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.
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