anason final report

Studying the Effects of Repeated Impacts on Titanium 6Al-4V Alloy

by

Christopher Anason

An Engineering Project Submitted to the Graduate Faculty

of Rensselaer Polytechnic Institute

in Partial Fulfillment of the Requirements

for the degree of

MASTER OF ENGINEERING

Major Subject: Mechanical Engineering

Approved:

______________________________________________

Ernesto Gutierrez-Miravete, Engineering Project Adviser

Rensselaer Polytechnic Institute

Hartford, CT

December, 2010

(For Graduation December, 2010)

ii

© Copyright 2010

by

Christopher Anason

All Rights Reserved

iii

CONTENTS

LIST OF TABLES ............................................................................................................. v

LIST OF FIGURES .......................................................................................................... vi

NOMENCLATURE ....................................................................................................... viii

ACKNOWLEDGMENT .................................................................................................. ix

ABSTRACT ...................................................................................................................... x

1. Introduction .................................................................................................................. 1

1.1 Problem Description........................................................................................... 2

2. Methodology ................................................................................................................ 3

2.1 Static Analysis of Dynamic Impacts .................................................................. 3

2.1.1 Impact Pressure as a Function of Radius ............................................... 4

2.2 Axis-Symmetric Assumption ............................................................................. 5

2.3 Single and Multiple Impact Simulations ............................................................ 6

2.4 Saint-Venant’s Principle .................................................................................... 8

3. Results and Discussion .............................................................................................. 10

3.1 Determination of the Hertzian Pressure Distribution ....................................... 10

3.2 Modeling the First Impact ................................................................................ 13

3.2.1 Mesh, Boundary Conditions, Material Conditions, and Loads ............ 13

3.2.2 First Impact - Loading and Unloading ................................................. 14

3.3 Analysis of Four Loading/Unloading Cycles for Specified Case .................... 17

3.3.1 Analysis of Subsequent Impacts .......................................................... 19

3.4 Analysis of Maximum Stress/Strain Values For All Cases ............................. 26

4. Conclusions................................................................................................................ 32

4.1 Suggestions and Recommendations for Future Work ...................................... 33

5. References .................................................................................................................. 35

6. Appendices ................................................................................................................ 36

iv

6.1 Appendix A ...................................................................................................... 36

6.2 Appendix B ...................................................................................................... 38

6.3 Appendix C ...................................................................................................... 41

v

LIST OF TABLES

Table 1 - Steel Shot Radius and Velocity used by (5) ....................................................... 7

Table 2 - Projectile Size and Velocity to be Simulated by COMSOL Model ................... 7

Table 3 - Featured case analyzed by MATLAB (R=0.00025m) ..................................... 11

Table 4 - Mesh Statistics of COMSOL Model “Ti-6AL-4V Impact.mph” ..................... 14

Table 5 - Parameters for Specific Case Discussed in Section 3.3 ................................... 18

Table 6 - Calculated values of amax and pmax for each tested combination of R and v ..... 27

Table 7 - Stress/Strain Results for specific case (R = 0.00025m, v = 35 m/s) ................. 28

Table 8 - Maximum Compressive Radial Stresses After Impacts 1-4 with Incremental

and Percent Changes ........................................................................................................ 31

Table 9 - Material Properties for Ti 6Al-4V and Steel .................................................... 36

Table 10 - Derived Parameters used in COMSOL Analyses .......................................... 37

vi

LIST OF FIGURES

Figure 1 - Sketch representing the axis-symmetric model and 2-dimensional projection

to be modeled ..................................................................................................................... 5

Figure 2 - Sketches showing the "Projectile" and "Target" prior to the first impact ......... 6

Figure 3 - Sketches showing the second projectile and the deformed target before and

during the second impact ................................................................................................... 6

Figure 4 - Mesh Showing Finer Element Resolution Near the Impact Site ...................... 8

Figure 5 - Deformed shape and effective plastic strain (plotted by color), after first

impact (R = 0.0003m, v = 60 m/s) ..................................................................................... 9

Figure 6 - Plot of Pressure vs. Radius for R = 0.00025m ................................................ 12

Figure 7 - Boundary Conditions and Load applied to COMSOL model ......................... 13

Figure 8 - Surface Plot of Radial Stress After First Impact (load active) ........................ 15

Figure 9 - Photograph of Meteor Crater, Arizona – from (7) .......................................... 16

Figure 10 - Surface Plot of Radial Stress After First Impact (unloaded) ........................ 17

Figure 11 - Radial Stress and Radial Displacement plotted for the Loading and

Unloading of Impacts 1-4 (2 plots), Top Surface ............................................................ 20

Figure 12 - Radial Stress and Axial Displacement plotted for Impacts 1-4 (2 plots),

Along Axis of Impact ...................................................................................................... 22

Figure 13 - Axial Strain plotted for Impacts 1-4 (2 plots) ............................................... 24

Figure 14 - Plastic Strain plotted for Impacts 1-4 (2 plots) ............................................. 25

Figure 15 - Plastic Strain for Impacts 1-4 ........................................................................ 29

Figure 16 - Maximum Tensile Radial Stress - Impacts 1-4 Loading/Unloading Cycles . 30

Figure 17 - Maximum Compressive Radial Stress - Impacts 1-4 Loading/Unloading

Cycles .............................................................................................................................. 30

Figure 18 - Maximum Compressive Radial Stresses After Impacts 1-4 ......................... 31

vii

LIST OF EQUATIONS

[1] Maximum Force Developed at Impact

((2), Equation 2.8) ………….……………………………………............... 4

[2] Maximum Radius of the Circle of Contact ((2), Equation 2.9) …………... 4

[3] Maximum Normal Pressure in the Circle of Contact

((2), Equation 2.10) ………………………………………………………... 4

[4] Normal Pressure Distribution ((2), Equation 2.2)………………………….. 5

[5] Applied Pressure Load to Boundary 3…………………………………….. 14

viii

NOMENCLATURE

Symbol Description Units

mP Maximum Force Developed During Impact N

ρ Density of the Projectile m

kg

R Radius of the Impacting Projectile m

r Radius from the Axis of Impact m

v Velocity of the Impacting Sphere m/s

ν Poisson’s Ratio [ ]

ma Maximum Radius of the Circle of Contact m

a Radius of the Circle of Contact m

z Distance from Top Surface (in z-direction) m

E Young’s Modulus Pa

mp′, maxp′ Maximum Normal Pressure in the Circle of Contact Pa

p Normal Pressure Pa

p′ Normal Pressure in the Circle of Contact Pa

yσ Yield Strength Pa

a Coefficient of Thermal Expansion Co

610 −

plE Modulus of Plasticity Pa

yε Strain at Onset of Yielding [ ]

max,plε

Maximum Plastic Strain [ ]

max,,trσ Maximum Tensile Radial Stress Pa

max,,crσ Maximum Compressive Radial Stress Pa

zε Axial Strain [ ]

plε

Plastic Strain [ ]

zu

Axial Displacement m

A Johnson-Cook Yield Strength Pa

B Johnson-Cook Hardening Coefficient Pa

n Johnson-Cook Strain Hardening Exponent [ ]

ix

ACKNOWLEDGMENT

I would like to thank my family and friends for their support throughout my graduate

study and through the completion of this Engineering Project. I would also like to thank

the faculty and staff of Rensselaer for their assistance throughout these past three years.

