig 2. Similar

equations can be written for rotational displacement by replacing the linear quantities by

their rotational equivalents- mass by moment of inertia, velocity by angular velocity (ω) and

displacement by angular displacement.

The friction co-efficient, co-efficient of damping between particles are also inputs given and

are incorporated in calculating forces or positions of the particle. These figures also

characterise the materials modelled in DEM along with the parameters like stiffness and

bond strength.

In PFC- 2D and other DEM software, walls can be used as physical constraints for unbound

systems like a system of marbles. These walls can also move independently and their

positions are also calculated in the same manner as described above.

Fig 2 Time step

are the various time steps (also equals the total

number of calculation cycles). T is the total time over which the event happens and is the

time step over which a calculation is carried out.

The time step over which the element positions are calculated is a critical issue in DEM

simulations. In DEM, only the neighbouring particles in contact with a particle are

considered for computing the resultant force on the particle. This is done based on the

assumption that the time step is so small that a disturbance (a force or displacement)

cannot travel any further than the immediate neighbours of a particle. If the time-step over

which the calculation performed is large, the computed values will be erroneous. This time-

step above which the results blows up in DEM is called critical time step.

Fig 3 Calculation cycle in DEM

2.2 Critical time step determination:

The time-step over which the position of the particles is updated is very crucial in terms of

the computed forces and positions as mentioned earlier. The critical time step is the largest

time-step for which a disturbance or a force is not transmitted beyond the neighbours of a

particle.

For an undamped system of multiple masses and springs, the critical time step is computed

using the following relation which has been derived in PFC-2D user manual [7].

(Translational motion) (2.6)

(Rotational motion) (2.7)

In a system of particles, the critical time step is calculated for each degree of freedom for

every particle assuming that degrees of freedom are uncoupled. The stiffnesses are

estimated by summing the contribution from all contacts of a particle. The minimum of the

, ,i i ix x [ ] [ ],C C

i ix n

iF

critical time steps is taken for the calculation. In PFC-2D, a fraction of this critical time step

can also be specified.

2.3 Stability in explicit integration

Integration schemes that require the use of a time step are

called conditionally stable. If the solution for any initial condition does not grow without

bound for any time step i.e. when

is large then the integration scheme is called

unconditionally stable.

A numerical model with N degrees of freedom contains N natural frequencies and

corresponding node shapes. Mathematically natural frequencies and mode shapes are eigen

values and eigen vectors .Since explicit integration is conditionally stable (disturbance must

not travel beyond the neighbours) the theory of spectral stability shows that the time step

should satisfy

(2.8)

for linear viscous damping where ξ is the fraction of critical damping at ωmax which is the

highest natural frequency of the mesh.

2.4 Assumptions in PFC-2D:

1. Particles have circular cross-section with finite mass and are rigid.

2. Particles may translate or rotate independently of each other.

3. Particles interact only at contacts and a contact comprises of only two particles.

4. Particle overlap is negligible compared to particle size.(model deformation is through

these particle overlap)

5. Bonds exist at contacts which break when load on them exceeds bond strength.

6. Generalised force displacement laws relate particle motion to force and moment at

that contact.

A contact model describes the physical behaviour occurring at a contact. Contact models

can be defined apart from the existing contact models in PFC-2D. The other features and

details of PFC have been described in detail in PFC-2D manual [7].

Chapter 3

ISSUES IN PFC-2D SIMULATIONS

Solving Problems in PFC-2D involves material modelling, loading the model and

interpretation of the results. After modelling the material, the model is generally loaded by

movement of walls or applying a gravity force whose magnitude can be specified. The

results in PFC-2D are interpreted from plots between required parameters like velocity,

displacement against time steps.

3.1 Material modelling in PFC-2D

Fig 4 Material modelling in PFC-2D

Material modelling involves particle generation, assigning the material characterising

parameters-stiffness and bond strengths and arrangement or bringing the particular

assembly to an initial stress condition, and applying the boundary conditions. The boundary

conditions specify the state of the model at its peripheries and initial conditions refer to its

state prior to an event (loading).

In DEM any material will be modelled as an assembly of particles in an ordered or random

fashion. In case of a random arrangement, the particles will be generated at random

positions within a container constructed with walls. These randomly generated particles are

then compacted such that a desired porosity is achieved. In case of an ordered

arrangement, the position of each ball must be specified at the time of generation so that

Particle generation

Assigning material properties

Arrangement or compaction

Applying initial and boundary

conditions

the model forms the required pattern which may be hexagonal or orthogonal array of

particles.

