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All Graduate Theses and Dissertations
Graduate Studies
2014
Uncertainty Analysis of Mechanical Properties
from Miniature Tensile Testing of High Strength
Steels
Deepthi Rao Malpally
Utah State University
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UNCERTAINTY ANALYSIS OF MECHANICAL PROPERTIES FROM
MINIATURE TENSILE TESTING OF HIGH STRENGTH STEELS
by
Deepthi Rao Malpally
A thesis submitted in partial fulfillment
of the requirements for the degree
of
MASTER OF SCIENCE
in
Mechanical Engineering
Approved:
Dr. Leijun Li
Major Professor
Dr. Thomas H. Fronk
Committee Member
Dr. Steven L. Folkman
Committee Member
Dr. Nicholas A. Roberts
Committee Member
Dr. Mark R. McLellan
Vice President for Research and
Dean of the School of Graduate Studies
UTAH STATE UNIVERSITY
Logan, Utah
2014
ii
Copyright
c Deepthi Rao Malpally 2014
All Rights Reserved
iii
Abstract
Uncertainty Analysis of Mechanical Properties from Miniature Tensile Testing of High
Strength Steels
by
Deepthi Rao Malpally, Master of Science
Utah State University, 2014
Major Professor: Dr. Leijun Li
Department: Mechanical and Aerospace Engineering
Boat samples extracted from scheduled maintenance shutdowns of piping and pressure
vessels provide opportunities for testing for mechanical properties of the service exposed components. However, it is not clear whether testing of miniature specimens machined from boat
samples which are about 2 in. long can be a viable replacement for the standard-sized mechanical testing. Three steels, stainless steel Type 304, sensitized Type 304, and SA516 Grade
70 carbon steel, are tested by standard-sized specimen and miniature specimen tensile tests.
Mechanical properties as affected by the specimen geometry and tensile testing procedure for
miniature specimen testing are compared to that of conventional specimens tested according
to ASTM A370-10. The miniature tensile testing results are analyzed by using Monte Carlo
Method (MCM) for uncertainty estimation in order to quantify the probability distribution of
mechanical properties. For the steels under study, miniature specimens with a cross-sectional
area of 3 mm2 and 12 mm gauge length are found to produce equivalent mechanical properties
as tested from standard-sized specimens.
(59 pages)
iv
Public Abstract
Uncertainty Analysis of Mechanical Properties from Miniature Tensile Testing of High
Strength Steels
by
Deepthi Rao Malpally, Master of Science
Utah State University, 2014
Major Professor: Dr. Leijun Li
Department: Mechanical and Aerospace Engineering
This Miniature mechanical testing study is concerned with the use of miniature specimens to identify the mechanical properties of stainless steel Type 304, sensitized Type 304
and SA516 Grade 70 carbon steel as a viable replacement for the standard sized mechanical
testing. The study aims at obtaining suitable specimen geometry and tensile testing procedure for miniature mechanical testing whose mechanical properties are comparable to that of
conventional specimens of ASTM A370-10 of the same steel. All specimens are flat and the
gauge length cross section will be varied to obtain suitable geometry. The miniature tensile
testing results are further validated by using Monte Carlo Method (MCM) for uncertainty
estimation in order to know the probability distribution of mechanical properties. Miniature
specimens with a cross section of 3 mm2 and 12 mm gauge length are found to produce equivalent mechanical properties as tested from standard-sized specimens. If a reasonable agreement
is received, it will provide us with a very useful tool to evaluate mechanical properties of degraded materials, which cannot be removed from service for standard testing, for repair and
service life evaluation.
v
Dedicated to my dearest parents and brother...
vi
Acknowledgments
I would like to express my highest regards and gratitude to my major professor, Dr. Leijun
Li, for his continued guidance and advice despite several constraints. I will be forever indebted
to him for believing in a novice like me right from the beginning of my career as a research
assistant at USU. I would like to thank my committee members, Dr. Thomas H. Fronk, Dr.
Steven L. Folkman and Dr. Nick Roberts, for their invaluable suggestions and also Dr. Barton
Smith for his advice in uncertainty analysis. I would like to express my sincere gratitude to the
graduate advisor, Christine Spall, for her constant encouragement and for helping in fulfilling
all the requirements to accomplish graduate studies. I would like to thank all past members of
the Materials Processing & Testing Laboratory, Andrew for helping me prepare the specimens,
Jacob, Zhifen, Yin and Bishal for their friendship. Special thanks to Dayakar Naik for being
my mentor academically and emotionally.