Lastly, I would like to thank Professor Gutierrez-Miravete for his guidance and advice

with the intricacies of this project, not the least of which were technical issues with

COMSOL, plastic deformation, and finite element analysis technique. Thank you!

x

ABSTRACT

The goal of this project was to model and analyze the repeated impact of steel projectiles

onto a sample of Titanium 6Al-4V (Ti64) Alloy. In lieu of a dynamic finite element

analysis, the effects of a dynamic impact were represented as a Hertzian pressure

distribution. The Ti64 sample was loaded and unloaded statically, to simulate the effect

of an impact without the costly dynamic analysis. The analysis was run for 4 impacts of

25 combinations involving 5 projectile sizes and 5 projectile velocities. For each of the

cases, the residual stresses and strains were analyzed after each impact. The condition

was quantified by recording the maximum plastic strain, and the maximum tensile and

compressive radial stresses. After completion of COMSOL analysis of several

combinations of projectile size and projectile velocity, the resulting data suggests that

the majority of the impact effects (residual stresses and plastic strains) were inflicted

during the first impact, and to a smaller extent, the second impact. Subsequent impacts

produced negligible effects in the stress/strain fields, suggesting that additional impacts

will have a decreasing effect as the number of impacts increase.

1. Introduction

In all fields of engineering, components must be designed with a certain level of

robustness so that in normal operating conditions (and sometimes adverse conditions),

the component will function as designed. This includes components that must be able to

withstand impacts, which include objects that are in motion (automobiles, airplane jet

turbines, etc.), objects that are stationary but at risk of being hit by moving objects

(bridge pilings, highway guardrails, etc.), and objects that are designed to protect

humans from workplace dangers (Kevlar® bulletproof vests, safety glasses, sports

helmets, etc.). During the design of these products, thousands of hours of painstaking

research, analysis, and testing were put into designing the component so that it could

survive one impact; however whether they are designed for more than one impact

remains to be seen.

When discussing the example of safety glasses, the product is designed to protect

the wearer from one major impact event, and then the products are disposed of. Once

the glasses are cracked, the structural integrity of the plastic is compromised, and the

glasses are replaced. Similarly, when automobiles are subjected to high-speed impacts

(on road or in crash testing), the metal portions of the frame and body are more than

likely scrapped, as large deformations are difficult to repair.

For components that do not see catastrophic impacts, gradual effects from small-

magnitude impacts must be analyzed to determine what happens to a material when it is

subjected to repeated impact loadings. Parts that are difficult to manufacture, parts that

are built to be durable, and parts that are too expensive to be disposable could all benefit

from the analysis of repeated impact damages.

Examples of the usefulness of analysis of repeated impacts are many in number

and varied in type, and include examples such as:

• Civil Engineers must be able to say with certainty that a bridge support can

withstand numerous small impacts throughout its service life.

• Sports protective helmets must be durable enough to withstand many impacts

throughout their lives, which may span several seasons. In addition,

2

motorcycle helmets must conform to D.O.T. regulations in regards to safety

requirements.

• Kevlar® bulletproof vests must be capable of withstanding multiple bullet

impacts, in the case that a police officer or military soldier is hit multiple

times.

• Shot peening treatment of metals for fatigue life enhancement

1.1 Problem Description

Through this project, as will be explained in the following chapters, finite element

analysis will be utilized to analyze trends in how certain materials will respond to single

and repeated impacts. Numerous impact simulations involving a spherical element

(henceforth referred to as the “projectile”) hitting a flat plate (henceforth referred to as

the “target”) will be performed. Additionally, the effects of variables such as:

• Projectile Velocity

• Projectile Size

will be studied in order to determine trends in repeated impact situations.

(2) shows that through experiment, the forces and stresses imparted by a steel

ball onto a steel plate (whether being pressed into the plate or being dropped onto the

plate) can be approximated by a pressure distribution over the impact area. Thus, a static

simulation using this “approximation” pressure distribution can be performed for the

purposes of this project, using a static analysis Finite Element solver such as COMSOL,

in lieu of a solver more tailored towards the more cumbersome and costly dynamic

analysis, such as LS-DYNA. As a result, more analyses can be performed, allowing the

testing of the variables mentioned in the previous paragraph.

3

2. Methodology

Impacts can happen in an infinite number of situations and geometries; however this

study will analyze the residual stresses and plastic strains based upon one specific

geometry. A spherical projectile will be impacting a flat plate target, with the dynamic

impact being approximated using static analysis. This impact simulation will then be

repeated in order to extract various trends in stresses or strains as a function of projectile

size or impact velocity.

2.1 Static Analysis of Dynamic Impacts

Dynamic analysis of an impact event is complicated when using Finite Element solvers,

due to non-linear factors such as:

1. Large deformations occurring in both objects

2. Changing boundary conditions due to contact between the two objects

3. Dissipation of momentum in the projectile

For these reasons, a dynamic impact simulation is cumbersome and costly (both in time

and computing power), and would make it especially difficult to run repeated impacts

models while varying certain parameters of the impact, as is this project’s goal.

Alternately, performing a static analysis can be achieved many times with

relatively small run-times, easily changing variables between runs, and quickly getting

large amounts of data that could reveal trends during repeated impacts. Approximation

of a dynamic impact can be performed using a static analysis because it has been shown

experimentally that impacts of a spherical projectile impart a stress distribution on the

base material that is a function of radius from the center of impact, the material

properties of both objects, and dimensional properties of the projectile.

(2) shows that experimentally, the pressure imparted by the steel sphere onto a flat

plate can be well represented by a system of equations that involve projectile size,

projectile velocity, and material properties of the sphere. For the purposes of this

analysis, the pressure as a function of radius from impact location can be determined for

a given projectile size, velocity, or material, as is shown in Section 2.1.1.

4

2.1.1 Impact Pressure as a Function of Radius

In (2), it is shown that dynamic impacts can be equated to a distributed load applied as a

static force, through an extrapolation of Hertz’s theory of contact between two elastic

spheres. Davies assumes that one elastic sphere is large enough to be considered “semi-

infinite” when compared to the other, thus equating the system to a sphere impacting a

flat plate.

It follows that for a Hertzian impact between an elastic sphere and an elastic

plate, the initial contact between the two bodies will produce a force of zero, but as the

elastic bodies begin to deform, the force of impact will increase until it reaches a

maximum. This maximum force is the point at which the bodies stop flattening (see

Equation [1]), and begin to spring back to their original shapes. The contact force

between the sphere and flat plate will therefore decrease after reaching its maximum.

