Minor Report.pdf

8/10/2019 Minor Report.pdf

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MINOR REPORT

Modelling and Simulation of Steady State Penetrationof Rigid Perfectly Plastic Targets

Under the guidance of

Dr. Punit Kumar

Associate Professor

Department of Mechanical Engineering

NIT Kurukshetra

Submitted by:

Prabhat Vashishth 110811

Sachin Chaudhary 110437

Gaurav Nagpal 110655

Somdeep Ahlawat 110724

Ishaan S. Gyan 110509

Asheet Pradhan 110589


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ACKNOWLEDGEMENT

We acknowledge the support, the encouragement, extendedfor this study by our guide Dr. Punit Kumar. We greatly

appreciate the motivation and understanding extended for

the project work by sir.

We are also thankful to my college Librry Staff and

Administrative Staff, who directly or indirectly have been

helpful in some or the other way.

We thank our Dearest parents, who encouraged me to

extend my reach. We are indebted to all of them, who did

their best to bring improvements through their suggestions.

Prabhat Vashishth

Sachin Chaudhary

Gaurav Nagpal

Somdeep Ahlawat

Ishaan S. Gyan

Asheet Pradhan


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Content

1.Introduction

2.Problem formulation2.1 Mathematical description

2.2 Governing Equations

3.Fem modelling

4.Simulation

4.1 Geometry

4.2 Simulation parameters

4.3 Material properties

5.Results and Discussion

6.Future Scope

7.References


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1.

Introduction

Ballistic perforation is an exceedingly complex mechanical process that has

been examined for the past 200 years. At the present time, there are three

reasonably distinct directions for these investigations: derivation of empirical

formulas based on extensive testing, development of relatively 'simple' models

of the perforation process and applying the relevant equations of motion and

material behaviour, and full numerical solutions based on solving all the

governing equations over a spatial grid. Because of the computer resources

required and the expense in performing a large variety of parametric studies,there has been considerable interest in the intermediate or engineering

modelling approach.

A number of review articles on ballistic perforation have been published in the

past few years. Some of these contain detailed descriptions and give

appropriate reference to the various engineering models and numerical

techniques that have been proposed up to the time of the surveys. Activity in

the field of developing new models of the perforation process is now strong; a

number of investigations are currently in progress, and new results are

appearing continually in the literature.

In this article a detailed simulation solution in COMSOL Multiphysics 4.3a to an

idealized penetration problem is presented . The approach taken is as follows.

Suppose that the penetrator is in the intermediate stages of penetration so

that the active target/penetrator interface is at least one or two penetrator

diameters away from either target face, and the remaining penetrator is still

much longer than several diameters and is still travelling at a speed close to itsstriking velocity.

2.Problem Formulation

The problem of steady penetration by a semi-infinite, rigid penetrator into an

infinite, rigid/perfectly plastic target has been studied. The rod is assumed to

be cylindrical, with a hemispherical nose, and the target is assumed to obey

the Von-Mises yield criterion with the associated flow rule. Contact between

target and penetrator has been assumed to be smooth and frictionless. Results


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computed and presented graphically include the velocity field in the target and

pressure contours.

2.1. Mathematical description

This situation is idealized here in several ways. First, it is assumed that the rod

is semi-infinite in length and that the target is infinite with a semi-infinite hole.

Furthermore, it is assumed that the rate of penetration and all flow fields are

steady as seen from the nose of the penetrator. These approximations are

reasonable if the major features of the plastic flow field become constant

within a diameter or so of the nose of the penetrator. Next, it is assumed that

no shear stress can be transmitted across the target/penetrator interface. Thisis justified on the grounds that a thin layer of material at the interface either

melts or is severely degraded by adiabatic shear. This assumption, together

with the previous one, makes it possible to decompose the problem into two

parts in which either a rigid rod penetrates a deformable target or a

deformable rod is upset at the bottom of a hole in a rigid target. Of course, in

the combined case the contour of the hole is unknown, but if it can be chosen

so that normal stresses match in the two cases along the whole boundary

between penetrator and target, then the complete solution is known

irrespective of the relative motion at the boundary. Finally, the deforming

material is assumed to be rigid/perfectly plastic. This assumption should be

adequate for examining the flow and stress fields near the penetrator nose,

but will lose accuracy with increasing distance, since it forces the effects of

compressibility and wave propagation to be ignored. In this study only the case

of the deforming target and a rigid penetrator is considered, where the

penetrator is assumed to have a circular cylindrical body and a hemispherical

nose.

2.2. Governing Equation

With respect to a set of cylindrical coordinate axes fixed to the centre of the

hemispherical nose of the rigid cylindrical penetrator, equations governing the

deformations of the target are

0

z

w

r

u [1]


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Equation 1 expresses the balance of mass and implies that the target

undergoes only volume preserving deformations so that the mass density of

the target stays constant.

z

uwr

uuzr

rzr

[2]

z

ww

r

wu

rz

rzz

[3]

Equation 2 and 3 expresses the balance of linear momentum in the absence of

body forces and holds in all Galilean coordinate systems. In particular it holds

in one that translates at the constant velocity of the penetrator.

z

w

r

w

z

u

r

w

z

u

r

u

Ip

po

zrz

rzr

2

1

2

1

3

[4]

Equation 4 expresses flow rule for rigid perfectly plastic and is based

on von- mises yield criterion.

