the investigation of projectile impacts by adib …
TRANSCRIPT
THE INVESTIGATION OF PROJECTILE IMPACTS
ON GRANITES
BY
ADIB HAMDANI BIN ROSLI
A thesis submitted in fulfilment of the requirement for
the degree of Master of Science in Mechanical
Engineering
Kulliyyah of Engineering
International Islamic University Malaysia
AUGUST 2014
ii
ABSTRACT
It is essential to further research extensively in rock fracture mechanics due to widely
used granite in the construction as well as safety industry. Less stiff projectile has
higher impact duration with its target, thus more kinetic energy from the projectile is
expected to be transferred in damaging the target. Granite rock plate specimens having
18 mm thickness were subjected to 9.5-mm diameter steel, copper and lead spherical
projectile impacts. Thus, the research was conducted to study the effect of varying
projectile stiffness on the damage size generated in the granite target. Two
experimental configurations were adopted for impact on rock specimen, namely
confined surface impact and unconfined edge impact. In the first category, the impact
was conducted on the flat surface of the rock specimen while in the second category,
the impact was conducted on the specimen edge. Each configuration consisted of three
modes of impact, which are steel-on-rock (SOR), copper-on-rock (COR) and lead-on-
rock (LOR) where the projectile stiffness increases from lead to steel materials. The
experimental outcomes were then numerically predicted by using mesh-based
elements and mesh-free smooth particle hydrodynamics (SPH) methods with power
law plasticity and Johnson-Holmquist material models were utilized to model the
ductile and brittle material behavior respectively. Steel projectile was found to cause
higher damage on the impacted area, while copper projectile produced larger radial
cracks in confined surface impact and extended crater in unconfined edge impact.
Longer contact time between copper and rock specimen allowed more kinetic energy
to be transferred to generate higher damage in the rock target. The least stiff lead
projectile however, produced the smallest damage because of its low yield strength
instigated the projectile to absorb most of the kinetic energy in its massive
deformation. Edge impact on the other hand, produced mainly double-layered crater
formation subjected to SOR and COR modes of impact. Experimental findings
revealed the copper projectile produced significantly larger craters compared to SOR
impact, showing the superiority of copper over steel projectile in causing higher
damage and fracture of rock specimen. Nevertheless, lead projectile did not follow the
trend of projectile stiffness as demonstrated by the copper projectile. Therefore,
threshold value of projectile stiffness was proposed for the extent of damage failure to
be strongly dependent upon the projectile stiffness. Explicit finite element was used to
simulate the experimental work. Crater dimensions were modeled efficiently by using
refined mesh models for rock specimen. SPH method was capable to predict the
formation of double-layered craters on impacted edge and had close proximity with
experimental evidence. Mesh-based elements method, on the other hand was able to
represent irregular crater formation obtained from the impact tests.