( ) 2

3

25

2

25

3

m1

5.23

2P vR

E

−

⋅⋅=ν

ρπ [1]

The radius of the “circle of contact” between the sphere and the flat plate can be

calculated based on geometrical and material factors, and is used to calculate the normal

pressure. See Equation [2].

5

25

1

215.2 Rv

Eam

−⋅⋅=

νρπ

[2]

Additionally, the maximum normal pressure, mp′ , in the circle of contact can be

determined by Hertz’s equations of contact, particularly Equation [3], shown below:

( ) 5

25

4

25

1

15.2

1v

Epm ⋅

−

⋅⋅=′ν

ρππ [3]

Using the relation represented by Equation [4], the distribution of normal pressure as a

function of radius, )(rp , from the center of impact can then be determined.

Furthermore, using Equation [1] through Equation [4], the normal pressure distribution

can thus be shown to be functions of projectile velocity, size, and material, and the

radius from the center of impact.

5

( )a

rapp

22 −′=

[4]

The static representation of a dynamic impact explained above, in addition to using an

assumption of axis symmetry (See Section 2.2), this simplifies the problem at hand.

2.2 Axis-Symmetric Assumption

In order to further simplify the model in order to reduce run-time, increase the number of

simulations that can be run, and ultimately increase the number of variables that can be

tested, the analyses will be run on an axis-symmetric target (see Figure 1). In lieu of

having an object set in a coordinate system such as Cartesian, the use of a model with

axial symmetry allows for reduction of one variable in the coordinate system. Of the

three coordinates (radial r, circumferential θ, and axial z), the analysis parameters only

vary with changes in r and z. Nothing varies with changes in θ. Essentially, as (1)

explains it: “the problem is mathematically two-dimensional”.

Figure 1 - Sketch representing the axis-symmetric model and 2-dimensional projection to be

modeled

Since the assumption of axial symmetry requires all boundary conditions and

loads to be symmetric about the main axis, the model to be analyzed will simulate the

impact to occur at the center of the top of the target “disk”. The cross section of the

cylinder, extending from the axis of symmetry to the outside edge, will therefore be a 2-

dimensional model, to be oriented such that the center of the impact will occur at the

top-left corner of the cross-sectional view (see Figure 4).

6

This simplification of the model from 3-dimensions to a 2-dimensional cross-

section, and the additional reduction of taking one-half of the planar cross-section, will

drastically reduce the complexity of the model and run-time of the analysis, and allow

for a more refined mesh (where it is needed – see Section 2.4) for the same computing

power used.

2.3 Single and Multiple Impact Simulations

In order to study the effects of repeated impact simulations, multiple separate impacts

must be modeled in succession in order to analyze the effects after each impact (See

Figure 2 and Figure 3). Initially, the target will be undamaged, and the first impact will

be analyzed (as a static load approximation of an impact load). When the applied load is

removed from the model, the elastic deformation will be eliminated, leaving the plastic

(permanent) deformation and residual stresses. The target will have been permanently

deformed by the first impact (Figure 2). Using the deformed shape of the target as the

initial state, another impact is simulated (Figure 3), resulting in additional plastic

deformation.

Figure 2 - Sketches showing the "Projectile" and "Target" prior to the first impact

Figure 3 - Sketches showing the second projectile and the deformed target before and during the

second impact

7

Post-impact parameters to be analyzed include plastic deformation at locations

around the impact zone and residual stresses in the deformed region. These parameters

will be obtained for each impact simulation, allowing for comparison from impact to

impact. The goal of this repeated analysis is to reveal and analyze trends in residual

stresses and strains after several impacts.

Other factors that will be analyzed by this impact simulation include the size of

the projectile and the velocity of the projectile. Increasing the size (i.e. diameter of the

sphere) of the projectile will increase the area that is affected by the impact, however the

affected regions will remain within the enhanced mesh determined using Saint-Venant’s

Principle (see Section 2.4). Increasing or decreasing the velocity of the projectile as it

impacts the target will affect the penetrating power of the projectile, thus affecting how

deep the impact effects are felt each time.

In the shot-peening analysis performed in (5), projectile radius and velocity were

chosen to be a particular value, which is commonly used in shot-peening applications.

These values, from Table 2 of (5), are shown in Table 1. In order to study the effect of

each of these parameters, for this analysis, the radius and velocity will vary. The

projectile sizes and velocities to be simulated by this project are shown in Table 2.

Table 1 - Steel Shot Radius and Velocity used by (5)

Description Value

Radius, R 0.18 mm

Impact Velocity, v 18-50 m/s

Table 2 - Projectile Size and Velocity to be Simulated by COMSOL Model

Size of Projectile (m) Velocity of Projectile (m/s)

0.00010 10

0.00015 20

0.00020 35

0.00025 50

0.00030 60

2.4 Saint-Venant’s Principle

Invoking Saint-Venant’s

mesh, and will allow for quicker run

and nodes from the model. In

“If an actual distribution of f

distribution of stress and strain throughout the body is altered only near the regions of

load application.”

In the context of

a long beam are unlikely to significantly affect the other end. However, in

project’s finite element model to study the effects of impact, it means that areas far from

the impact zone are unlikely to see the same amounts of stresses and strains than the

areas closer to the impact zone. As a result, fewer elements and nodes need to be located

at areas far from the impact site, as represented in

can be utilized nearest the impact site to provide the resolution needed for an accurate

analysis.

Figure 4 - Mesh Showing Finer Element Resolu

8

Venant’s Principle

Venant’s Principle allows for additional simplification of the model’s

mesh, and will allow for quicker run-times and the elimination of unnecessary elements

and nodes from the model. In (6), Saint-Venant’s Principle is stated as:

If an actual distribution of forces is replaced by a statically equivalent system, the


In the context of (6), the definition above means that loads applied to one end of

a long beam are unlikely to significantly affect the other end. However, in


are unlikely to see the same amounts of stresses and strains than the


at areas far from the impact site, as represented in Figure 4. Additionally, a finer mesh

tilized nearest the impact site to provide the resolution needed for an accurate

Mesh Showing Finer Element Resolution Near the Impact S

Principle allows for additional simplification of the model’s

times and the elimination of unnecessary elements

orces is replaced by a statically equivalent system, the


the definition above means that loads applied to one end of

a long beam are unlikely to significantly affect the other end. However, in using this


are unlikely to see the same amounts of stresses and strains than the


. Additionally, a finer mesh

tilized nearest the impact site to provide the resolution needed for an accurate

tion Near the Impact Site

The use of the adjusted mesh to

when the initial impact is sim

deformation and stress eff

the axis of impact). Even for the combination of the largest projectile (

and highest impact velocity (

of the refined mesh. This

in distant corners of the model would do little towards increasing the accuracy of the

simulation.