Solving above four equations we get-

0u w

r z

[5]

12

2 3

p u u w u wu w

r r r z z r r z I

[6]

12

2 3

p w u w w wu w

z z z r z r r zI

[7]

Where,2

0

0

v

In order to solve the problem numerically, it is possible to consider only a finite

region of the target, and since deformations of the target are axisymmetric,

only the target region shown in Fig. 1 is studied.


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Fig.1: The region to be studied

Whether the region considered is adequate or not can be easily decided by

solving the problem for two different values of the parameter . If the two

solutions so obtained are essentially equal to each other in the vicinity of the

penetrator, then the region studied is sufficient and the effect of boundary

conditions at the outer surface EFA has a negligible effect on the deformations

of the target material in close proximity to the penetrator.The boundary conditions imposed on the finite region are

0, 0zz rv on the bottom surface AB [8]

t.n = 0, v.n = 0 on the common interface BCD [9]

0, 0rz rv on the axis of symmetry DE [10]

0, 1r zv v

on the boundary surface EFA [11]

3.FEM modelling

The finite element analysis is a numerical technique. In this method all the

complexities of the problems, like varying shape, boundary conditions and

loads are maintained as they are but the solutions obtained are approximate.

Because of its diversity and flexibility as an analysis tool, it is receiving much

attention in engineering. The fast improvements in computer hardware

technology and slashing of cost of computers have boosted this method, sincethe computer is the basic need for the application of this method. A number of


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popular brand of finite element analysis packages are now available

commercially. Some of the popular packages are STAAD-PRO, GT-STRUDEL,

NASTRAN, NISA and ANSYS. Using these packages one can analyse several

complex structures. The finite element analysis originated as a method of

stress analysis in the design of aircrafts. It started as an extension of matrixmethod of structural analysis. Today this method is used not only for the

analysis in solid mechanics, but even in the analysis of fluid flow, heat transfer,

electric and magnetic fields and many others. Civil engineers use this method

extensively for the analysis of beams, space frames, plates, shells, folded

plates, foundations, rock mechanics problems and seepage analysis of fluid

through porous media. Both static and dynamic problems can be handled by

finite element analysis. This method is used extensively for the analysis and

design of ships, aircrafts, space crafts, electric motors and heat engines.

4.Simulation

This model demonstrates the fluidstructure interaction interface for studying

the ballistic perforation phenomenon for a given set of material properties.

The geometry is in two dimensions and the model is isothermal. This model

can be used to investigate the influence of various design parameters such as

the choice of materials, dimensions, flow stresses, density etc. COMSOL

Multiphysics 4.3a have been used for simulation work

4.1. Geometry

Fig 2. Geometry of the region to be studied in COMSOL Multiphysics 4.3a


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Table 1. unit used in geometry

Length unit m

Angular unit deg

Table 2. Geometry statisticsProperty Value

Space dimension 2

Number of domains 3

Number of boundaries 12

Table 3. Dimensions of Circle 1 (c1)

Name Value

Position {0, 0}

Sector angle 90

Table 4. Dimensions of Rectangle 1 (r1)

Name Value

Position {0, 0.3}

Height 0.3

Size {1, 0.3}

Table 5. Dimensions of Circle 2 (c2)

Name Value

Position {0, 0}

Radius 0.2

Sector angle 90

Table 6. Dimensions of Rectangle 2 (r2)

Name Value

Position {0, 0.5}

Width 0.2

Height 0.5

Size {0.2, 0.5}


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4.2. Material used and their properties

For simulation purpose structural steel as the target and cast iron as the

penetrator have been used. The material properties of structural steel have

a great influence on the parameter .The value of is taken as four in

order to compare results with Batra et al.

Fig 3. Domain 2 showing cast iron

Table 7:Selection of domain for cast iron

Geometric entity level Domain

Selection Domains 12

Table 8: Material parameters of cast iron

Name Value Unit

Density 7000 kg/m^3

Young's modulus 140e9 Pa

Poisson's ratio 0.25 1


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Fig 3. Domain 2 showing structural steel

Table 9:Selection of domain for structural steel

Geometric entity level Domain

Selection Domain 3

Table 10:Material parameters of structural steel

Name Value Unit

Density 7850 kg/m^3

Flow stress/Strain rate 1700 Pa*s

Youngs Modulus 200e9 Pa

Poissions Ratio 0.33

4.3. Meshing Parameters

Table 11: Meshing statistics

Property Value

Minimum element quality 0.3601

Average element quality 0.8747

Triangular elements 319

Quadrilateral elements 20

Edge elements 62

Vertex elements 10


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Fig 4. Meshing obtained using COMSOL Multiphysics

5.Results and Discussion

Figure 4 shows the velocity field in target material relative to the hemispherical

penetrator. In target points that lie to the rear of the centre of the penetrator

nose, the flow quickly becomes essentially parallel to the axis of the

penetrator. Target points that lie ahead of the penetrator nose and within one

penetrator diameter from it have a noticeable radial component of velocity.