iii
خلاصة البحث
إجراء مزيد من البحوث على نطاق واسع في الميكانيكا كسر الصخور بسبب الجرانيت المستخدمة على ومن الضروريقذيفة أقل قاسية لديه مدة تأثير ارتفاع مع هدفه ، وبالتالي المزيد من . نطاق واسع في البناء، فضلا عن صناعة السلامة
وتعرض الجرانيت صخرة العينات وجود . الإضرار الهدفالطاقة الحركية من يتوقع أن قذيفة على أن يتم تحويلها في وهكذا، وقد أجريت . ، والنحاس ، وتؤدي الآثار قذيفة كروية mm-9.5سماكة إلى الصلب قطرها 18mm لوحة
تم اعتماد اثنين من .البحوث ل دراسة تأثير متفاوتة صلابة قذيفة على حجم الضرر ولدت في الهدف الجرانيتفي الفئة . ية لل تأثير على العينة الصخرية ، تأثير السطح وهي المحصورة و غير المحصورة تأثير الحافةتكوينات التجريب
.الأولى ، وأجريت تأثير ذلك على سطح مستو من العينة الصخرية بينما في الفئة الثانية ، أجريت تأثير على حافة العينة، والنحاس و على (SOR)هي من الصلب على صخرةوتألفت كل تكوين من ثلاثة أنماط من تأثير ، والتي
ثم . حيث الزيادات صلابة قذيفة من الرصاص على المواد الفولاذية (LOR)و الرصاص على صخرة (COR)الصخور تم توقع النتائج التجريبية عدديا باستخدام العناصر المستندة إلى شبكة وخالية من شبكة الهيدروناميكا الجسيمات ناعمة
(SPH) طرق اللدونة مع القانون والسلطة، و نماذج المواد Johnson-Holmquist استخدمت لنمذجة سلوك الموادتم العثور على قذيفة الصلب يسبب ارتفاع الضرر على المنطقة المتأثرة، في حين تنتج . الدكتايل وهشة على التوالي
يعد الاتصال . فرة ممتدة في تأثير حافة غير المحصورةقذيفة النحاس الشقوق شعاعي أكبر الأثر في سطح المحصورة و الحمرة بين النحاس و العينة الصخرية يسمح المزيد من الطاقة الحركية على أن يتم تحويلها لتوليد الضرر العالي في الهدف
لى أقل قاسية قذيفة الرصاص ومع ذلك ، تنتج أصغر الضرر بسبب قوتها العائد المنخفض حرض قذيفة ع . الصخورتأثير الحافة من ناحية أخرى ، ينتج بشكل أساسي تشكيل الحفرة . امتصاص معظم الطاقة الحركية في تشوه الضخمة
وكشفت النتائج التجريبية أنتجت قذيفة النحاس الحفر .وسائط التأثير COR و SORالمزدوجة الطبقات تعرض ل، والتي تبين تفوق النحاس على قذيفة الصلب في التسبب في الضرر العالي و كسر SORأكبر بكثير بالمقارنة مع تأثير
.ومع ذلك ، لم قذيفة الرصاص لا تتبع هذا الاتجاه من صلابة قذيفة كما يتضح من قذيفة النحاس . من عينة الصخورتم استخدام عنصر . فةلذلك ، اقترح قيمة عتبة صلابة قذيفة ل مدى فشل الضرر أن تعتمد بقوة على صلابة قذي
. وعلى غرار أبعاد الحفرة بكفاءة باستخدام نماذج شبكة المكرر ل عينة الصخور . محدود صريحة لمحاكاة العمل التجريبيقادرعلى التنبؤ تشكيل الحفر المزدوجة الطبقات على حافة أثرت وكان القرب الوثيق مع الأدلة SPHكان أسلوب
ن الأسلوب القائم على عناصر شبكة قادرة على تمثيل تشكيل حفرة غير النظامية التي تم التجريبيةمن ناحية أخرى كا . الحصول عليها من الاختبارات الأثر
iv
APPROVAL PAGE
I certify that I have supervised and read this study and that in my opinion, it conforms
to acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a thesis for the degree of Master of Science in Mechanical Engineering
…………………………………..
Qasim H. Shah
Supervisor
…………………………………..
Syed M. Kashif
Co-Supervisor
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a
thesis for the degree of Master of Science in Mechanical Engineering
…………………………………..
Judha Purbolaksono
External Examiner
…………………………………..
Jaffar Syed Mohamed Ali
Internal Examiner
This thesis was submitted to the Department of Mechanical Engineering and is
accepted as a fulfilment of the requirement for the degree of Master of Science in
Mechanical Engineering
…………………………………..
Meftah Hrairi
Head, Department of Mechanical
Engineering
This thesis was submitted to the Kulliyyah of Engineering and is accepted as a
fulfilment of the requirement for the degree of Master of Science in Mechanical
Engineering
…………..………………………
Md. Noor bin Hj Salleh
Dean, Kulliyyah of Engineering
v
DECLARATION
I hereby declare that this thesis is the result of my own investigation, except where
otherwise stated. I also declare that it has not been previously or concurrently
submitted as a whole for any other degrees at IIUM or other institutions.