The modelled material will have to be brought to an initial condition which may be an initial

stress, initial velocity or residual strain as dictated by the material whose response is to be

simulated. For instance, if the failure of a pre-stressed rod is to be studied, then the model

must be brought to that state of stress before loading. However this initialising can be done

only after the material properties have been assigned to the model.

The modelled material properties are characterised by the following parameters

Density

Normal stiffness (kn) and tangential stiffness (ks) represent elastic characteristics

(like Young’s modulus) of the materials in PFC-2D.

The bond strengths signify the failure criterion of the material.

Friction co-efficient and co-efficient of damping.

These input properties cannot be measured physically and have to be determined so that

the simulated material replicates the real material behaviour.

3.1.1 Determination of the stiffness parameters -kn and ks:

The stiffnesses kn and ks can be found for any random arrangement by the ad hoc process

of validating a numerical simulation of a standard laboratory test with an actual

experimental result. This process is done by applying a strain on a rectangular DEM model

with a given kn and ks. Different values of Poisson’s ratio ν are measured by varying ks

keeping kn constant, which will give Poisson’s ratio as a function of kn/ks. According to the

Poisson’s ratio of the material, kn/ks is determined and Kn is varied till the simulated

material gives the required Young’s modulus E.

Tavarez[2] employed a unit cell approach, for a hexagonal close packing of elements, to

derive kn and ks as functions of Young’s modulus(E) and Poisson’s ratio (ν) for two

conditions of plane stress and plane strain. The relations given below are for plane strain.

(3.1)

(3.2)

Where kn is normal stiffness of simulated material

ks is shear or tangential stiffness of simulated

Physical properties of the material:

E is Young’s modulus

T is Thickness

ν is Poisson’s ratio

Tavarez has derived kn and ks assuming that only one normal spring and one shear spring

exists at a contact between two elements. But in PFC-2D a contact consists of two springs in

series- one for each contact entity. Therefore the kn calculated above will be the equivalent

stiffness (kn/2) of the springs in series for a PFC-2D model. Therefore the following

equations have been used to calculate the stiffness for the simulations.

(3.3)

(3.4)

For an orthogonal arrangement of elements Potyondy [4] has derived kn by assuming the

material at each contact to be an elastic beam with stiffness equivalent to kn/2.

(3.5)

Ks is calculated assuming that the relation between kn and ks is analogous to the relation

between Young’s modulus and Shear Modulus.

(3.6)

3.1.2 Determining the failure criterion- bond strength

In general the failure criterion of the material will be considered as the yield strength or

ultimate tensile strength under gradual loading conditions. In case of impact, fracture

toughness is an appropriate failure criterion as it is a measure of the amount of energy that

a material can absorb before fracture.

The failure criterion in PFC-2D is specified as normal and shear bond-strengths for contact

bonds which are point contact bonds. PFC-2D also has an additional option of assigning

parallel bonds to particles which will have contact area between the particles. They can be

used to model the adhesives keeping the different particles of a composite together.

Tavarez[2] has calculated bond strength for different failure criteria for modelling concrete

impact. The bond strengths based upon yield Strength and fracture toughness are

calculated.

(3.7)

Fracture toughness is the amount of stress required to propagate a pre existing flaw which

may have occurred while machining or welding. When fracture toughness is taken as the

failure criterion, the bond strength is computed as shown below

(3.8)

Where f is the bond-strength

R is Radius of the particle

t is Thickness of the particle

S- Yield strength or Ultimate Tensile Strength

KIc –the fracture toughness of the material

3.1.3 Friction co-efficient and damping constant

Friction is a property of a system of materials as well as the surfaces. In a particle model

friction is the parameter that determines slip or relative sliding between particles. In PFC-

2D, friction is specified by the co-efficient of friction for the elements (balls). The friction is

not present between particles bonded with contact bonds. However the exclusive presence

of parallel bonds does not deactivate friction. The co-efficient of friction is chosen according

to the surface and materials being modelled.

Damping refers to the dissipation of kinetic energy. The damping of motion between

particles is governed by the damping co-efficient. The PFC-2D standard contact model offers

two modes of damping – local damping which acts on each ball and viscous damping that

acts at each contact.

Local damping applies a force with a magnitude proportional to unbalanced force to each

ball and is used in quasi-static simulations. In PFC-2D it is specified as a fraction of critical

damping. Viscous damping adds a normal and a shear dashpot which act in parallel to the

existing spring at each contact and is used for solving impact and free fall problems. Viscous

damping, in PFC-2D is specified as critical damping ratio (ratio of damping constant to

critical damping) in normal and shear directions.

