Florida State University Libraries
2015
Characterization of Sapphire: for ItsMaterial Properties at High TemperaturesHarman Singh Bal
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FLORIDA STATE UNIVERSITY
COLLEGE OF ENGINEERING
CHARACTERIZATION OF SAPPHIRE:
FOR ITS MATERIAL PROPERTIES AT HIGH TEMPERATURES
By
HARMAN SINGH BAL
A Thesis submitted to theDepartment of Mechanical Engineering
in partial fulfillment of therequirements for the degree of
Master of Science
2015
Copyright c© 2015 Harman Singh Bal. All Rights Reserved.
Harman Singh Bal defended this thesis on November 9, 2015.The members of the supervisory committee were:
William S. Oates
Professor Directing Thesis
Rajan Kumar
Committee Member
Eric Hellstrom
Committee Member
The Graduate School has verified and approved the above-named committee members, andcertifies that the thesis has been approved in accordance with university requirements.
ii
To my friends and family
iii
ACKNOWLEDGMENTS
First, I want to thank my advisor, Dr. William Oates for his guidance and support during
my graduate study and research. I would also like to thank my co-advisor and committee
member Dr. Rajan Kumar for his constructive criticism and motivation throughout. I would
also like to thank my committee member Dr. Eric Hellstrom. Additionally, I would like to
thank my colleagues and fellow graduate students for their constant help with the research.
Finally, I would like to thank my family and friends for their unconditional love through
good as well as tough times.
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TABLE OF CONTENTS
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
1 Introduction 1
1.1 Sapphire - Ancient to Ceramic Era . . . . . . . . . . . . . . . . . . . . . . . 11.2 Present Day Applications of Sapphire . . . . . . . . . . . . . . . . . . . . . . 1
1.2.1 Optical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Mechanical Applications . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Electrical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.4 Chemical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.5 High Temperature and High Pressure Applications . . . . . . . . . . 2
1.3 Structure and Properties of Sapphire . . . . . . . . . . . . . . . . . . . . . . 31.3.1 Optical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Mechanical Characteristics . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Sapphire - In High Temperature Pressure Transducers . . . . . . . . . . . . . 61.5 Light-Matter Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Experimental Setup 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Tensile Testing Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Bending Test Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Design of the Bend Bar Set-up . . . . . . . . . . . . . . . . . . . . . . 162.3.2 Validation of Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.3 Baseline Experiments and Results . . . . . . . . . . . . . . . . . . . . 21
3 Results and Observations 26
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Cross-polarized Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.1 Bayesian Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . 343.4 Confocal Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.5 Conclusions and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.5.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
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LIST OF TABLES
1.1 Coefficients of reflection of the surface depending on the roughness and theangle of ray incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Properties of Sapphire [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1 Strength comparison between pristine and laser machined specimen . . . . . . 33
3.2 Modulus comparison between pristine and laser machined specimen . . . . . . 33
vi
LIST OF FIGURES
1.1 Different crystallographic planes of sapphire crystal . . . . . . . . . . . . . . . 3
1.2 (a) Dependence of refractive index on wavelength, (b) Dependence of refractiveindex on temperature[7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 (a)Temperature dependence of the absorption coefficient, K[1],[8], (b) Trans-mission(T) of sapphire [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Indentation Fracture in (a)pristine specimen versus (b) laser machined speci-men[16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Testing apparatus with tensile set-up inside the furnace. . . . . . . . . . . . . 14
2.2 Dog-bone specimen hinged at inverted-L aluminum piece and ceramic tube withdimensions of the specimen mentioned . . . . . . . . . . . . . . . . . . . . . . 15
2.3 (a) Bend Bar set-up depicting SS-410 block and SS pins (b) 2D view of thebend bar set-up with dimensions in millimeters . . . . . . . . . . . . . . . . . 17
2.4 Displacement(µm) of bend-bar specimen in COMSOL. . . . . . . . . . . . . . 18
2.5 Displacement(µm) of cantilever in COMSOL. . . . . . . . . . . . . . . . . . . 19
2.6 Displacement(µm) of stainless steel cantilever in COMSOL. . . . . . . . . . . 20
2.7 Load vs. displacement curves for polycrystalline alumina . . . . . . . . . . . . 21
2.8 Modulus vs. displacement curves for polycrystalline alumina. . . . . . . . . . 22
2.9 Load cell under no loads vs. Load cell under compressive Load. . . . . . . . . 23
2.10 Final Setup including capacitor probe . . . . . . . . . . . . . . . . . . . . . . 24
2.11 (a) Load vs. displacement curves from the MTS data. (b) Load vs. displace-ment curve from capacitor probe data . . . . . . . . . . . . . . . . . . . . . . 25
2.12 (a) Modulus vs. displacement curve from MTS data. (b) Modulus vs. displace-ment curve from capacitor probe data . . . . . . . . . . . . . . . . . . . . . . 25
3.1 (a) Pristine specimen configuration (b) laser machined specimen with milledregion configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Cross-polarized images of pristine and laser machined sapphire . . . . . . . . 28
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3.3 (a) Cross-polarized image of laser machined region in 20x optical zoom aftertest at 950◦C (2nd specimen). (b)Cross-polarized image of laser machined partof the specimen with 20x optical zoom after test at 950◦C (3rd specimen). . . 29
3.4 (a) Cross-polarized image of laser machined region in the 20x optical zoom aftertest at 1300◦C (2nd specimen). (b)Cross-polarized image of laser machined partof the specimen with 20x optical zoom after test at 1300◦C (3rd specimen). . 29
3.5 (a) Load vs. displacement curve for pristine sapphire at room temperature. (b)Load vs. displacement curve for laser machined sapphire at room temperature.Strength based on (3.2) as given in the legend. . . . . . . . . . . . . . . . . . 31
3.6 (a) Load vs. displacement curve for pristine sapphire at 950◦C. (b) Load vs. dis-placement curve for laser machined sapphire at 950◦C. Strength based on (3.2)as given in the legend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.7 (a) Load vs. displacement curve for pristine sapphire at 1300◦C. (b) Loadvs. displacement curve for laser machined sapphire at 1300◦C. Strength is basedon (3.2) as given in the legend. . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.8 (a) Strength comparison from previous studies (b) Strength comparison be-tween laser machined and pristine sapphire at different temperatures. . . . . 34
3.9 (a) Parameter chain obtained with 5×104 realizations of beam the beam theorymodel to demonstrate burn in. (b) 95%Credible(dark gray) and 95%prediction(lightgrey) intervals for the beam theory model with fixed thickness . . . . . . . . 35