Adib Hamdani Bin Rosli
Signature…………………. Date …..................
vi
INTERNATIONAL ISLAMIC UNIVERSITY MALAYSIA
DECLARATION OF COPYRIGHT AND AFFIRMATION
OF FAIR USE OF UNPUBLISHED RESEARCH
Copyright ©2014 by International Islamic University Malaysia. All rights reserved.
THE INVESTIGATION OF PROJECTILE IMPACTS ON GRANITES
No part of this unpublished research may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, electronic, mechanical, photocopying,
recording or otherwise without prior written permission of the copyright holder except
as provided below.
1. Any material contained in or derived from this unpublished research may
be used by others in their writing with due acknowledgement.
2. IIUM or its library will have the right to make and transmit copies (print
or electronic) for institutional and academic purposes.
3. The IIUM library will have the right to make, store in a retrieval system
and supply copies of this unpublished research if requested by other
universities and research libraries.
Affirmed by Adib Hamdani Bin Rosli
……..……..…………… …………………..
Signature Date
vii
ACKNOWLEDGEMENT
In the name of Allah, the most gracious and the most merciful.
All praise, gratitude and appreciation are for Allah, the most merciful and the most
compassionate for granting me strength, knowledge and opportunity to complete this
task.
I am highly obliged to the Research Management Center (RMC), International
Islamic University Malaysia for providing research grant [EDW A11-117-0908] for
the completion of the project.
I would like to express my gratefulness to my supervisor, Dr. Qasim Hussain
Shah for scholarly guiding and leading me remarkably through the tunnel of hope.
I would like to thank my co-supervisor, Dr. Syed Mohamad Kashif for his
suggestion and assistance in completing the thesis.
Unforgettable my colleagues at the automotive workshop of the Kulliyyah of
Engineering, IIUM. Also all others whether academic and non-academic staff for
assistance and keeping asking me on my completion of my study.
My deepest gratitude towards my mother, Hamidah and father, Rosli for their
love, prayers and support throughout the turbulence period.
Finally, my supportive wife, Shahirah who is irreplaceable.
viii
TABLE OF CONTENTS
Abstract ......................................................................................................................... ii Abstract in Arabic ......................................................................................................... iii
Approval page ............................................................................................................... iv Declaration Page ............................................................................................................ v Copyright Page.............................................................................................................. vi Acknowledgement ....................................................................................................... vii List of Tables ................................................................................................................. x
List of Figures .............................................................................................................. xii List of Abbreviations ................................................................................................... xv
List of Symbols .......................................................................................................... xvii
CHAPTER 1: INTRODUCTION ............................................................................... 1 1.1 Background ................................................................................................... 1 1.2 Problem statement and its significance ........................................................ 6
1.3 Research objectives ...................................................................................... 8
1.4 Research methodology ................................................................................. 8 1.5 Scope of research ........................................................................................ 10 1.6 Thesis organization ..................................................................................... 10
CHAPTER 2: LITERATURE REVIEW ............................................................... 11 2.1 Introduction ................................................................................................ 11
2.2 Experimental work ..................................................................................... 11 2.2.1 Brittle fracture mechanics ................................................................. 11
2.2.2 Brittle strength and toughness ........................................................... 14 2.3 Numerical techniques ................................................................................. 15
2.3.1 Discrete Element Method (DEM) ..................................................... 15
2.3.2 Lagrange Solid Elements .................................................................. 17
2.3.3 Smooth Particle Hydrodynamics (SPH) ........................................... 19 2.4 Brittle material models ............................................................................... 20
2.4.1 Taylor-Chen-Kuszmaul (TCK) ......................................................... 20 2.4.2 Johnson-Holmquist (J-H) .................................................................. 21