Figure 5 - Deformed shape and

9

The use of the adjusted mesh to invoke Saint-Venant’s Principle is

the initial impact is simulated. As shown in the deformed mesh

deformation and stress effects are limited to the upper-left corner of the mesh (i.e. near

Even for the combination of the largest projectile (

and highest impact velocity (v = 60 m/s), the impact effects are limited to within

This suggests that the mesh is sufficient, and that refining the mesh


eformed shape and effective plastic strain (plotted by color), after first impact

0.0003m, v = 60 m/s)

Venant’s Principle is validated

ulated. As shown in the deformed mesh in Figure 5, the

left corner of the mesh (i.e. near

Even for the combination of the largest projectile (R = 0.0003m)

ects are limited to within the area

refining the mesh


after first impact (R =

10

3. Results and Discussion

3.1 Determination of the Hertzian Pressure Distribution

Using equations from (2) (Equation [1] through Equation [4]) to find the pressure

distribution imparted on the target by the projectile, the static approximation of the

impact force can be determined. Using the MATLAB code “findpressure_R00025.m”

included in Appendix B, the following steps were performed for the various

combinations of Projectile Radius (R) and Projectile Velocity (v):

1. The material properties for the steel projectile are established

2. The projectile radius and projectile velocity are established (Note: For the

ease of data visualization, only one of these variables remain constant – the

other varies between the values shown in Table 2)

3. The Maximum Radius of the Circle of Contact ( ma ) is determined per

Equation [2]

4. The Maximum Normal Pressure ( mp′ ) in the circle of contact is determined

per Equation [3]

5. The Pressure Distribution ( p ) is determined per Equation [4]

6. By using many data points for the Radius from the Center of Impact (r)

incremented by 0.000001m, the pressure distribution is plotted against the

radius. The different cases of the variable iterated in Step 2 are plotted on

the same set of axes

7. The matrix of data (“p”) created by the m-file is written to a MS Excel

worksheet, for further analysis if needed

The process of analyzing the various combinations of R and v involves several

related MATLAB m-files, because testing combinations of 5 different projectile radii

and 5 different velocities will result in 25 different combinations of the two variables.

Testing each one of these by the same m-file would result in a very large matrix of data

that would cause difficulty for the computer (and user) to analyze and would result in

excessive file sizes.

11

Instead, for each case, one of the two variables was held constant, and the other

iterated within the specified values of Table 2. This results in 10 different cases, and 10

different m-files. The particular case shown in this results section will be shown in

Table 3 below. All other M-files, and all MS Excel results files and MATLAB

comparative plots are included in the Resource CD of this report.

Table 3 - Featured case analyzed by MATLAB (R=0.00025m)

Radius

(m)

Velocity

(m/s)

0.00025

10

20

35

50

60

By running the MATLAB M-file shown in Appendix B, the following outputs are

obtained:

1. Microsoft Excel Worksheet containing columns for:

a. Projectile Radius (R) – in this case, R = 0.00025m.

b. Projectile Velocity (v) for the last loop – in this case, v = 60.

c. Radius from the Center of Impact (r) – varies from 0.0000m to 0.0025m

d. Pressure for the 1st case: v = 10 m/s

e. Pressure for the 2nd

case: v = 20 m/s

f. Pressure for the 3rd

case: v = 35 m/s

g. Pressure for the 4th

case: v = 50 m/s

h. Pressure for the 5th

case: v = 60 m/s

NOTE: Due to worksheet size (~2,500 rows), the Microsoft Excel outputs are

included in the Resource CD for this project.

12

2. MATLAB Plot of all five pressure distributions (see d through h above) against

the radius from the center of impact. This plot for the featured case is shown in

Figure 6. All other plots are included in the Resource CD for this project.

Figure 6 - Plot of Pressure vs. Radius for R = 0.00025m

The MATLAB plot in Figure 6 indicates that for a given projectile radius, an

increase in the velocity will create a larger affected area and higher localized pressures at

all points of the affected area. For example, the top data series (for s

mv 60= ) has a

higher pressure at the impact location, roughly 17.8 GPa, than the data series for the next

smallest velocity (for s

mv 50= ), which is roughly 16.6 GPa. Additionally, for

smv 60= , the pressure distribution is non-zero up to 0.000062m, compared to only

0.000058m for s

mv 50= .

13

As a result, it can be shown that for a given projectile radius, the pressure

distribution imparted by a projectile is larger and more penetrating if the velocity of the

projectile is larger. Likewise, the constant-velocity calculations show that as the radius

increases, the pressure and penetrating power of the impact is also larger.

3.2 Modeling the First Impact

3.2.1 Mesh, Boundary Conditions, Material Conditions, and Loads

Starting with the mesh defined for the impact analysis (shown in Figure 4), the

appropriate boundary conditions and loads were applied to the mesh to simulate an

object being hit on its upper surface (see Figure 7). Recall from Section 2.2 that the

model is being simulated as axis-symmetric; therefore a COMSOL model of type “Axial

Symmetry (2D)” will be used.

Figure 7 - Boundary Conditions and Load applied to COMSOL model

14

Table 4 - Mesh Statistics of COMSOL Model “Ti-6AL-4V Impact.mph”

Parameter Value

Number of Degrees of Freedom 26156

Number of Mesh Points 3307

Number of Elements 6465

Triangular 6465

Quadrilateral 0

Number of Boundary Elements 147

Number of Vertex Elements 4

Minimum Element Quality 0.6028

Element Area Ratio 0.0064

After specifying the mesh (see Table 4 for mesh statistics), boundary conditions

and material conditions, the load is applied for the case being studied. After determining

the values of a and p′ for the specific values of R and v, the pressure distribution can be

determined. To input into COMSOL, the pressure distribution is applied as a piecewise

analytical function:

ar

ara

raprp

>

<−

⋅′=,0

,)(

22

[5]

The piecewise analytical function shown in Equation [5] ensures that the pressure

distribution is applied to the upper boundary at locations inside the circle of impact

(where ar < ). All other locations ( ar > ) along the upper boundary will receive no

external pressure, even if the effects of the applied pressure are felt at those locations.

3.2.2 First Impact - Loading and Unloading

After the setup of the model was completed, the finite element analysis was run. The

resulting surface plot (see Figure 8) that was produced provides valuable information as

to the physical effects of the projectile impact on the target material. The resulting plot

is a surface plot showing the radial stress of the target, while also representing the

displacement by the shape of the model.

maximum and minimum plastic stains.

Figure 8 - Surfa

As can be seen by the surface plot coloration, both compressive (blue) and tensile

(red) radial stresses occur. The majority of the compressive radial stresses occur

region underneath the projectile impact, while

affected area of impact. Simply speaking, the impact is pushing lower

laterally away from the impact up to a certain point, where the effects are pulling top

surface material towards

a ridge of material around the edge of t

in meteorite impacts on the earth (see

factors, such as the very high speed and high energy of the projectile, and the physical

and chemical reactions of the impact explosion, contribute to the additional complexity

of the dynamic impact.

15

displacement by the shape of the model. Also shown are the locations and values of the

maximum and minimum plastic stains.