3.10 (a) Parameter chain obtained with 5 × 104 (b) Marginal posterior density forthe thickness of specimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.11 Marginal posterior densities depicting variation in modulus with thickness isalso varied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.12 Load vs. displacement curves with Bayesian uncertainty analysis, Dark greyrepresents the credible interval, Light grey represents 95%prediciton interval . 38
3.13 Posterior densities for pristine and laser machined spceimen . . . . . . . . . . 39
3.14 Confocal microscope images for the pristine specimen at 10x zoom which wastested at room temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.15 Confocal microscope images for the pristine specimen at 20x zoom which wastested at room temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.16 Confocal microscope images for the laser machined specimen at 20x zoom whichwas tested at 950◦C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
viii
3.17 Confocal microscope images for the laser machined specimen at 20x zoom whichwas tested at 1300◦C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
ix
ABSTRACT
There are numerous needs for sensing, one of which is in pressure sensing for high tem-
perature application such as combustion related process and embedded in aircraft wings
for reusable space vehicles. Currently, silicon based MEMS technology is used for pressure
sensing. However, due to material properties the sensors have a limited range of approxi-
mately 600◦C which is capable of being pushed towards 1000◦C with active cooling. This
can introduce reliability issues when you add more parts and high flow rates to remove
large amounts of heat. To overcome this challenge, sapphire is investigated for optical based
pressure transducers at temperatures approaching 1400◦C. Due to its hardness and chemical
inertness, traditional cutting and etching methods used in MEMS technology are not appli-
cable. A method that is being investigated as a possible alternative is laser machining using
a picosecond laser. In this research, we study the material property changes that occur from
laser machining and quantify the changes with the experimental results obtained by testing
sapphire at high-temperature with a standard 4-point bending set-up. Keywords: Sapphire,
Bayesian analysis, thermomechanics, alumina
x
CHAPTER 1
INTRODUCTION
1.1 Sapphire - Ancient to Ceramic Era
Sapphire, over the ages has demonstrated its importance for various civilizations over
different periods of time. The blue sapphire signified the height of celestial hope and faith,
and was believed to bring protection in ancient and medieval worlds. The Greeks wore it for
wisdom of Delphi, Buddhists believed it brought devotion along with spiritual enlightenment
while Hindus considered it to the be greatest offering in the temples for worship. And today,
it’s consider a stone of love, fidelity and commitment that every woman cherishes to have.
However, for the scientific community with recent pace of development and application of
ceramics, sapphire offers potential solutions to the toughest problems engineers face today.
As most ceramics, sapphire has superior wear and abrasive resistance relative to metals,
practically do not show any traces of corrosion, and are great thermal insulators carrying
a low coefficient of thermal expansion. Sapphire has added advantage of being the hardest
known ceramic of all and thus capable of surviving high temperatures and thermal shocks.
Therefore, sapphire has its valid advocates when it comes to materials applications at high
temperatures; however, there are quite a few challenges that still need to be surmounted by
the present day engineers in order to make sapphire practical for sensing applications.
1.2 Present Day Applications of Sapphire
1.2.1 Optical Applications
Sapphire is highly transparent at wavelengths of light between 0.18µm and 5.5µm, also 5-
10 times higher toughness than glass which makes it superior material for optical applications.
High optical transmission of sapphire combined with its wear resistance and the ability to
withstand high temperatures has made sapphire a leading material for high temperature
1
optical sensing and optical spectroscopy that must operate in harsh environments. Further,
sapphire is often used as endoscopes lenses due to its durability in contact with tissues and
in sterilization environments.
1.2.2 Mechanical Applications
Because of the high hardness of sapphire, it has now been widely used for scratch proof
windows in high quality watches, scanner applications and precision mechanical components
designed to undergo high wear.
1.2.3 Electrical Applications
Sapphire also has a high and stable dielectric constant and thus provides wide use for
electronic substrates. Electric properties of sapphire are dependent on crystallographic ori-
entation. Sapphire wafers are used in the semiconductor industry as a substrate for the
growth of gallium nitride based devices light emitting diodes (LED).
1.2.4 Chemical Applications
Chemically inert to common acids and alkali at temperatures up to 1000◦C, sapphire
tubes, crucibles and other sapphire optics are irreplaceable for different applications for the
chemical industry. This makes sapphire excellent for sprays and nozzles of all types, products
subjected to high temperatures or in contact with acids. Sapphire replaced quartzware to
improve durability and reduce contamination while offering good UV transmission.
1.2.5 High Temperature and High Pressure Applications
Sapphire has a unique combination of physical, chemical and optical properties allowing
it to withstand high pressure and thermal shock. Sapphire has a melting point of over
2000◦C, making it ideal for high-temperature applications. High pressure sapphire tubes
and windows, high strength and vacuum windows, high temperature sapphire windows are
small part of large range of optics in this specific field.
2
1.3 Structure and Properties of Sapphire
Sapphire, which is a single crystal form of α-alumina (corundum), is an anisotropic mate-
rial and has a hexagonal/rhombic structure. In Al2O3, the preferred coordination number for
Al+3 ions is 6 and that of O−
2 is 4[1]. This structure thus obtained is nearly hexagonal close
packing of the oxygen ions, with aluminum ions filling two-thirds of the octahedral sites.
The atomic structure leads to anisotropic physical, mechanical, and thermal properties of
sapphire. These properties will be discussed in more detail later in this chapter. Since the
properties differ depending on the planes, the applications must be carefully considered with
respect to crystal cut. The most commonly used planes for sapphire are depicted in Fig-
ure 1.1, these are C,M,A,R which in Bravias classification are noted as (0001), (101̄0), (112̄0)
and (101̄1) respectively. For our research, the plane to be used was determined primarily by
the market availability. The r-plane sapphire was thus used for the study.
Figure 1.1: Different crystallographic planes of sapphire crystal
3
1.3.1 Optical Properties
The Refractive Index of sapphire is higher than most optical materials as the oxygen
ions in the lattice are closely packed. The refractive index for different wavelengths was
presented by Malitson[2] in between the interval 0.2-6µm. The maximum is reached in
the vacuum ultraviolet at λ =0.1425µm with value of 2.5 and it increases with decrease in
wavelength. This behavior is depicted in Figure 1.2 (a). Refractive index is a temperature
dependent quantitiy and Figure 1.2 (b) shows the change in refractive index with increasing
temperature[3].
(a) (b)
Figure 1.2: (a) Dependence of refractive index on wavelength, (b) Dependence ofrefractive index on temperature[7].