2.5 Other issues on numerical modeling .......................................................... 22 2.6 Summary ..................................................................................................... 24
CHAPTER 3 : METHODOLOGY.......................................................................... 25 3.1 Introduction ................................................................................................ 25
3.2 Experimental work ..................................................................................... 25 3.2.1 Confined surface impact ................................................................... 29 3.2.2 Unconfined edge impact ................................................................... 31
3.3 Numerical simulation ................................................................................. 32 3.3.1 Material model characterizations ...................................................... 33
3.3.1.1 Johnson-Holmquist constitutive equations ........................... 33 3.3.1.2 Power Law Plasticity constitutive equations ........................ 36
3.3.2 Confined surface simulation ............................................................. 37 3.3.3 Unconfined edge simulation ............................................................. 38
ix
3.4 Summary ..................................................................................................... 42
CHAPTER 4: RESULTS AND FINDINGS ............................................................ 43 4.1 Introduction ................................................................................................ 43 4.2 Confined surface impact ............................................................................. 43
4.2.1 Experimental results .......................................................................... 44
4.2.1.1 Ductile damage on projectile ................................................ 45 4.2.1.2 Brittle damage on granite rock ............................................. 48
4.2.2 Lagrange Mesh-based Finite Element simulation ............................. 51 4.2.2.1 Impact duration ..................................................................... 52 4.2.2.2 Projectile model damage ...................................................... 54
4.2.2.3 Rock model damage ............................................................. 59 4.2.3 Summary on confined surface impact ............................................... 62
4.3 Unconfined edge impact ............................................................................. 65
4.3.1 Experimental and SPH method ......................................................... 65 4.3.1.1 Projectile damage ................................................................. 66 4.3.1.2 Rock edge damages .............................................................. 67
4.3.2 Mesh-based Elements and SPH comparative case study .................. 76 4.3.2.1 Mesh density variation ......................................................... 77
4.3.2.2 Comparison between meshed solid with SPH ...................... 83
4.3.3 Summary of unconfined rock edge impact ....................................... 87
CHAPTER 5: CONCLUSION AND RECOMMENDATION ............................. 90 5.1 Conclusion .................................................................................................. 90 5.2 Major contribution ...................................................................................... 94 5.3 Recommendations for future studies .......................................................... 94
REFERENCES ........................................................................................................... 96
APPENDIX A: SAMPLES OF LS-DYNA CODE ................................................... 102
APPENDIX B: PUBLICATIONS ............................................................................. 107
x
LIST OF TABLES
Table No. Page No.
3.1 Modes of impact and projectile velocities recorded in the confined surface
impact 31
3.2 Modes of impact and projectile velocities recorded in the unconfined edge
impact 32
3.3 Johnson-Holmquist parameters for granite (*MAT_110) after Tuomas (2004) 36
3.4 Material parameters for power law plasticity 37
3.5 Summary of mesh configurations for Lagrange simulation 39
4.1 Standard deviation (s.d.) between velocity data recorded in the confined
surface impact 44
4.2 Damage measurement on copper projectiles 47
4.3 Damage measurement on lead projectiles 48
4.4 Crater dimensions subjected to SOR impact 49
4.5 Normalized projectile kinetic energy for confined surface impact simulation 52
4.6 Range of impact duration against the kinetic energy 54
4.7 Final longitudinal deformation with rates 57
4.8 Final lateral deformation with average rate 58
4.9 Comparison of numerical projectile damage with experimental result 59
4.10 Percentage difference for numerical projectile against experimental work 59
4.11 Numerical rock’s damage measurement 62
4.12 Standard deviation (s.d.) between velocity data recorded in the unconfined
edge impact 66
4.13 Comparison of kinetic energy carried by different projectiles 66
4.14 Post-impact state of projectiles 67
xi
4.15 Comparison of experimental size of radial craters on the impacted edges with
numerical results 71
4.16 Comparison of experimental size of median craters on the impacted edges
with LS-DYNA results 71
4.17 Measurement of ‘a’, ‘r’, and ‘d’ for all nine simulation sets, given different
mesh density 80
xii
LIST OF FIGURES
Figure No. Page No.