Surface Plot of Radial Stress After First Impact (load active


(red) radial stresses occur. The majority of the compressive radial stresses occur

projectile impact, while the tensile stresses are just outside the

affected area of impact. Simply speaking, the impact is pushing lower

laterally away from the impact up to a certain point, where the effects are pulling top

surface material towards the edge of the impact. On a very large scale, this would create

a ridge of material around the edge of the impact site, which in principle is

meteorite impacts on the earth (see Figure 9). In a meteorite impact, many dynamic


chemical reactions of the impact explosion, contribute to the additional complexity

Also shown are the locations and values of the

load active)


(red) radial stresses occur. The majority of the compressive radial stresses occur in a

tensile stresses are just outside the

affected area of impact. Simply speaking, the impact is pushing lower-surface material

laterally away from the impact up to a certain point, where the effects are pulling top-

On a very large scale, this would create

in principle is what happens

In a meteorite impact, many dynamic


chemical reactions of the impact explosion, contribute to the additional complexity

16

Figure 9 - Photograph of Meteor Crater, Arizona – from (7)

Since the COMSOL model was explicitly defined to be a static loading of the

target, the stresses and strains represented in Figure 8 are not the residual effects. In

order to calculate the residual stress/strain field of the material following impact, the

applied load must be removed, so that the elastic deformation and recoverable stresses

will leave the material, and the condition of the post-impact target can be analyzed. By

removing the pressure distribution, and running the simulation again with the “Current

Solution” as the base model, this allows the model to partially “bounce back” to the state

without elastic deformation – see Figure 10 for the unloaded shape.

Figure 10 - Surface Plot of Radial Str

After the load is removed, there are

(compressive stresses are blue, tensile stresses are red), along with the accompanying

strains. The vast majority of the model has neglig

effects being felt in the small sq

efficiency of the mesh being used.

between the two states, particularly in their size. The loaded state has a very well

defined and very small area of high stresses, whereas the unloaded surface plot shows a

larger area of lower stress. A

magnitude larger in the loaded state than in the unloaded state.

numerical results of an impact (including three additional impacts) will be discussed as it

relates to progressive impact effects.

3.3 Analysis of Four Loading/Unloading Cycles for Specified Case

In analyzing the numerical results of multiple impacts, it is important to determi

parameters that will be recorded and compared between the loaded and unloaded states

17

Surface Plot of Radial Stress After First Impact (unloaded)

load is removed, there are distinct regions with residual radial stresses


strains. The vast majority of the model has negligible residual stresses, with

effects being felt in the small square of high-density mesh – these results highlight

efficiency of the mesh being used. The areas of compressive stress are very different



larger area of lower stress. Additionally, the maximum compressive stress is an order of

magnitude larger in the loaded state than in the unloaded state. In Section

impact (including three additional impacts) will be discussed as it

relates to progressive impact effects.

Analysis of Four Loading/Unloading Cycles for Specified Case

In analyzing the numerical results of multiple impacts, it is important to determi


mpact (unloaded)

s with residual radial stresses


ible residual stresses, with most of the

these results highlight the

The areas of compressive stress are very different



dditionally, the maximum compressive stress is an order of

In Section 3.3, the

impact (including three additional impacts) will be discussed as it

Analysis of Four Loading/Unloading Cycles for Specified Case

In analyzing the numerical results of multiple impacts, it is important to determine the


18

of each impact. This section will analyze one combination of projectile radius and

velocity, resulting in the values of maxa and

maxp shown in Table 5.

Analysis of the first impact using the methods described in Section 3.2.2 results

in deformed shapes which have elevated interior stresses and strains. Using the post-

processing features of COMSOL the following quantities of interest are determined after

each run:

• Radial stress ( )rσ along the top surface, z = 0.000250 m

• Radial stress ( )rσ , along the axis of impact, r = 0

• Axial displacement ( )zu , along the top surface, z = 0.000250 m

• Axial displacement ( )zu , along the axis of impact, r = 0

• Axial strain ( )zε along the axis of impact, r = 0

• Plastic Strain ( )plε

along the axis of impact, r = 0

These sets of data are exported from COMSOL into “.txt” format, so that they

can be plotted against each other for multiple impact cycles. MS Excel will be used to

compile data and generate comparative plots.

Table 5 - Parameters for Specific Case Discussed in Section 3.3

Parameter Value Unit

Projectile Radius, R 0.00025 m

Projectile Velocity, v 35 m/s

Radius of the Circle of

Impact, maxa

5.059E-5 m

Maximum Normal

Pressure in the Circle of

Contact, maxp

1.435E10 Pa

19

3.3.1 Analysis of Subsequent Impacts

The second impact began with the deformed shape from the first impact after it was

unloaded – meaning that the plastic deformation and residual stress field from the first

impact was incorporated the model. Similar to the way that the first impact was run

again with zero-load to “unload” it, the appropriate pressure distribution was then re-

applied, and the model was run again. The results were analyzed and upon unloading a

second time, the stress/strain field was analyzed once more.

The data files for each residual stress and displacement state, in addition to the

MS Excel worksheets are included in the Resource CD of this report.

20

Figure 11 - Radial Stress and Radial Displacement plotted for the Loading and Unloading of

Impacts 1-4 (2 plots), Top Surface

The radial stresses taken along the upper surface of the model show that there are

distinct regions of compression and tension, as was previously discussed. The data sets,

in order, represent loading and unloading stresses for impacts 1, 2, 3, and 4. Negative

stresses indicate compression, whereas positive stresses indicate tension. From a high

-1.60E+10

-1.40E+10

-1.20E+10

-1.00E+10

-8.00E+09

-6.00E+09

-4.00E+09

-2.00E+09

0.00E+00

2.00E+09

4.00E+09

6.00E+09

0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003

Str

ess

(P

a)

Radius from the Axis of Impact (m)

Radial Stresses of Top Surface - Impacts 1-4

sr1

sr2

sr3

sr4

sr5

sr6

sr7

sr8

-2.78E-04

-2.76E-04

-2.74E-04

-2.72E-04

-2.70E-04

-2.68E-04

-2.66E-04

-2.64E-04

-2.62E-04

0 0.000002 0.000004 0.000006 0.000008 0.00001

Str

ess

(P

a)

Radius from Axis of Impact (m)

Axial Displacement Along the Top Surface

- Impacts 1-4

ztop1

ztop2

ztop3

ztop4

ztop5

ztop6

ztop7

ztop8

21

level view of the entire top surface (Top plot of Figure 11), it appears that there are no

differences within any of the loaded stresses or any of the unloaded stresses. This would

indicate either:

1. There is no change in the target when it is hit repeatedly

2. There is an issue with the finite element analysis in that the load is not

accurately applied upon the previously deformed model

However, the impact effects of the material represented by the displacement in the axial

direction ( )zu , show that four distinct plots can be seen (see the bottom plot of Figure

11). These represent (from top to bottom), the level of compression displacements after

unloading from the 1st, 2

nd, 3

rd, and 4

th impacts, respectively.

From the initial state of zero-displacement, the largest change occurs during the

first impact. In that impact, the displacement line shifted to the red line (labeled “sr2”).