The Reflection Coefficient of sapphire at n=1.768 is equal to 7.8%. Since reflection is
dependent on wavelength and the state of the surface, it therefore changes with certain me-
chanical or thermal treatments. In the IR region, reflection coefficient increases at λ=11 µm
and the maximum is observed at 13.5µm and 22µm. Table 1.1 shows the reflective coeffi-
cient of reflection of a sapphire surface subjected to different types of mechanical and thermal
treatments which lead to different surface roughness.
4
Table 1.1: Coefficients of reflection of the surface depending on the roughness andthe angle of ray incidence
The Absorption Coefficient , as other optical properties is also temperature depen-
dent and its dependency on temperature is shown in Figure 1.3(a) [1],[8]. The structural
imperfection and and the impurities present in the crystal make sapphire the pretty stones
they are known as. The presence of Cr+ ions in the crystal is the reason for the appearance
of the red rubies. Similarly, the presence of the iron and titanium lead to the blue sapphire.
Transmission in sapphire is described in Figure 1.3(b). The curve was obtained for
pure, sufficiently perfect crystals. Transimission can be increased by high quality polish.
The quality of the surface finish is especially significant for λ <250µm. The Transmission
and other optical properties of sapphire, such as Scattering and Luminescence do vary
with different physical conditions, but their dependence on temperature is not extensively
known. While transmission does depend on surface finishing, scattering in sapphire depends
on the wavelength. Luminescence is entirely caused by the impurities or lattice defects.
The intensity of these luminescence bands depend on the kind of impurity. Emittance ,
another optical property of sapphire has been observed to change with both wavelength and
temperature. Doping sapphire with impurities such as Ti+ and Mg does affect the emittance
and luminescence of the material.
5
(a) (b)
Figure 1.3: (a)Temperature dependence of the absorption coefficient, K[1],[8], (b)Transmission(T) of sapphire [1]
1.3.2 Mechanical Characteristics
Density, hardness, elastic constants, strength, thermal expansion coefficient and other
material constants were gathered from various studies and tabulated by Dobrovinskaya et.
al in Table 1.2. This table also includes the electrical and thermal properties. On quick
observation, one would realize the wide range in the table for almost all the properties,
mainly because of sapphire’s anisotropic property attributes. Different researchers using
different methods to find such properties results in uncertainty which may also contribute
to the wide ranges reported.
1.4 Sapphire - In High Temperature Pressure
Transducers
As the world seeks to design more efficient turbines, engineers around the world have
focused attention on understanding in detail the fundamental thermal-fluid phenomena, and
6
Table 1.2: Properties of Sapphire [1]
7
also the implementation of active control methods. In order to understand the thermal
fluid phenomena, real time measurements are necessary upon which actuators and feedback
controls can be designed to elucidate the effects that lower the efficiency for turbines and
control of ultra-high speed aircraft(i.e. hypersonic vehicles and re-entry space vehicles).
Such equipment needs to be robust to survive high temperatures. The modern day turbines
have the increased inlet temperatures up to 1600◦C in order to improve turbine peak power
and efficiency. The Integrated Coal Gasification Combined Cycle (IGCC) converts coal to
a fuel gas by mixing it with an oxygen containing gas and then burning the mixture in
combustion turbine generator. The waste heat from the turbine is used to generate steam
that is, in turn, used by steam turbine for additional power generation. The reliability
of the gasifier is key to the performance but the sensing issues are currently a plague to
the system. The thermocouples currently in use last 30-45 days. Thermocouples mainly
fail because of the corrosion resulting from slag penetration into the refractory and stresses
caused by temperature cycles. Therefore, new instrumentation which is capable of handling
and operating in adverse conditions with a lifetime of over a year are required.
Future gas turbine systems will require direct measurements within in the hot section,
even within in the flame zone, as a wider range of fuel enters the supply mix. The appli-
cations of such robust devices can also be found in space re-entry vehicles, jet engines and
internal combustion engines as well. Broadband high temperature sapphire optical pressure
sensors show great potential to revolutionize measurements in harsh environments as these
sensors can be placed in close proximity to fluid-structure interfaces of interest. Specifically,
the development of the high-temperature sensors that measure instantaneous, fluctuating
and mean pressure will greatly extend the spatial and temporal resolution of existing mea-
surement capabilities as well as increase the overall accuracy of high temperature technology.
Substantial amount of research has been focused on fabricating devices using silicon car-
bide (SiC), diamond and sapphire. Each of these materials exhibit excellent material prop-
erties including surviving at high temperatures, mechanical strength, and corrosion/erosion
resistance. Since all these sensors are either piezo-resistive or capacitive, it makes them
highly temperature dependent and/or contingent upon close proximity electronics which
8
limit the high-temperature capabilities. Sapphire-based sensors fabricated using optical de-
tection schemes will not suffer from many of the limitations of other harsh environment
devices.
1.5 Light-Matter Interactions
Surface treatments of brittle materials can affect the fracture toughness and strength of
the material[4]. Both of these properties are crucial for the application in high temperature
pressure transducers. Thermal quenching, annealing or surface coatings are often used to
increase the strength. Literature exists which proves that these effects can lead to improved
tolerance of nano scale flaws where the strength is a function of flaw geometry, shear faults
and surface residual zones[5]. Laser machining is capable of achieving similar effects and is
the preferred way to machine sapphire. However, the damage evolution from irradiation and
thermo-mechanical coupling is still not well understood. The high temperature annealing
and crack healing has been quantified in sapphire [6], but there is lack of measurements
describing how laser irradiation can alter the mechanical strength and elastic properties of
the material.
(a) (b)
Figure 1.4: Indentation Fracture in (a)pristine specimen versus (b) laser machinedspecimen[16]
Recent studies at Florida State University illustrated the formation of dislocations and
amorphous zones in a thin surface region (approximately 5µm) from laser machining of sap-
9
phire. The behaviour showed some interesting implications on enhanced fracture toughness
based on indentation tests as shown in Figure 1.4.
These studeis have provided insight into toughening effects, but no known data exist with
respect to strength measurements. Therefore, experimental characterization of the material
strength as a function of laser irradiation and thermo-mechanical loads are conducted in
order to understand the change in material properties as the pressure transducers must
sustain cyclical mechanical loadings over a broad temperature range. This is necessary for
understanding the material behaviour for integration into load bearing devices and structures
such as high temperature sensors.
10
CHAPTER 2
EXPERIMENTAL SETUP
2.1 Introduction
In Chapter 1, it was explained why is it necessary to understand the thermo-mechanical
material behaviour of sapphire at high temperatures. Specifically, it is desired to understand
the affect of laser machining processes on sapphire, over a broad range of temperatures. In
this chapter, the design of the set-up used to quantify sapphire thermoelastic strength and
elastic properties is explained.