1.1 Flow diagram of research methodology 9
3.1 Layout of the air gun experiment 26
3.2 Trigger with two tubes 26
3.3 Initial position of projectile 27
3.4 Experimental laboratory and equipment 27
3.5 Chronograph with its two sensors 28
3.6 Granite specimen 28
3.7 Projectile materials 29
3.8 Front and side views of a confined granite specimen 30
3.9 The unconfined edge of granite specimen with point of impact 31
3.10 Side view of clamped granite 31
3.11 Numerical model with direction of projectile indicated by arrow 38
3.12 2S (blue ball), 4S (top block) and 8S (bottom block) models 40
3.13 SPH and 16S models of granite 40
3.14 General configuration with SPH model 41
4.1 Damage on steel projectile from confined surface impact 45
4.2 Dimensional damage on copper projectile from confined surface impact 46
4.3 Dimensional damage on lead projectile from confined surface impact 47
4.4 Crater damage on front granite surface due to SOR impact 49
4.5 Front damage surface due to (a) COR impact and (b) LOR impact 50
4.6 Rear damage surface due to (a) SOR impact and (b) COR impact 51
4.7 Impact duration between steel projectile and rock model 53
xiii
4.8 Numerical and experimental steel projectiles 55
4.9 Numerical and experimental copper projectiles 55
4.10 Numerical and experimental lead projectiles 55
4.11 Longitudinal deformation of projectiles 56
4.12 Lateral deformation of projectiles 56
4.13 Sequence of crater formation in numerical SOR impact 60
4.14 Sequence of crater formation in numerical COR impact 61
4.15 Double-layered localized and extended craters formed on granite specimens
after steel (left) and copper (right) projectile impact 68
4.16 A shallow crack on front surface of granite after LOR impact 68
4.17 Damage measurement for median crater lengths = h1 and h2, and radial crater
lengths = d1 and d2 69
4.18 Measurement for localized and extended median/radial crater lengths in SPH
rock model 70
4.19 Localized radial and median crater lengths due to SOR and COR impacts 72
4.20 Extended radial and median crater lengths due to SOR and COR impacts 73
4.21 Effective plastic strain shown by 4S, 8S and 16S COR models 77
4.22 Effective plastic strain shown by 4S, 8S and 16S SOR models 78
4.23 Effective plastic strain shown by 4S, 8S and 16S LOR models 79
4.24 Damage measurement on rock edge based on crater depth, radial and axial
lengths 80
4.25 Projectile off-contact from rock edge and the radial crater length oscillation 81
4.26 Distorted element lines in (a) LOR-4S and (b) LOR-8S. The extension of
depth crater is shown in (b) 83
4.27 Close-up views on damaged zone for (a) COR and (b) SOR between 16S
model (left) and SPH model (right) 84
4.28 Close-up views on damaged zone for LOR between 16S model (left) and
SPH model (right) 85
xiv
4.29 Globally damaged SPH models in (a) COR, (b) SOR and (c) LOR as
indicated by the arrows due to tensile instability 87
xv
LIST OF ABBREVIATIONS
ALE Arbitrary Lagrange-Euler
BPM Bonded-particle model
CCNBD Cracked chevron notch Brazilian disc
COR Copper-on-rock
DCA Dominant crack algorithm
DEM Discrete element method
EOI Edge-on impact
EOS Equation of States
FE Finite element
GPA Generalized particle algorithm
GPa GigaPascal
HEL Hugoniot Elastic Limit
HJC Holmquist-Johnson-Cook
HVI High velocity impact
J Joule
J-H Johnson-Holmquist
K.E. Kinetic energy
kg kilogram
LOR Lead-on-rock
mm Millimeter
ms Milliseconds
PFC Particle flow code
PPV Particle peak velocity
xvi
RC Reinforced concrete
RMR Rock mass rating
S.d. Standard deviation
SEM Scanning electron microscope
SHPB Split Hopkinson pressure bar
SOR Steel-on-rock
SPH Smooth Particle Hydrodynamics
TCK Taylor-Chen-Kuszmaul
UCS Uniaxial compression strength
xvii
LIST OF SYMBOLS
KIC Fracture toughness for mode I
v Impact velocity
Vp Critical velocity to perforation
X Penetration of projectile into rock
σ Equivalent stress
σi*
Intact material strength
σf* Fracture material strength
D Damage parameter
εfp Plastic strain to fracture
P* Normalized pressure
σ* Normalized stress
σY Yield stress
εyp Plastic strain to yield
E Young’s modulus
2S Steel, copper and lead projectile models
4S Coarse mesh rock
8S Medium mesh rock
16S Fine mesh rock
ρ Density
V Volume
1
CHAPTER ONE
INTRODUCTION
1.1 BACKGROUND
The space exploration leads to the study of rock mechanics due to the threat posed by
flying meteor rocks in the space endangering the floating spacecrafts. The space
project also includes firing bullets onto meteor rocks for sampling purpose of debris.