For impact problems, the critical damping ratio must be related to a measured value of

restitution co-efficient. The co-efficient of restitution is a measure of energy loss that occurs

with the collision of two masses and varies with temperature, impact velocity, size, material,

and body shape. It is defined as the ratio of relative velocity between two bodies after

impact to the relative velocity before impact [7].

Assigning material characterising parameters to a model in PFC-2D or any other DEM

simulation model is important and since these properties cannot be measured, they have to

be determined appropriately according to the required material and loading conditions.

Chapter 4

Simulation and Results

The procedure carried out to simulate the impact has been discussed and results have been

presented in this chapter. The impact simulation of 2024 T351 aluminium target with copper

projectile is carried out here.

The simulation setup is shown in fig 5.The dimensions of the target and projectile are given

below

`

Fig 5 Target and projectile

In PFC-2D, the particles can be defined as spheres or cylinders. Here both copper and

aluminium 2024 are modelled with cylinders of 5mm height (out of the plane thickness), as

the diameter of the projectiles used by Woodward [8] is around 5mm. The particles are

generated in hexagonal close packing and the medium is flawless, i.e. no initial cracks exist

for both the target and projectile. Hexagonal close packing of particles is used due to the

ease in calculating the stiffness values kn and ks. Only contact bonds are used between the

particles of the target.

The copper projectile is idealised to be a rigid body. The copper projectile is modelled as a

clump to make it rigid. A clump is a group of particles which have no relative motion

A B

C D

between each other. Bonds cannot be assigned between particles in a clump as the particles

cannot disintegrate unless the clump is deactivated.

The properties of both the materials are enlisted in the table 1

Table 1:

Property/Material Copper Aluminium

Young’s Modulus(E) 114GPa 73.1GPa

Poisson’s ratio(ν) 0.34 0.33

Yield Stress 70MPa 324MPa

Fracture toughness 37MPa-m1/2

The plane strain condition was used to calculate the stiffness for hexagonal packing due to

negligible strain of the target in the direction perpendicular to the motion of the projectile.

The spring stiffness for the hexagonal close packing is calculated using equations (3.3) and

(3.4). The bond strengths are calculated using equation (3.7) using yield stress as failure

criterion. The bond strengths for the projectile does not matter as it has been modelled as a

clump as mentioned before. Fixed boundary conditions are applied at the boundaries AB

and CD of the aluminium target as shown above in fig 5. The projectile is initialised with a

constant velocity.

PFC-2D calculates the critical time step and a fraction (0.7) of this time step is mentioned as

safety factor. A co-efficient of friction of 0.5 and a damping ratio of 0.157 are used for the

simulations. These values were chosen by trial and error.

The study of effects of variation of damping ratio and co-efficient of friction were carried

out by varying these parameters individually keeping other model parameters constant for

different particle sizes. It is found that the damping co-efficient does not affect the

simulated depth of penetration but affects time of penetration. The co-efficient of friction

affects the depth of penetration. A suitable value must be found so that the modelled

material replicates the actual material behaviour.

The modelling of the ballistic impact in PFC-2D involves material validation as the material

properties cannot be directly given as input to PFC-2D and has to be related to parameters

which cannot be measured experimentally. The static deflection of a DEM model of a

cantilever was done and compared with the analytical results. A hexagonal arrangement of

materials is employed along with the yield strength as the yield strength for particle sizes of

0.125mm radius. In fig 6 below, AC is fixed and a shear force is applied along the side BD.

The values are given below in Table 2. The deviation of values from the analytical values is

supposed to be due to the pronounced effect of shear due the small length of the beam

chosen.

Fig 6 cantilever beam (All dimensions are in mm)

Table 2:

Material Force (Analytical)deflection(mm) PFC-2D deflection(mm)

Aluminium 1N 5.47e-5 4.5e-5

2N 1.09e-4 9.2e-5

Copper 1N 3.5e-5 2.6e-5

2N 7.01e-5 5.5e-5

Fig 7 PFC-2D simulation of a cantilever deflection

1

10 1

A B

C D

The depth of penetration for a velocity of 420 m/s is studied for different discrete element

sizes- 0.25mm, 0.125mm and 0.0625 to check for convergence for yield as failure criteria.