Standardized tensile and bending tests, both were initially considered as potential ways
to measure thermo-mechanical properties. An MTS 1 kN load frame was integrated with
a block furnace Sentro-Tech ST-1600C-445-AG to construct the set-up for the tests. A
cantilever was used to extend the reach of the MTS to apply load inside the furnace through
its top. The cantilever made out of aluminum was mounted on a bracket attached to the
cross-head as showed in Figure 2.1. The vertical cross-head movement of the MTS was
used to apply the loads to the specimen through a tube that goes into the furnace via a
hole in the roof of the furnace. The tube is made of alumina ceramic and is capable of
handling temperature up to nominally 1800◦C. The tube is connected to the load cell which
is mounted on the end of the cantilever extension.The load cell is connected to the MTS
frame and provides data input to the software, and is used to control the MTS displacement
and velocity. The MTS controller can provide very fine control with cross-head speed as low
as one micron per minute. The furnace has rating of up to 1800◦C and is sufficient for the
desired thermo-mechanical testing. The average heating rate of furnace is about 8-12◦C per
minute.
Tensile tests are desired to quantify mechanical properties. Tensile loading is most com-
monly applied to simplify stress and strain measurements. The strength of a material often
is the primary concern which is simply calculated from load measurements over a known
11
cross-section area. Potential ductility can also be quantified by assuming uniform plastic de-
formation in the gauge region. Despite these advantages, the challenges exist when testing
brittle materials under tensile loads as discussed in Section 2.2. In case of tensile setup, the
relation in (2.1) is used to determine the modulus of the specimen where applied load and
change in length of gauge region are obtained from the experiments.
E =P × L
A×∆L(2.1)
where P is the applied load, A is the cross-sectional area, l is the length of the gauge region
and ∆L change in length of the gauge region.
Bending tests, may also be used to quantify thermo-mechanical properties through use
of a set of flexural equations. Bending test procedures and specimen preparation is much
simpler and tooling required is less complicated making it a much more convenient method
over tensile testing. However, there is experimental uncertainty that must be addressed. The
relation in (2.2) below is obtained from beam theory and is used to measure the modulus
for bend bar tests. In this equation, the modulus is calculation depends on the moment of
inertia which is highly sensitive to thickness of the specimen. Hence, the uncertainty of the
thickness of the specimen greatly influences the precision with which the modulus can be
determined. The modulus for a four-point bend bar set-up is
E =P × a× (3L2 − 4a2)
24× I × δ(2.2)
where P is the applied load, a is the distance between load and beam supports, L is the
length of the beam between the supports, I is the moment of inertia and δ is the beam
deflection.
The integration of the furnace and the testing mechanism also introduces more elements
of uncertainty which can be in the form of added compliance or additional noise to the signal.
Hence, careful analysis was carried out to minimize the uncertainty and possible sources of
compliance. The following bullets highlight the most important considerations taken during
the design of the set-up.
12
• Undesirable compliance because of the metallic cantilever to extend the reach of the
testing frame as explained in Section 2.2
• Finite element analysis was conducted to understand the potential sources of compli-
ance introduced into the system
• Materials selection based on finite element analysis to minimize the compliance with
respect to the specimens
• Validating the design by via elastic modulus comparisons of test results relative to
published results
• Noise induced in the signal because of the vibrations of the furnace
• Validation of bend bar modulus estimates using separate resonance experiments
In the following sections, these issues are explored and addressed to provide the final
set-up used for all experimental analysis.
2.2 Tensile Testing Set-up
Since tensile tests provide simple methods to quantify stress and strain, initial attempts
were focused on making the equipment capable of tensile testing inside the furnace to min-
imize the complexities associated with bending measurements. However, the complexities
related to such tests proved more challenging than initially anticipated. Machining the spec-
imen into dog-bone dimensions was one challenge which could be addressed by using laser
machining. The primary challenge was gripping the specimen at the ends in the furnace at
elevated temperatures above 1000◦C. Instead of gripping the specimen, which at high tem-
peratures could lead to high stress in grip region which is undesired, specimen were hinged
between grip regions as shown in Figure 2.2. A prototype of the set-up was made using
aluminum hardware to test acrylic specimen to validate the design at room temperature.
The specimen geometry is shown in the Figure 2.2 and was designed using ASTM D638-02a,
but length and thickness were limited by the market availability. The set-up is shown in
Figure 2.1. The ceramic tube is used to reach into the furnace, one end of the ceramic tube
13
Figure 2.1: Testing apparatus with tensile set-up inside the furnace.
is hinged to the flat edges of the dog-bone specimen and the other end into the load cell at
the end of cantilever which will move with motion of the MTS cross-head. The other end
of the dog-bone specimen is attached to an inverted L-shaped hinge made of aluminum as
shown in Figure 2.2. This aluminum part is held down by a heavy block of stainless steel
410(SS-410). As the MTS moves in the vertical direction upwards, the ceramic tube pulls
the dog-bone specimen upwards which also forces the aluminum piece to move in vertical
direction. The SS-410 block is thus employed to restrict this upward motion of the block
and transmit the tensile load to the specimen. The SS-410 block has its top touching the
roof of the furnace and the bottom is used to hold the aluminum piece down. This way the
aluminum piece cannot move upwards and the entire tension is restricted to the dog-bone
specimen. It is emphasized that the aluminum piece is used only for the prototype to be
tested at room temperature, for high temperature experiments, aluminum will be replaced
14
Figure 2.2: Dog-bone specimen hinged at inverted-L aluminum piece and ceramictube with dimensions of the specimen mentioned
by SS-410 with same dimensions. The range of loads for experiments were determined using
a COMSOL model having material properties on the order of earlier established studies.
The tests were done to obtain load vs displacements curves and the data was processed
to extract moduli for acrylic. The loading rate was .05in/minute and data was acquired
at 10Hz. The comparisons were made with established values for the material. The data
obtained from the experiments was not reliable. The expected modulus values were around
1.5-2.5 GPa. From all the tests, the maximum value for modulus extracted was 15-17 MPa.
This meant there is very significant and undesirable amount of compliance in the system. On
carefully analysing where the added displacements might be coming from, it was observed
that roof of the furnace is compliant in the upward direction when loads were applied and
was not perfectly constrained. Attempts to make the roof and the furnace more stiff were
15
deemed infeasible because of the risks involved which could completely break the insulation
of the furnace and delay the project.