One of common types of robust rocks is granite, which is an intrusive and igneous
rock with interconnected coarse-grained crystals with no preferred orientation (de
Vallejo and Ferrer, 2011). Plutonic rock, such as granite is also called as ‘intrusive’, as
the magma was cooled down much slower beneath layers of older rocks. As a result of
longer time of cooling which consequently influencing the crystallization time, the
formed individual grains become larger to the extent they are visible to naked eyes.
The main compositions are relatively low-temperature minerals such as quartz and
feldspar (Handy and Spangler, 2007), in addition of micas and mafic minerals (de
Vallejo and Ferrer, 2011). Two predominant minerals, i.e. Quartz can be identified
from its colorless appearance while feldspar gives main color for particular granite.
Biotite or hornblende which is ferromagnesian mineral contributes dark colour to the
granite. Muscovite mica, meanwhile, exists only in several types of granites as a clear
reflective mineral which sparkles from reflective light (Cotta, 2008).
The classification of granite falls under the family of quasi-brittle rocks which
exhibit fracturing in non-linear mechanical pattern. This pattern can be identified from
kinetics of pre-existent and loading initiated meso-cracks. The term meso-crack is
intermediary between macro- and micro-crack, which represents visible defects
through naked eyes as well as representative volumes, i.e. a material point of structure
2
body (Dragon, 2008). The mixture of minerals and geological history of granite create
structural ‘defects’ which determine the granite’s mineral orientation, porosity,
microfissures or degree of weathering (de Vallejo and Ferrer, 2011). Rock’s hardness
is inhomogeneous and determined by dominant hardness of component mineral(s).
Thus, granite’s hardness is in between the hardness of quartz and feldspar.
Nevertheless, mica grains can simply flake out according to the size and exposure on
the granite’s surface (Cotta, 2008).
Manchao et al. (2010) identified two main regions in granite as the weakest
area through scanning electron microscope (SEM) imaging technique while analyzing
microscopic characteristics and crack pattern of mineral grains. The weakest area was
found to be feldspar mineral while the second one was bonding between quartz grains.
In comparison between rockburst test (dynamic loading) and uniaxial load test (static
loading), the intra-granular microcracks in both weakest areas is lower under blast
loading in terms of fractal dimensions. This finding indicates larger amount of energy
is stored during rockburst compared to uniaxial static test (Manchao et al., 2010).
Nasseri and Mohanty (2008) defined material’s fracture toughness, KIC as the
material property associated with the ability to carry loads or resist deformation in the
presence of a crack. This property is essential in evaluating granite’s strength since
granite is a type of brittle material with pre-existing cracks. In granite constituents,
microcracks and irregular grain size are two main sources of the presence of cracks
leading to the study of fracture toughness. The former with its orientation was
recognized as the dominant role influencing fracture toughness than of the latter with
its orientation, given similar dimension (Nasseri and Mohanty, 2008).