Fig 8 Convergence results

The depth of penetration for velocities ranging from 230m/s to 738m/s are obtained for a

mesh with element radius of .125 mm and the results are plotted in fig 9. The results of 1-D

DEM from [10] have also been plotted for comparison.

Fig 9 Depth of penetration for hexagonal packing

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

0 0.05 0.1 0.15 0.2 0.25 0.3

De

pth

of

pe

net

rati

on

(mm

)

Element Radius (mm)

0

1

2

3

4

5

6

7

0 200 400 600 800 1000 1200

De

pth

of

pe

net

rati

on

(mm

)

Velocity(m/s)

experimental

2-D DEM with PFC-2D

1-D DEM

The PFC-2D visualisation for the impact is shown in fig 9.

Fig 10 PFC-2D simulation of impact

Fig 11 Impact crater formed during penetration of Aluminium target

Chapter 5

Conclusion

A simulation of the impact of the copper projectile on an Aluminium target was performed

using DEM software PFC-2D. The depth of penetration for various velocities was compared

with experimental results and the values obtained from the simulation deviate from

experiment at higher velocities. This discrepancy is attributed to lack of material property

validation and not considering the variation in strength and stiffness due to the increase in

temperature due to heat dissipation during impact. Yield strength is used as failure criterion,

but fracture toughness will be more appropriate for the event of impact. Yield strength was

used to compare with the results obtained in [10].

The effect of simulation results with various DEM parameters was also carried out. Varying

the damping ratio keeping other parameters constant does not affect the depth of

penetration, but it affects the simulation time. The friction co-efficient affects the depth of

penetration and a suitable co-efficient must be assigned to the materials. A convergence

study was also performed to analyse the effect of DEM particle size. A suitable damping

constant must also be assigned to the material in case of a temporal study of the event.

5.1 Scope for future work:

The simulation has been performed assuming that the discrete elements are hexagonally

packed. Performing the simulation for an irregular arrangement or any other arrangement

and verifying that the depth of penetration is independent of the arrangement will be useful

in proving usefulness of DEM in impact simulation as the Depth of penetration is only

dependent on the material properties and velocity of the projectile. Studying the effect of

porosity will provide further insight to material modelling for impact in DEM.

The simulation discussed above has assumed idealisations such as a rigid projectile and a

flawless medium of the target and projectile. Performing a simulation for a medium with

flaws and incorporating a deformable projectile would be required for the study of the

projectile deformation and disintegration.

The event of impact involves heat transfer and consequent variation of material properties

like strength and stiffness which have not been considered in these simulations. Tavarez [2]

has discussed the consideration of thermodynamic aspects for concrete impact.

Acknowledgements

First and foremost I would like to thank the Department of Applied Mechanics, Indian

Institute of Technology which granted me an opportunity to participate in the Summer

Fellowship Program 2011. Sincere thanks to my guide Dr C. Lakshmana Rao for accepting me

and allowing me to work in the Piezoelectric Sensors and Actuators lab. His expertise in

computational methods, constant guidance, support and motivation made this project an

impeccable experience.

Sincere thanks to Rajesh P.Nair for being with me throughout this endeavour, helping me

with relevant reference material, in an exemplary way.

Special thanks to research scholar Srinivasan who helped me document and compile this

report.

I am grateful to research scholars Sandeep Jose, Sunir Hassan, Bharat, Athmanathan,

Sandhya, Ramarao and Akilesan for their encouragement and numerous conversations

which were intellectually enriching and inspiring.

Lastly I would like to express my sincere gratitude to my parents for their constant support

and understanding.

References

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Geotechnique(1979);

2. Fedrico A Tavarez, Michael E Plesha. Discrete element method for modelling solid and

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properties. Proceedings of the 3rd international conference on Discrete Element Methods

(2002)

5. F.Camborde, C.Mariotti, F.V. Donze. Numerical study of rock and concrete behaviour by

discrete element modelling (2000)

6. S.A. Magnier, F.V.Donze.“Numerical simulation of impacts using a discrete element

method”

7. PFC-2D manual version 3.1 Theory and Background, Itasca.

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Int.J.Mech.Sci.Vol 24.No. 2, 1982

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ammunition

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Impact on Thick Targets using Discrete Element Method: 1-D and 2-D models.

11. Irving H. Shames, G.Krishna Mohana Rao. Engineering Mechanics: Statics and Dynamics

12. Kurtaran H, Buyuk M, Eskandarian A, “Ballistic impact simulation of GT model vehicle

door using Finite Element Method,” Theoret. Appl. Fracture Mech., 40, 113– 121(2003)