2.3 Bending Test Set-up
2.3.1 Design of the Bend Bar Set-up
A bend bar set-up was designed to overcome the deficiencies of the high temperature
tensile test set-up. The set-up is much simpler and the specimen dimensions much easier
to machine. The geometry of the specimen was determined by the the ASTM C1161-94
testing standards and also subject to material availability in the market. Sapphire sheets of
25.4×25.4×0.1 mm3 were purchased. The dimensions of the specimen were designed for easy
manipulation and elastic response with respect to the load cells available. The most feasible
bend-bar dimensions found out were 16×6×0.1 mm3, and 5 specimens could be cut from
one sapphire sheet. The supports for the four point bend bar were set at 10 mill imeters
and loading pins are 4 millimeters apart. The loading pins were super glued to the end of
ceramic tube as shown in Figure 2.3. The bottom pins were super glued on the alumina
block part as depicted in Figure 2.3. The diameter of the bottom pins is 3.18 mm and that
of the load applying pins is 1.6 mm. The pins are also made of alumina.
2.3.2 Validation of Design
In order to validate the design of the set-up, a working prototype was made for room
temperature tests after carefully making a comsol finite element model for the added parts
of the setup. Since sapphire is expensive, polycrystalline alumina was used to test the set-
up. The polycrystalline alumina dimensions were same as the sapphire specimen but the
thickness was 0.63 mm. Since mechanical properties of alumina are similar to sapphire,
alumina can be used to perform benchmark tests at high temperature. The polycrystalline
alumina was purchased from 2 different sources, Insaco,Inc and Mcmaster-Carr. The data
obtained from the tests is expected to align with the material data sheets provided by both
the suppliers.
16
(a) (b)
Figure 2.3: (a) Bend Bar set-up depicting SS-410 block and SS pins (b) 2D view ofthe bend bar set-up with dimensions in millimeters
The specimens were cut into the above mentioned dimensions using diamond saw. The
setup to be used must be capable of handling temperatures up to 1600◦C. The MTS 1 kN
was integrated with a block furnace ST-1600C-445-AG to construct the setup for the tests.
A cantilever was used to extend the reach of the MTS to apply the loads. The cantilever
was mounted on a bracket attached to the crosshead as showed in Figure 2.1. The vertical
crosshead movement of the MTS was to apply the necessary loads through a tube that goes
into furnace via a hole on the roof of the furnace. One end of the tube is attached to a load
cell connected to the MTS and the other end has the two grooves where the two alumina
pins are superglued. The axis of pins was perpendicular to axis of the tube as shown in
Figure 2.3. The pins make contact with the specimen and are responsible for applying force
on the specimen. The two pins are used to take advantage of four point bend bar test
relative to three point bend bar test. The peak stress produced in the three point flexure
fixture is at the specimen mid-point (or point contact) with reduced stress elsewhere. On the
other hand, the four point flexure fixture produces peak stresses along an extended region
17
Figure 2.4: Displacement(µm) of bend-bar specimen in COMSOL.
of the specimen surface. Hence, exposing a larger area of the specimen is possible with more
potential for defects and flaws leading to failure in this regime.
The deflections in the bend bar specimens are clearly different than tensile specimens.
Using COMSOL, a finite elements model was designed to determine the expected deflections
and load regimes for the specimen, both for polycrystalline alumina and sapphire. Since
the MTS-1kN machine was modified with a cantilever arm to extend the reach, it was
possible that major source of compliance could exist in the cantilever arm; therefore, another
COMSOL model was developed to understand the cantilever deflection under the same load
applied to the specimen. The model was set up with dimensions of the specimen with a
free tetrahedral mesh with maximum and minimum size as 1 mm and 18 µm respectively.
Edge loads were applied in the model equal to the maximum values used in the experiments,
which in this case was 50 N. The load is acting in the Z direction and the edges of the tensile
surface are constrained in all directions. The modulus of the model was set to 300 GPa to
18
Figure 2.5: Displacement(µm) of cantilever in COMSOL.
relate to established values for alumina. The model for the cantilever connected to MTS
load frame was modelled to estimate deflections of aluminum 6061 of 65 GPa. The mesh
was custom defined with maximum elements size as 3 mm and minimum element size as
400µm. Figure 2.5 shows the comsol model for the cantilever with load acting in the +Z
direction. The fixed edge of the cantilever constrained in all directions. A comparison of the
displacement solutions for the specimen and the cantilever will help clarify potential sources
of compliance in the set-up.
Figures 2.4 and 2.5 show the displacements in microns for an alumina specimen and the
aluminum cantilever respectively under the same loads. The maximum bend bar specimen
displacement was about 26.5 µm while displacement in the aluminum cantilever was about
70 µm. The larger displacements in the cantilever results in undesirable load frame com-
pliance. Ideally, the specimen displacements should be significantly larger in order to get
quality load-displacement data from an approximately rigid test frame. This information
19
Figure 2.6: Displacement(µm) of stainless steel cantilever in COMSOL.
was then utilized to make a stiffer system. The aluminum was decided to be replaced with
stainless steel 416 and its thickness was increased. The deflection at the free end of cantilever
is inversely proportional to the cube of thickness, therefore, on doubling the thickness, pro-
duces an 8-fold increase in rigidity. The model was then edited to simulate stainless steel
characteristics with added thickness in order to confirm increased bending rigidity. Fig-
ure 2.6 shows the results of final model which helped determine the final cantilever to be
used in the experiments. The maximum displacement at the free end was only 0.8 µm in
contrast to 70µm in the aluminum cantilever. This was deemed suffciently small relatively
to the specimen deflection of 26.5 µm at 50N. After making the final changes, the parts were
fabricated for room temperature tests for the alumina and sapphire.
20
2.3.3 Baseline Experiments and Results
After the design of the set-up was finalized, the first series of tests were done using
polycrystalline alumina at room temperature. The load and deflection measurements were
implemented using (2.2) to measure the modulus and comparison were drawn against the
established values for the polycrystalline alumina. The Figure 2.7 shows the load vs. dis-
placement curves for alumina at room temperature with data obtained from MTS machine.
The interesting thing to note here is the displacements under the loads go up to 800µm. How-
ever, the finite element model in Figure 2.4 estimates much smaller displacement 26.5µm for
the same loads. The two plausible reasons for the error in experimental data could be an
unrecognised source of machine compliance or the compliance of the specimens.