The rock’s strength is not a single intrinsic value. The range of rock strength
varies due to intrinsic properties of rock, its cohesion and angle of friction as well as
3
external factors, for instance the magnitude of stress acting upon it. Compressive
strength is the most celebrated measured property for intact rock due to its simple test
to obtain a particular value. Failure of rock is usually assumed occurring at peak
strength, though it is in fact an over-simplification and sometimes even less accurate.
By definition, de Vallejo et al. (2011) stated the fracture as “The formation of planes
of separation within the rock as the cohesion between particles is broken and new
surfaces are formed”. Before rock fracture occurs, both cohesive and frictional forces
exist in between particles and after fractured, only frictional forces remain.
Shear stress failure is considered as the most common and important type of
failure where one face of surface slips over another (de Vallejo and Ferrer, 2011).
Continuous slipping causes slip surface to convert to slip zone, which includes more
grains in the process. Rock’s sliding resistance comes mostly out of internal friction
between grains. The internal friction, though, does not solely depend on sliding
friction, as the irregular arrangement of grains also plays significant role in defining
sliding resistance (Handy and Spangler, 2007).
Strength is defined as the resistance of a body to deformation under a
particular stress regime. Fracture of rock exhibits complex process, where the
phenomenon occurs when fractured planes are generated mainly due to plane
shearing. Cohesion between minerals plays part in defining the rock strength, which is
defined as the bonding force between mineral particles which compose a rock (de
Vallejo and Ferrer, 2011).
The earth experienced impact processes over a long period before. Ahrens and
O’Keefe (1987) inferred that impact accretion is now forming the planetary systems
like what had occurred in our own system approximately 4.5 million years ago. In an
impact phenomenon, shock waves are generated and travelled in both the projectile as
4
well as the target. The waves propagate since early of initial contact is made between
the striker and the target. Jetting is caused when rarefaction waves unload the shocked
zones from available free surfaces. The compression stage terminates when backward
propagating shockwaves meet the back of the projectile (Ahrens and O’Keefe, 1987).
Three regimes are depicted in an impacted target for the peak shock pressure.
The first regime is impedance match regime, second is pressure decay regime and last
one is called as elastic regime. Impedance match regime is when peak shock pressure
is defined by planar impedance match method, extending to amount of projectile radii
into the target. Pressure decay, P in regime 2 is given in the form of:
Equation 1.1
where the exponent of pressure decay, i.e. n is expressed as:
Equation 1.2
where v is the impact velocity. In elastic decay regime, the shock decays below the
strength of target, which is Hugoniot Elastic Limit (HEL). In practical terms, a lot of
target material is expelled in regime 1, resulting in crater formation. Next, the shock
energy deposition occurs in the zone of regime 2. The process continues with regime 3
where crater’s radius exceeds HEL (Ahrens and O’Keefe, 1987).
Equation of states is crucial in describing the relationship between state
variables (pressure and changing volume) especially when a material is under
compression. Among required conditions in equation of states (EOS) are entropy
values, internal energies and specific volumes. These are the conditions for incipient
melting (IM), complete melting (CM), incipient vaporization (IV) and complete
vaporization (CV) to occur during impact process (Ahrens and O’Keefe, 1987).
5
Dynamic crack propagation in rocks is a very haphazard process. Traditionally
the rock breaking tools like pick axe and hammers produce irregular shaped final
mass. Dynamite has also been used extensively to break apart the hard rock mass. The
process is called as blast loading, where energy is suddenly released and dissipation of
accumululated energy occurs in various ways, for instance in blast waves (Kinney and
Graham, 1985). Recently some chemicals have been employed to be inserted into
drilled rock holes which crack the rocks on hardening. Non-explosive demolition
agents are used to break the rocks but the chemicals used are unsafe for human usage.
This technique breaks the rocks slowly and silently where required final rock shapes
are obtainable to some extent.