Figure 5 shows the pressure contour in the target. The pressure is maximum at

the tip of nose of penetrator. The maximum pressure is almost 3 times the

value of flow stress0

at the nose tip.


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Fig 5. Velocity distribution in the target

Fig 6. Pressure contour in the penetrator/target

6.Future scope

The simulated results can be brought closer to Batra et al. by solving the

equations numerically . Thus a computer code for solving the equations

numerically can be made and used for the design parameters given above. So

we are trying to make a computer code using MATLAB and MATHEMATICA. Till

now we have generated the mesh using MATLAB. The code used, mesh


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generated and the coordinate matrix obtained in the MATLAB is given below.

In order to solve the equation using finite element method, six noded iso-

parametric triangular elements have been used.

Fig7. Six noded iso parametric triangular element

>> x=[];

>> y=[];

>> for j=1:5

for i=0:5

if(mod(j,2)==0)

a=j*sin((pi/12)*i);

b=(j+1)*sin((pi/12)*i);

c=(j+1)*sin((pi/12)*(i+1));

d=j*sin((pi/12)*i);

e=j*sin((pi/12)*(i+1));

f=(j+1)*sin((pi/12)*(i+1));

p=j*cos((pi/12)*i);

q=(j+1)*cos((pi/12)*i);


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r=(j+1)*cos((pi/12)*(i+1));

s=j*cos((pi/12)*i);

t=j*cos((pi/12)*(i+1));

u=(j+1)*cos((pi/12)*(i+1));

x=horzcat(x,a,b,c,d,e,f);

y=horzcat(y,p,q,r,s,t,u);

else

a=j*sin((pi/12)*(6-i));

b=j*sin((pi/12)*(5-i));

c= (j+1)*sin((pi/12)*(6-i));

d= (j+1)*sin((pi/12)*(5-i));

e=j*sin((pi/12)*(5-i));

f= (j+1)*sin((pi/12)*(6-i));

p=j*cos((pi/12)*(6-i));

q=j*cos((pi/12)*(5-i));

r=(j+1)*cos((pi/12)*(6-i));

s=(j+1)*cos((pi/12)*(5-i));

t=j*cos((pi/12)*(5-i));

u=(j+1)*cos((pi/12)*(6-i));

x=horzcat(x,a,b,c,d,e,f);

y=horzcat(y,p,q,r,s,t,u);

end

end

end

>> plot(y,x)

>> plot(x,y)


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Fig 8. Meshing obtained after MATLAB code.

Coordinates matrix and its code

coordinatesX=[];

coordinatesY=[];

for j=1:5

for i=1:12

x=[];

y=[];

if(mod(i,2)==0)

s=i/2;

else

s=((i-mod(i,2))/2);

end

if(mod(i,2)==0)

a=j*sin((pi/12)*(s-1));

e=j*sin((pi/12)*(s));

c=(j+1)*sin((pi/12)*(s));

p=j*cos((pi/12)*(s-1));

g=j*cos((pi/12)*s);

r=(j+1)*cos((pi/12)*s);


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x=horzcat(x,a,e,c);

y=horzcat(y,p,q,r);

coordinatesX=vertcat(coordinatesX,x);

coordinatesY=vertcat(coordinatesY,y);

else

a=j*sin((pi/12)*s);

b=(j+1)*sin((pi/12)*s);

c=(j+1)*sin((pi/12)*(s+1));

p=j*cos((pi/12)*s);

q=(j+1)*cos((pi/12)*s);

r=(j+1)*cos((pi/12)*(s+1));

x=horzcat(x,a,b,c);

y=horzcat(y,p,q,r);

coordinatesX=vertcat(coordinatesX,x);

coordinatesY=vertcat(coordinatesY,y);

end

end

end

dlmwrite('myfile.dat', coordinatesX, 'delimiter', '\t', ...

'precision', 6)

type myfile.dat

dlmwrite('myfile1.dat', coordinatesY, 'delimiter', '\t', ...

'precision', 6)

This generates a matrix for the x coordinates:


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The y-coordinates are given below


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7.References

1. A. Tate, J. Me& Phys. Solids 15, 387 ,19672. A. Tate,J.Mech. Phys. Solids17, 141, 19693. R.C. Batra and T.W. Wright,

InI. J. Engn Sci.241, 41-54, 19864. Chales E. Anderson,Jr. and SOL R. Bodner,Int. J. Impact Engng 71, 9-35, 19885. David S. Burnett, Finite Element Analysis, Addison-Wesley Publication,19876. P.Seshu, Finite Element Analysis, PHI leaning private limited, 2013

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