Figure 2.7: Load vs. displacement curves for polycrystalline alumina
The modulus calculated from the data using (2.2), is shown in Figure 2.8. These values
are significantly lower than the established values for alumina which are nominally around
300 GPa. The one additional way to determine the true elastic properties of the material
was to estimate the modulus from an entirely different method. For this purpose, it was
21
Figure 2.8: Modulus vs. displacement curves for polycrystalline alumina.
decided to evaluate the modulus of the specimen by determining its resonance frequency. Six
different polycrystalline alumina specimens were used. The specimens were attached to the
piezoelectric actuator as a cantilever and the displacement at the free end was measured with
the help of a capacitor probe. The equation ω=1.016(h/l2) [E/ρ]1/2 [9] was used to determine
the modulus E, where ω is the resonant angular frequency, ρ is the material density, h and
l are thickness and length of the specimen respectively. The change in displacement at the
free end gives the resonant angular frequency. The modulus obtained with this method was
at the higher range of the the data shown in Figure 2.8. This confirms that the material
properties were well off from the established material values of the alumina.
Upon further review of the set-up, it was observed that load cell undergoes non-negligible
compression when subjected to loads comparable to or less than the experiments. Figure 2.9
shows the load cell under compression vs load cell under no loads. It was determined that dis-
22
Figure 2.9: Load cell under no loads vs. Load cell under compressive Load.
placements within the load cell were non-negligible requiring an additional probe to quantify
displacements closer to the specimen.
This additional source of compliance was avoided by using a capacitor probe to measure
the displacements of the alumina rod used to apply the load on the specimen by means of a
metallic flange attached to the rod. Figure 2.12 shows the final setup used to estimate the
true specimen displacements. In this setup the only potential source of compliance is the
alumina rod assembly and support insulation in the furnace. Figure 2.11 and Figure 2.12
show the comparison between the MTS and capacitor probe data. Figure 2.11 shows the
difference in the displacement readings from MTS and capacitor probe under the same loads.
Figure 2.12 shows the estimation of modulus based on load-displacement readings. The data
from capacitor probe itself had alot of noise, and method of non-linear least squares with
3rd order guassian equation generated in Matlab were used for fitting to the data. The
data in both Figure 2.11(b) and 2.12(b) is the curve fitted data. The data from capacitor
23
probe measurements and the resonant frequency method align along each other and provid-
ing support that the setup is a reliable source to perform tests required in understanding
thermomechanical material behavior. The data shown in Figure 2.12 gives the values of
the modulus over a significant range and the average of the values is considered as the true
modulus of the material. Statistical analysis of this data and additional data using Bayesian
uncertainty analysis is described in the next chapter.
Figure 2.10: Final Setup including capacitor probe
24
(a) (b)
Figure 2.11: (a) Load vs. displacement curves from the MTS data. (b) Load vs.displacement curve from capacitor probe data
(a) (b)
Figure 2.12: (a) Modulus vs. displacement curve from MTS data. (b) Modulus vs.displacement curve from capacitor probe data
25
CHAPTER 3
RESULTS AND OBSERVATIONS
3.1 Introduction
As mentioned in Section 1.5, this study is focused on measuring mechanical properties
after laser irradiation to understand the effects of laser machining on sapphire. This chapter
explains the experiments and the results obtained for laser machined sapphire with compar-
ison to pristine sapphire. The specimens were machined to induce a laser damage zone in
the area of uniform moment as show in Figure 3.1(b). The middle region has been milled
using a picosecond laser at University of Florida. The machined depth is 20µm. A laser
fluence of 3.81J/cm2 was used with frequency of 100 kHz and scan speed of 100 mm/s. The
laser equipment and further details of the set-up have been discussed elsewhere[12]. The
Figure 3.1(b) shows the cross-section schematic of the specimen with a milled out zone in
the center with width 2 mm. Both pristine and laser machined specimen were tested at room
temperature, 950◦C, and 1300◦C. The heating rate inside the furnace was 8-12◦C/minute
in all the cases. The furnace was kept at the required temperature for 15 minutes before
the mechanical testing was initiated. The change in the microstructure of sapphire after
experiments at high temperatures were evident and shown later in Section 3.2. The results
will be discussed in Section 3.3 along with the mathematical tools used in analyzing un-
certainty. Finally, confocal microscopy was used to analyze the new surfaces created after
fracture described later in Section 3.4.
The laser induced damage has been reported in several studies and the change in micro-
structure is quite apparent as described by Utezaa and Y. Jianga et. al[10],[11]. Earlier
studies have shown a change in strength of sapphire with change in temperature as well as
crystal orientation. The compression along the c-axis leads to twinning on the rhombohedral
plane(r-plane) of sapphire which is well documented. However, there is lack of research on
the failure modes when specimen are cut on the r-plane.
26
(a) (b)
Figure 3.1: (a) Pristine specimen configuration (b) laser machined specimen withmilled region configuration
3.2 Cross-polarized Images
Figure 3.2 shows cross-polarized images of the laser machined specimen under 20X op-
tical zoom. Figure 3.2(a) has the pristine part(bottom half) of the specimen in focus and
Figure 3.2(b) shows the laser machined region in focus. The difference in the two regions
is clearly visible. The laser machined region shows the characteristic horizontal lines in the
direction of laser machining. Figures 3.2(c) and (d) show the pristine and laser machined
region in focus respectively after the specimen was tested at 950◦C. The material is notably
different in reference to Figure 3.2. The laser machined region shows signs of recovery after
laser damage as depicted by changes in color towards white (the original form of pristine
sapphire). The other two specimens that were tested at 950◦C are shown in Figure 3.3 with
the focal plane along the laser machined region. There is an observable difference between
this specimen and the one in Figure 3.3. The specimen in Figure 3.3 shows a little sign of
recovery and also have a coarser surface. This change in roughness is comparable to that
of the pristine specimen. Figure 3.4(a) and (b) show the pristine and laser machined region
for 1300◦C. There are stark similarities between Figure 3.4(b) and Figure 3.2(f) in laser
machined region. The Figure 3.4(a) shows a similar surface structure relative to that of
the pristine specimen and the laser machined region in Figure 3.3(a) and (b), whereas the
Figure 3.4(b) still shows the laser machining pattern. This change in surface roughness will
be compared to the strength measurements later in the chapter.
27
(a) (b)
(c) (d)
(e) (f)
Figure 3.2: Cross-polarized images of pristine and laser machined sapphire
28
(a) (b)
Figure 3.3: (a) Cross-polarized image of laser machined region in 20x optical zoomafter test at 950◦C (2nd specimen). (b)Cross-polarized image of laser machinedpart of the specimen with 20x optical zoom after test at 950◦C (3rd specimen).
(a) (b)
Figure 3.4: (a) Cross-polarized image of laser machined region in the 20x opti-cal zoom after test at 1300◦C (2nd specimen). (b)Cross-polarized image of lasermachined part of the specimen with 20x optical zoom after test at 1300◦C (3rdspecimen).