High velocity weapons (kinetic projectiles) could also be employed to
demolish rock structures, which is known as impact loading. It contradicts to
aforementioned blast, since impact is a localized circumstance for the damage done.
When a striker collides with a target during impact, some or all kinetic energy from
the striker is transformed to perform damage on both bodies in contact, where wave
propagation takes place in governing the process (Zukas, 1994). This research
involves projectile impact configuration, which is one of main impact problems.
In impact phenomenon, the damage done might be local or global. Global
damage occurs when it involves all or almost every part of the impacted target, while
local damage exists near or around the impacted area only. For low-velocity projectile
impact, membrane forces are such a crucial energy absorbing mechanism, thus should
be retained in the governing equations, regardless whether perforation occurs or not
(Jones, 1989). Another important aspect in impact dynamics is stress wave
propagation, which is derived from conservation of mass, momentum and energy
principles (Nicholas and Recht, 1990).
6
Modelling of failure mechanism falls into three categories, which are
instantaneous, time-dependent and micromechanical. The former, i.e. instanteneous
model is usually avoided in order to obtain accurate model since it is physically
incorrect and a well-known fact that material failure associates closely with time-
dependent process (Oscarson and Graff, 1968). The model assumes the failure or
separation of materials (spalling) happens right at the time of stress or strain reaching
its critical value. Nicholas and Rajendran (1990) explains that its failure condition
does not require any damage computation. Time-dependent model, on the other hand,
deals with time history effect (Zukas, 1994) with a cumulative damage model
proposed by Johnson and Stryk (1986) as one of its model. The model solves for
fracture strain and damage with material constants D1 up to D5 are listed in a table
built out of the experiments. Nevertheless, for brittle materials, the failure is in the
form of inelastic deformation, primarily due to nucleation and growth of microcracks,
which contradicts the plasticity of metal’s failure (Nicholas and Rajendran, 1990).
Finally, the latter, i.e. micromechanical model is a complex form though it follows
exactly the physical failure process of materials.
1.2 PROBLEM STATEMENT AND ITS SIGNIFICANCE
Dynamic crack propagation in rocks is a very haphazard process. Granite was the
most commonly used building stone for structural buildings in Britain and most world
cities during nineteenth and twentieth century (Cotta, 2008). This fact suggests the
essence of further studying granite’s behavior in particular related to impact and blast
loadings. Though granite exhibits variation of characteristics like other types of rocks,
nevertheless it has less properties variation than the rest (Cotta, 2008). Due to less
variation in inhomogeneous granite’s material behavior especially the rock’s fracture
makes the granite to be among popular choices for building construction.
7
The case of perforation of plates by striker or projectile is a much noted
complex phenomenon (Jones, 1994). Jones and de Oliveira (1980) predict a critical
velocity leading to perforation as:
Vp k σyd(1 )
Equation 1.3
where the important parameter to be noted in this study is the diameter of striker,
which is denoted as ‘d’ in the above equation. From the equation, it is crystal-clear
that the diameter of striker has direct impact on the determination of critical velocity
of perforation, thus this research is proposed to study the effect of sudden increment
of contact area between the projectile and target, in order to improve the previous
finding. From the tests conducted by Corran, Shadbolt and Ruiz (1983), the
perforation energy increases as the nose radius decreases from ∞ (flat-ended) to d/2
(hemispherical).
Depending upon the shape, density, and the projectile material, the end results
for broken rocks may differ significantly. It is well known fact that when two stiff
materials come into contact at high velocity they may separate instantly retaining most
of their kinetic energy. But when a projectile is made up of a less stiff material which
strikes a comparatively brittle targe, for instance granite rock, it is expected that the
projectile would deform plastically with very little rebound. That may impart most of
the kinetic energy of the projectile into the brittle target causing more damage to the
target material. This phenomenon is one of the important factors to be investigated in
the present research work so that the effect of varying projectiles stiffness on the
damage and failure process of brittle materials can be determined, which was
preceded by Dotoli et al. (2008) for numerical simulation.