After the 2nd

impact, the displacement made another jump to the purple “sr4” line. As

the residual stresses progressed to the 3rd

(“sr6”) and 4th

(“sr8”) impacts, however, the

changes became smaller and smaller. If more and more impacts were plotted on this

chart, it is likely that the changes would reduce in magnitude until the plots were on top

of each other. To summarize, the majority of the effects are felt in the first two impacts,

after which the additional changes are negligible.

It should be noted that the previously discussed radial stresses and axial

displacements were along the top surface of the target (at z = 0). Also recorded were the

radial stresses and axial displacements along the axis of impact (at r = 0), which are

shown in Figure 12. While the effects of the stress increments between impacts are

consistent with the previous discussion, the results themselves are different for the

stresses along the axis of impact. Instead of having areas of compressive stresses and

tensile stresses as the top surface had, the cross section along the axis of impact is in

varying stages of compression. In the “loaded” states, the stresses exceed 1.20 x 1010

Pa,

whereas the “unloaded” states do not exceed 2.50 x 109 Pa.

22

Figure 12 - Radial Stress and Axial Displacement plotted for Impacts 1-4 (2 plots), Along Axis of

Impact

The radial stresses along the top surface of the target are important to study

because if cracks were to begin spreading throughout the material, they would most

likely be initiated at the top surface. The top surface is also the location where cracks

can most easily be inspected and evaluated. The radial stresses within the material must

be studied because once a crack forms, it must have the proper stress field in order to

-1.40E+10

-1.20E+10

-1.00E+10

-8.00E+09

-6.00E+09

-4.00E+09

-2.00E+09

0.00E+00

2.00E+09

0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003

Str

ess

(P

a)

Depth (m)

Radial Stress Along Axis of Impact

Impacts 1-4

rsz1

rsz2

rsz3

rsz4

rsz5

rsz6

rsz7

rsz8

-2.70E-04

-2.68E-04

-2.66E-04

-2.64E-04

-2.62E-04

-2.60E-04

0 0.000001 0.000002 0.000003 0.000004 0.000005

z-d

isp

lace

me

nt

(m)

Distance From Top Surface (m)

Axial Displacement Along Axis of Impact

- Impacts 1-4

zaxis2

zaxis4

zaxis6

zaxis8

23

propagate throughout the material. By determining whether the stress field necessary for

crack creation and/or propagation is bred from the repeated impact scenario, analyses of

fatigue life will be enhanced.

It should be noted that in the top plot of Figure 12 that at a depth of

mz 610250 −×= , the stress values for both the loaded and unloaded states have a

magnitude that is negligible when compared to the peak compressive and tensile

stresses. This shows that the mesh is adequate as it pertains to Saint Venant’s Principle.

Theoretically, the stresses would be zero at that part of the model, but practically, the

results show that the mesh is suitably assembled for these simulations.

In addition to the radial stress, the values for the effective plastic strain and axial

strain were obtained for each of the four impacts. These data sets were taken along the

axis of impact and exported to “.txt” format and imported to MS Excel same as the

residual stresses. The data files and MS Excel worksheets (“Axial Strain - Impacts 1 and

2.xls” and “Plastic Strain – Impacts 1 and 2.xls”) are included in the Resource CD of this

report. The figures resulting from the plotting the cross-sectional axial strain ( )zε and

cross-sectional plastic strain ( )plε against the radius from axis of impact are shown in

this report. See Figure 13 for cross-sectional axial strain (2 plots) and Figure 14 for

cross-sectional plastic strain (2 plots).

24

Figure 13 - Axial Strain plotted for Impacts 1-4 (2 plots)

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003

Distance from Top Surface (m)

Axial Strain - Impacts 1-4(R = 0.00025m, v = 35 m/s)

ez1

ez2

ez3

ez4

ez5

ez6

ez7

ez8

-4.2

-4.15

-4.1

-4.05

-4

0.000015 0.00002 0.000025 3E-05 3.5E-05


Axial Strain - Impacts 1-4 (Enlargement of Peak Strain Region)

ez1

ez2

ez3

ez4

ez5

ez6

ez7

ez8

25

Figure 14 - Plastic Strain plotted for Impacts 1-4 (2 plots)

Through the analysis of the COMSOL axial strain results in Figure 13, it can be

seen that the axial strain within the target varies between the loaded and unloaded states

of impacts 1-4. For each successive impact, both the loaded and unloaded states exhibit

an increase in the axial strain. This passes the “sanity check” in that an impacted piece

of metal will be increasingly deformed the more it is hit. Provided there is plastic

deformation after impact, the metal will never bounce back to its original state. Similar

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003


Plastic Strain - Impacts 1-4(R = 0.00025m, v = 35 m/s)

epz2

epz4

epz6

epz8

-4.1

-4.08

-4.06

-4.04

-4.02

-4

0.000015 0.00002 0.000025 3E-05 3.5E-05


Plastic Strain - Impacts 1-4(Enlargement of Peak Strain Region)

epz2

epz4

epz6

epz8

26

to the radial stresses previously discussed, however, the axial strain increases at a

decreasing rate – there will come a point where additional repeated impacts will have

negligible effects. Likewise, the plastic strains increase after each impact, as is shown in

Figure 14. Between the loading and unloading cycles of each impact, the plastic strain

does not change, since no additional loading is being applied to the model. As a result,

the loaded and unloaded plots share the same line for each impact.

3.4 Analysis of Maximum Stress/Strain Values For All Cases

After examining the cross-sectional strain and stress results for the one case discussed in

Section 3.3, the effects of multiple impacts was studied for the remainder of the cases.

In order to quickly access the values of p′ and a for use in testing the different

combinations, a MATLAB code “find_amax_pmax.m”, included in Appendix B, was

written to explicitly calculate each of the values and export them into a MS Excel

worksheet. The resulting MS Excel file (with added column headings) is shown in Table

6.

27

Table 6 - Calculated values of amax and pmax for each tested combination of R and v

For each case, the impact loading / unloading cycle was performed four times,

and the following information was recorded at each step (8 steps per case):

• Maximum Plastic Strain ( )max,plε

• Maximum Tensile Radial Stress ( )max,,trσ

• Maximum Compressive Radial Stress ( )max,,crσ

28

Although all of the cases shown in Table 6 were analyzed, the results from only

one case will be discussed. For consistency sake, the same case previously discussed in

Section 3.3 (R = 0.00025m, v = 35 m/s) will be discussed. Due to the similarity of the

data sets for each run, the discussion and conclusions for all other cases would be

similar, albeit with different stress and strain values. The trends, however, are

consistent. All COMSOL results for the four-impact analysis are included in Appendix

C, and the specific case (R = 0.00025m, v = 35 m/s) is shown in Table 7.