29
3.3 Experimental Results
Since the geometry for the beam was modified, so was the equation to calculate modulus.
The relation given by (2.2) was then modified to (3.1) using solid mechanics and beam theory
according to
E =1
δ
[
Pa3
3I1+
Pa2
2
(
1
I2
(
b−L
2
)
−b
I1
)]
. (3.1)
From beam mechanics, we can also calculate the stress from load and geometry using
σmax =3Pa
wtx2. (3.2)
In these two equations, P is the applied load, a is the distance between load and beam
supports, L is the length of the beam between the supports, I1 is the moment of inertia
of the pristine region, I2 is the moment of inertia of the laser machine region with smaller
thickness, b is the distance between the supports and the machined region, w is the width
of specimen, tx is the thickness of the specimen the break point.
Since the loading pins were super glued to the alumina rod, it was essential to make sure
the pins were in contact with the specimen as the furnace is ramped up to the necessary
temperatures as the super glue will evaporate at a certain temperature and the pins will no
longer be in the right place to conduct the experiments. The rod and pins are in contact
with the specimen before the furnace is switched on. The rod has to be incremented upwards
as the temperature rises to avoid preloading on the specimen due to thermal expansion.
This can lead to fracture before even reaching the prescribed temperatures at which point
the specimen is loaded mechanically. Therefore, the load is constantly monitored and the
alumina rod is moved upwards manually to maintain a near zero load on the specimen. Also,
the location of the specimen is vital since the geometric parameters in the (3.1) and (3.2)
influence the modulus and strength measurements. The alumina block is carefully placed
with reference to the back wall of the furnace. The alumina rod was guided by the port on
the top of furnace and was at a fixed location inside the box furnace.
30
Figures 3.5 to 3.7 show the results from the load-displacement experiments. Each figure
shows the load displacement curve with the dialogue box on top left of each graph showing
the strength of each specimen based upon (3.2). The Matlab curve fitting tool was used
to find the linear curve fit which produces deterministic values for the modulus. A total
of nine pristine and nine laser machined specimen were tested and comparisons are drawn
between laser machined and pristine specimen at each temperature. Figure 3.5(a) and (b)
show room temperature data for pristine and laser machined data, respectively. The highest
and lowest value for strength are both observed in the case of the laser machined specimen.
At 950◦C the laser machined specimens have higher strength by almost a factor of two
relative to the pristine specimens. However, the variability in strength is significant. The
two specimen with higher strength correspond to the ones shown in Figure 3.3(a) and (b).
At 1300◦C, the pristine specimen exhibit higher strength than the laser machined specimen
but the difference is less compared to the ones for the specimen tested at 950◦C. While the
strength was measured for every individual specimen, the modulus was calculated from the
curve fit for all three specimens for each condition and hence one average modulus value was
calculated for every three specimens tested at the same conditions.
(a) (b)
Figure 3.5: (a) Load vs. displacement curve for pristine sapphire at room tem-perature. (b) Load vs. displacement curve for laser machined sapphire at roomtemperature. Strength based on (3.2) as given in the legend.
31
(a) (b)
Figure 3.6: (a) Load vs. displacement curve for pristine sapphire at 950◦C. (b)Load vs. displacement curve for laser machined sapphire at 950◦C. Strength basedon (3.2) as given in the legend.
(a) (b)
Figure 3.7: (a) Load vs. displacement curve for pristine sapphire at 1300◦C. (b)Load vs. displacement curve for laser machined sapphire at 1300◦C. Strength isbased on (3.2) as given in the legend.
32
Table 3.1: Strength comparison between pristine and laser machined specimen
Temperature Pristine LaserMachined
(◦C) (MPa) (MPa)25◦C spec 1 248 286
spec 2 227 290spec 3 216 499
950◦C spec 1 313 300spec 2 351 659spec 3 341 665
1300◦C spec 1 367 349spec 2 397 300spec 3 - 298
Table 3.2: Modulus comparison between pristine and laser machined specimen
Temperature Pristine LaserMachined
(◦C) (GPa) (GPa)25◦C 375 245950◦C 245 2251300◦C 270 132
Table 3.1 summarizes the strength of the pristine and laser machined specimens at dif-
ferent temperatures. The same is plotted in Figure 3.8(b) to be compared with an earlier
study by Schmid et. al[13] as shown in Figure 3.8(a). Schmid and fellow researchers did not
test r-plane sapphire in the study. The c, m and a plane crystal cuts were used. While all of
the specimens in the prior results showed loss in strength after 500◦C, the r-plane sapphire
in the present study exhibited an increase in strength rises from room temperature to 950◦C
after which it drops at 1300◦C. The laser machined specimen exhibit higher strength than
pristine specimen at room temperature and at 950◦C. However, at even higher temperatures,
the strength drop is more rapid than pristine specimen. Table 3.2 shows the modulus values
for pristine and laser machined specimens obtained from the slope of the fitted curve. The
pristine specimen show drop in modulus at 950◦C but increases at higher temperature. The
laser machined specimen exhibit a constant drop in the modulus values.
33
(a) (b)
Figure 3.8: (a) Strength comparison from previous studies (b) Strength comparisonbetween laser machined and pristine sapphire at different temperatures.
3.3.1 Bayesian Statistical Analysis
Bayesian statistics are used as a tool to quantify the model uncertainty in light of data
that affects the predictability in the load-displacement curves and strength[14,15]. Bayesian
statistics are based on the idea that calibration parameters exhibit uncertainty due to model
discrepancies and observation errors associated with data. Bayesian model calibration is
based on Bayes’ relation
π(θ | P data) =p(P | θ)π0(θ)
∫
Rp(P | θ)π0(θ)dθ
(3.3)
which quantifies the probability of observing parameters θ given a certain set of data Pdata.
π(θ | P data) is typically called the posterior density. The prior density, π0(θ), quantifies
the prior knowledge of the parameters without any regards of the new data. The prior is
updated using p(P | θ), which includes information from both the model and the data.
The denominator normalizes the density such that it has an area of unity. The calibration
parameters are denoted by θ and the parameter dependent model responses are found in the
following statistical model
34
P data(i) = P (i; θ) + ǫı, i = 1, ..., n, (3.4)
where ith experimental data point and error measurement are denoted by Pdata(i) and
ǫi, respectively. Bayes’ equation is implemented using the Markov Chain Monte Carlo
(MCMC)[14], a algorithm to sample the parameters. This method is used to sample param-
eters to reduce the sum of squares residual between the model and the data. The numerical
method employed was the Delayed Rejection Adaptive Metropolis(DRAM) algorithm. It is
assumed that certain model parameters are unknown and subject to uncertainty caused by
model discrepancies and measurement errors. In this study, two models used are based on
(2.2) and (3.1) for pristine and laser machined specimen respectively which originate from
beam theory. The following parameters are calibrated
θ = [E, t] (3.5)
where E is the modulus, and t is the thickness. In (2.2) and (3.1) t is contained within the
moment of inertia as I = (wt3)/12.