Table 7 - Stress/Strain Results for specific case (R = 0.00025m, v = 35 m/s)

R v a-m p-m State

Max

Plastic

Strain

Max Sig-r

(Tension)

Max Sig-r

(Compression)

0.00025 35 5.059E-

05 1.435E+10

Load 1 4.0596 3.734E+09 -1.316E+10

Unload 1 4.0596 2.058E+09 -2.328E+09

Load 2 4.0750 3.703E+09 -1.315E+10

Unload 2 4.0750 2.059E+09 -2.388E+09

Load 3 4.0874 3.658E+09 -1.314E+10

Unload 3 4.0874 2.048E+09 -2.409E+09

Load 4 4.0915 3.635E+09 -1.314E+10

Unload 4 4.0915 2.038E+09 -2.418E+09

At first glance, the data suggests that the majority of the plastic strain in the

target is due to the first impact, which imparts a plastic strain of roughly 4.05. In the

three subsequent impacts, this value of plastic strain only increase to 4.09, roughly 1% of

the initial strain. This means that even though the model was being exposed to the same

impact pressures for the second, third, and fourth impacts, little additional strain was

being caused. See Figure 15 for a graphical representation of the Maximum Plastic

Strain shown in Table 7.

29

Figure 15 - Plastic Strain for Impacts 1-4

While the values of the plastic strain increase between impact loadings (while not

changing between the loading and unloading of a given impact), the tensile and

compressive radial stresses fluctuate between loading cases and unloading cases, as is

shown in Figure 16 and Figure 17. Most importantly, however, are the unloaded

compressive radial stress values (i.e. Steps 2, 4, 6, and 8). These values show the true

residual stresses after each impact, since in reality, the time between loading and

unloading is very quick when the impact occurs. See Table 8 and Figure 18 for the

stress levels and graphical representation, respectively.

4.0550

4.0600

4.0650

4.0700

4.0750

4.0800

4.0850

4.0900

4.0950

0 2 4 6 8 10

Pla

stic

Str

ain

Step Number

Plastic Strain for Impacts 1-4 (R=0.00025m, v = 35m/s)

e-pl

30

Figure 16 - Maximum Tensile Radial Stress - Impacts 1-4 Loading/Unloading Cycles

Figure 17 - Maximum Compressive Radial Stress - Impacts 1-4 Loading/Unloading Cycles

0.000E+00

5.000E+08

1.000E+09

1.500E+09

2.000E+09

2.500E+09

3.000E+09

3.500E+09

4.000E+09

0 2 4 6 8 10

Ra

dia

l S

tre

ss (

Pa

)

Step Number

Tensile Radial Stress for Impacts 1-4(R=0.00025m, v = 35m/s)

SigR-T

-1.500E+10

-1.000E+10

-5.000E+09

0.000E+00

0 2 4 6 8 10

Ra

dia

l S

tre

ss (

Pa

)

Step Number

Compressive Radial Stresses

for Impacts 1-4(R=0.00025m, v = 35m/s)

SigR-C

31

Table 8 - Maximum Compressive Radial Stresses After Impacts 1-4 with Incremental and Percent

Changes

Impact #

Max Sig-r-C

[Pa] Change

Percent

Change

0 0.000E+00 - -

1 -2.328E+09 -2.328E+09 100.000%

2 -2.388E+09 -6.000E+07 2.513%

3 -2.409E+09 -2.100E+07 0.872%

4 -2.418E+09 -9.000E+06 0.372%

Figure 18 - Maximum Compressive Radial Stresses After Impacts 1-4

The maximum compressive radial stress increases after each of the impacts, but

in comparison to the magnitude of the stresses themselves, the changes are not

significant. As shown in Table 8, between impacts 1 and 2, there was a stress increase

of roughly 2.5%, between 2 and 3: 0.87%, and between 3 and 4: 0.37%. This suggests

that after the first two impacts, the effects resulting from subsequent impacts are

negligible.

-3.000E+09

-2.500E+09

-2.000E+09

-1.500E+09

-1.000E+09

-5.000E+08

0.000E+00

0 1 2 3 4 5

Ra

dia

l S

tre

ss (

Pa

)

Impact Number

Compressive Radial Stress After

Impact Unloading

Max Sig-r-C [Pa]

32

4. Conclusions

The goal of this project was to model and analyze the repeated impact of steel projectiles

onto a sample of Titanium 6Al-4V Alloy. In lieu of a dynamic finite element analysis,

the effects of a dynamic impact were represented by a Hertzian pressure distribution

which was loaded and unloaded statically to simulate an impact without the need for

dynamic analysis. This allowed for multiple runs so that the analysis could be run for 4

impacts of 25 combinations involving 5 projectile sizes and 5 projectile velocities.

When studying one case in particular, several important parameters such as radial

stress ( )rσ , axial displacement ( )zu , total axial strain ( )zε , and plastic strain ( )plε were

recorded. The goal of this data was to examine the effects of four impacts on the

material; by taking data along the top surface and along the axis of impact, the effects in

two directions were studied. When studying all of the chosen cases, the residual

condition was analyzed after each impact. The condition was quantified by recording

the maximum plastic strain ( )max,plε , the maximum tensile radial stress ( )max,,trσ , and the

maximum compressive radial stress ( )max,,crσ .

The data suggests that the majority of the impact effects were felt after the first

impact, and to a smaller extent, the second impact. For the specific case tested, the first

impact produced compressive stress increases roughly 40 times larger than the second

impact. Subsequent impacts produced negligible effects both in the plastic strain and

radial stresses. In the specific case studied, the third impact produced a mere 0.872%

increase in peak stress, while the fourth impact produced a 0.372% increase in peak

stress. These impacts pale in comparison to the first two impacts.

The results obtained by this study suggest that in an impact situation between two

specific metals (steel and titanium), once a few impacts have affected and deformed a

certain area, then that area is essentially impervious to additional deformation and/or

damage. To an extent, due to strain hardening of the material and the deformed

geometry of the new target (i.e. the projectile impacting straight at the target is no longer

hitting the surface normal to the surface – a “glancing blow”), this may be partly true.

On the other hand, the repeated impact in the same location and same size may

have skewed the results to be very dependent on the first and second impacts. This

33

effect is similar to a stamping machine for auto body panels, where a given pressure is

applied to a certain sized piece of sheet metal repeatedly in order to shape it as desired.

In this case, the sheet metal die is replaced by a steel sphere hitting the metal in the same

place at the same pressure.

4.1 Suggestions and Recommendations for Future Work

1. To prevent the “sheet metal die” effect as described above, additional

analyses should be run that vary the impact location, projectile size, and

projectile velocity from impact to impact. This will vary the pressure

distributions seen by the targets, allowing for a more varied and realistic

analysis. Conversely, the additional variables being tested will make it

difficult to test the effects of each parameter – in this study no parameters

changed from impact to impact, therefore the effect of projectile size and

projectile velocity could be compared from case to case. NOTE: varying

the location of impact will render the axis-symmetric assumption invalid

and introduce additional modeling complexity.

2. The long-term effects of repeated impacts should be analyzed for their

contributions to fatigue life. In this study only four impacts were

simulated, however many engineered components undergo many more

impacts tests for qualification purposes. The effects of hundred of

consecutive impact simulations would provide valuable information for

the study of component fatigue.