(a) (b)
Figure 3.9: (a) Parameter chain obtained with 5 × 104 realizations of beam thebeam theory model to demonstrate burn in. (b) 95%Credible(dark gray) and95%prediction(light grey) intervals for the beam theory model with fixed thick-ness
35
(a) (b)
Figure 3.10: (a) Parameter chain obtained with 5 × 104 (b) Marginal posteriordensity for the thickness of specimen.
The calculations for the modulus are most sensitive to the thickness of the specimen.
The specimen thickness was measured with the length probes provided by HEIDENHAIN
with a resolution of 100 nanometers. The measured values for thickness were between 114
and 116 µm. The uncertainty in thickness leads to larger uncertainty in modulus since
these two parameters cannot be uniquely identified using the beam equations. Figure 3.9
(a) shows the parameter chain with 5×104 iterations with parameter values which in this
case represents the modulus values. This distribution represents a converged posterior that
is not expected to change on increasing any number of iterations. Figure 3.9(b) shows the
load-displacement curve with prediction and credible intervals as obtained from Bayesian
analysis. The prediction interval takes into account the experimental or measurement errors
while the the credible interval takes into account the model uncertainties. Figure 3.10(a)
shows the chain for the thickness with 5×104 iterations. Figure 3.10(b) shows the posterior
density for thickness which is approximately uniform density over the limits 114-116 µm
prescribed in DRAM sampling as motivated by measurements. The densities for thickness
of the specimen are more or less the same for each case and will not be discussed further in
the chapter. Figure 3.11 shows the marginal posterior densities for modulus when thickness is
included as a random variable. The variation in the modulus is quite apparent and the drop in
modulus after laser machining is also depicted in the densities. The column on the left has the
densities for the pristine specimen and column on the right for the laser machined specimens.
In all the cases, laser machined specimen exhibit lower values for modulus than the pristine
36
specimen. There is drop in modulus going from room temperature to 950◦C for both pristine
as well as laser machined specimens where as at 1300◦C, the pristine specimens show increase
in the modulus while laser machined specimens continues to drop. The deterministic values
obtained by curve fits in Table 3.2 fall in between the bounds of the posterior densities.
Figure 3.12 shows the load vs. displacement curves generated with the model with prediction
and credible intervals for each case. Figure 3.13 shows the posterior densities on the same
plot for pristine and laser machined specimen to draw the comparison between different
cases.
(a) (b)
(c) (d)
(e) (f)
Figure 3.11: Marginal posterior densities depicting variation in modulus with thick-ness is also varied
37
(a) (b)
(c) (d)
(e) (f)
Figure 3.12: Load vs. displacement curves with Bayesian uncertainty analysis, Darkgrey represents the credible interval, Light grey represents 95%prediciton interval
38
(a) (b)
Figure 3.13: Posterior densities for pristine and laser machined spceimen
3.4 Confocal Microscopy
Confocal microscopy is an optical imaging technique for increasing optical resolution and
contrast of a micrograph by means of adding a spatial pinhole placed at the confocal plane
to the lens. It enables stitching of the images together to obtain a three-dimensional surface.
This tool helps understand the fracture surface of the bend bar specimen. Figures 3.14-3.17
show the confocal images of the specimen. Images in grayscale are the scanned images of a
certain region while the colored part of the figure show the 3D images of the same region.
The surface under tension is marked in the figures. The scale and color bar on the images
shows the heights and depths on the surface. Figure 3.14 is an image for a pristine specimen
at 10x while the Figure 3.15 is image of the specimen at 20x of the same specimen at some
other region. The 3D image show a certain river pattern suggesting crack propagation from
right to the left. Figures 3.16 and 3.17 are the laser machined specimen under 20x optical
zoom tested at 950◦C and 1300◦C respectively. Figure 3.16 does show similar river patterns
to that of Figure 3.15. These patterns can also be a result of acoustic shock wave propagating
during fracture. These patterns are noticeably missing from the specimen tested at higher
temperature. These images are the most accurate representation of all the specimen.
39
Figure 3.14: Confocal microscope images for the pristine specimen at 10x zoomwhich was tested at room temperature.
Figure 3.15: Confocal microscope images for the pristine specimen at 20x zoomwhich was tested at room temperature.
40
Figure 3.16: Confocal microscope images for the laser machined specimen at 20xzoom which was tested at 950◦C.
Figure 3.17: Confocal microscope images for the laser machined specimen at 20xzoom which was tested at 1300◦C.
41
3.5 Conclusions and Remarks
The aim of the study has been to design a test set-up capable of performing high temper-
ature material tests. The experiments were performed at desired conditions yielding results
which support understanding of the laser damaged sapphire. The laser ablated sapphire
yielded higher strength compared to pristine sapphire similar to data in the literature de-
picting larger fracture toughness laser damage on the surface of sapphire[16]. Laser ablated
sapphire exhibited higher strength at 25◦C and 950◦C but dropped down significantly at
1300◦C. This trend is different from what is observed for other planes of sapphire. The
modulus showed a consistent drop as present in Figure 3.13. The uncertainty with modulus
was also quantified and compared with the uncertainty in thickness.
3.5.1 Future Work
• Since sapphire is an anisotropic material, it becomes essential to understand its prop-
erties along different axes. The r-plane sapphire is used in the study, but the in-plane
crystal orientations need to be confirmed with x-ray measurements. This is an im-
portant step in the study to conclude the laser damage on sapphire specimen on the
r-plane.
• Also the uncertainty related to the parameters such as ’a’, ’L’ and ’b’ as described
in Section 3.3, should be considered in the Bayesian analysis to quantify potential
uncertainty associated with these parameters.
42
BIBLIOGRAPHY
[1]. Elena R. Dobrovinskaya, Leonid A., Lytvynov Valerian Pishchik. Sapphire-Material,Manufacturing, Applications. Springer, ISBN: 978-0-387-85694-0, 2009.
[2]. Malitson I.H. Opt. Soc. Am. 52 , p. 1377 .1962.
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BIOGRAPHICAL SKETCH
Harman Singh Bal, born in India, earned Bachelor’s degree in Mechanical Engineering soon
after which he moved to Tallahassee, Florida to further his education with a M.S. in Me-
chanical Engineering at Florida State University.
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