3. In addition to varying the projectile size and projectile velocity, the

materials being simulated should be analyzed. In this case, steel

projectiles were impacting a Titanium 6Al-4V alloy plate, however

analyzing the following material combinations (among others) would

enrich the understanding of repeated impacts for other engineering fields:

• Steel - Steel

34

• Steel - Concrete

• Steel - Wood

• Steel - Plastic

• Steel - Aluminum

• Steel - Composite

4. The results from the static simulations used in this study should be

compared to a finite element analysis using dynamic impact, using a

program such as LS-DYNA, to determine the validity of these results and

the suitability of the static approximation. Additionally, comparing these

results to real-life impact analysis (using strain gages) would lend

additional validation to the method.

35

5. References

(1) Cook, R. D. (2004). Concepts and Applications of Finite Element Analysis,

Fourth Edition. Singapore: John Wiley & Sons (Asia).

(2) Davies, R. M. (1949). The determination of static and dynamic yield stresses

using a steel ball. Royal Society of London , 416-432.

(3) Hibbeler, R. (2000). Mechanics of Materials. Upper Saddle River, New Jersey:

Prentice Hall.

(4) Lesuer, D. R. (2000). Experimental Investigations of Material Models for Ti-6Al-

4V Titanium and 2024-T3 Aluminum. Washington, D.C.: U.S. Department of

Transportation - Federal Aviation Administration - Office of Aviation Research.

(5) Meguid, S. A. (2007). Development and Validation of Novel FE Models for 3D

Analysis of Peening of Strain-Rate Sensitive Materials. Journal of Engineering

Materials and Technology , 129, 271-283.

(6) Ugural, A. C., & Fenster, S. K. (2003). Advanced Strength and Applied

Elasticity, Fourth Edition. Upper Saddle River, New Jersey: Prentice Hall.

(7) Terrestrial Impact Craters Slide Set. Accessed 12/3/10.

http://www.lpi.usra.edu/publications/slidesets/craters/slide_10.html

36

6. Appendices

6.1 Appendix A

Material Properties

Table 9 - Material Properties for Ti 6Al-4V and Steel

NOTE: Material Properties obtained from (3)

Property Ti 6Al-4V

Alloy

Steel (Structural

A36 Alloy)

Density (ρ) 4430

kg/m3

7860 kg/m3

Modulus of

Elasticity

(E)

120 GPa 200 GPa

Poisson’s

Ratio (ν) 0.36 0.286

Yield

Strength

( )yσ

924 MPa 250 MPa

Coefficient

of Thermal

Expansion

( a )

9.4 x

Co/10 6−

12.0 x

Co/10 6−

37

Table 10 - Derived Parameters used in COMSOL Analyses

Property Ti 6Al-4V

Alloy

A * 1.098 GPa

B * 1.096 GPa

n * 0.93

Modulus of

Plasticity

( )plE

1.016 GPa

Strain at

Onset of

Yielding

( yε )

9.24 x 10-3

* NOTE: A, B, and n are parameters for the

Johnson-Cook Material Model as determined by (4)

38

6.2 Appendix B

MATLAB M-File “findpressure_R00025.m” for R = 0.00025m:

% Chris Anason % Engineering Project % Find Pressure: Radius Constant clear all clc %% STEEL PROJECTILE MATERIAL CONSTANTS %% Poisson = 0.32; % Poisson's ratio of the Projectile [ ] E = 200e9; % Modulus of Elasticity for the Projectile [Pa] rho = 7860 ; % Density of Projectile [kg/m^3] %% DEFINE VARIABLES (SPHERE RADIUS AND VELOCITY) %% Radii = [.0001;.00015;.0002;.00025;.0003]; % Radius of the Projectile [m] Velocities = [10;20;35;50;60]; % Velocity of Projectile at Impact [m/s] Xmax = 2500e-6; % Maximum radius of target [m] R = 0.00025; for i=1:1:5 v = Velocities(i); %% Find 'a_max' (Maximum Radius of the "Circle of Contact") a_max = (2.5 * pi * rho * ((1-Poisson^2)/E))^(1/5) * R * v^(2/5); %% Find 'p_max' (Maximum Normal Pressure applied during impact) p_max = (1/pi) * (2.5 * pi * rho)^(1/5) * (E/(1-Poisson^2))^(4/5) * v^(2/5); %% Loop to solve for p(r) over 0<r<1 ii = 1; % Initialize Counter "ii" to 1 for r=0:0.000001:Xmax p(ii,1) = R; p(ii,2) = v; p(ii,3) = r; p(ii,3+i) = p_max * sqrt(a_max^2 - r^2) / a_max; %% Pressure Distribution as a function of r ii = ii+1; end i = i + 1; end

39

p; % Plot Pressure against Radius for various velocities plot(p(:,3),p(:,4),'-r') xlabel('Radius (m)') xlim([0 0.0001]) ylabel('Pressure (Pa)') title('Normal Pressure as a Function of Radius from the Center of Impact, R = 0.00025m') hold on plot(p(:,3),p(:,5),'-g') plot(p(:,3),p(:,6),'-b') plot(p(:,3),p(:,7),'-k') plot(p(:,3),p(:,8),'-c') h = legend('V=10 m/s','V=20 m/s','V=35 m/s','V=50 m/s','V=60 m/s',5); set(h,'Interpreter','none') %% Write data to Excel File xlswrite('R00025', p)

40

MATLAB M-File: “find_amax_pmax.m”

% Chris Anason % Engineering Project % MATLAB File: "find_amax_pmax.m" clear all clc %% STEEL PROJECTILE MATERIAL CONSTANTS %% Poisson = 0.32; % Poisson's ratio of the Projectile [ ] E = 200e9; % Modulus of Elasticity for the Projectile [Pa] rho = 7860 ; % Density of Projectile [kg/m^3] %% DEFINE VARIABLES (SPHERE RADIUS AND VELOCITY) %% Radii = [.0001;.00015;.0002;.00025;.0003]; % Radius of the Projectile [m] Velocities = [10; 20; 35; 50; 60]; % Velocity of Projectile at Impact [m/s] i = 1; for j=1:5 R = Radii(j); for k=1:5 v = Velocities(k); %% Find 'a_max' (Maximum Radius of the "Circle of Contact") a_max = (2.5 * pi * rho * ((1-Poisson^2)/E))^(1/5) * R * v^(2/5); %% Find 'p_max' (Maximum Normal Pressure applied during impact) p_max = (1/pi) * (2.5 * pi * rho)^(1/5) * (E/(1-Poisson^2))^(4/5) * v^(2/5); data(i,1) = R; data(i,2) = v; data(i,3) = a_max; data(i,4) = p_max; i = i+1; % Iterate "i" end end data; %% Write data to Excel File xlswrite('ImpactParameters', data)

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6.3 Appendix C

COMSOL Results of Four-Impact Analysis (MS Excel format)

anason final report

Documents

impact pressure

analysis of subsequent

mechanical engineering

impact loading

engineering major subject

christopher anason

multiple impact simulations

effects of repeated