Motivation and belief in oneself is of umpteen importance when one is far away from home.
I would like to thank my dear parents, Ravi and Anjana, for believing in me and boosting my
confidence. I hope to make you proud everyday. Thank you my late grandmother, Usha ajji,
for expressing immense happiness in all my endeavours. Whatever little accomplishments I
have today is all because of your blessings and encouragement. Thank you my little brother,
Sanjeev, for filling my absence at home and all my well wishers.
My experience as a graduate student at USU has been wonderful and fun filled only with
the presence of many fellow graduate students who eventually became friends for life. Thank
you Ravi, for helping me beilieve in myself, Neeraj, Bidisha, Saptarishi, Rajee, Ashish, Swati,
Ruchir, Manju, Joe Shope, Kurt, Scott and many others. I would like to thank my friends
back home, Prerita, Rahul and Danny, for helping me in many ways than I could describe. I
am thankful to God for showing me the right direction in life.
Deepthi Rao Malpally
vii
Contents
Page
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Public Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Overview of the field of inquiry . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Parameters effecting the mechanical properties of miniature specimens . .
2.3 Advances to miniature testing techniques . . . . . . . . . . . . . . . . . .
3
3
6
7
3 Experimental Procedure . . . . . . . . .
3.1 Component Element Properties . .
3.2 Sensitization . . . . . . . . . . . .
3.3 Tensile Specimen Preparation . . .
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4 Experimental Results and Discussion . . . . . . . . . . . . .
4.1 Conventional (Macro-sized specimen) Tensile Testing .
4.2 Miniature Tensile Testing . . . . . . . . . . . . . . . .
4.3 Optimum Specimen Size . . . . . . . . . . . . . . . . .
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5 Uncertainty analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Tolerance intervals in Sample Populations . . . . . . . . . . . . . . . . . .
5.2 Propagation of Mechanical Properties by Monte Carlo Method (MCM) . .
5.2.1 General Approach for MCM . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Propagation of Mechanical Properties . . . . . . . . . . . . . . . .
5.3 Convergence study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 29
. . . 29
. . . 32
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. . . 42
6 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
viii
List of Tables
Table
Page
3.1
Component element properties . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
4.1
Conventional (Macro-sized) results . . . . . . . . . . . . . . . . . . . . . . . . .
17
4.2
Results obtained from SEMTester 1000 EBSD . . . . . . . . . . . . . . . . . . .
18
4.3
Mechanical properties for Stainless Steel Type 304 miniature specimens . . . .
20
4.4
Mechanical properties for Sensitized Stainless Steel Type 304 miniature specimens 21
4.5
Mechanical properties for SA516 grade 70 carbon steel miniature specimens . .
22
5.1
Specimen area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
5.2
Load values for (YS) and (UTS) in specimens of gauge area (1x3) mm2 . . . .
36
5.3
Variables and their uncertainties - SS grade 304 . . . . . . . . . . . . . . . . . .
36
5.4
Final length of specimens of gauge cross section (1x3) after tensile testing - SS
grade 304 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
5.5
Variables and their uncertainties - Sensitized SS grade 304 . . . . . . . . . . . .
39
5.6
Final length of specimens of gauge cross section (1x3) after tensile testing Sensitized SS grade 304 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
5.7
Variables and their uncertainties - carbon steel SA516 grade 70 . . . . . . . . .
41
5.8
Final length of specimens of gauge cross section (1x3) after tensile testing carbon steel SA516 grade 70 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
ix
List of Figures
Figure
Page
2.1
Shear punch test fixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Indigenously developed table top ball-indentation test system . . . . . . . . . .
4
3.1
Microstructure of SA516 steel etched with 4% Nital
. . . . . . . . . . . . . . .
10
3.2
Microstructure of 304/304L steel etched with Vilella’s reagent . . . . . . . . . .
10
3.3
Microstructure of sensitized 304/304L steel etched with Vilella’s reagent . . . .
11
3.4
Microstructure of 304/304L steel . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3.5
Microstructure of sensitized 304/304L steel . . . . . . . . . . . . . . . . . . . .
12
3.6
SEMTester 1000 EBSD instrument for miniature testing . . . . . . . . . . . . .
14
3.7
Specimen cross section at gage length is (1x0.2) mm . . . . . . . . . . . . . . .
15
3.8
Variation in specimen gage thickness from 3 mm to 0.5 mm . . . . . . . . . . .
15
3.9
(a) Tinius Olsen H50KS (b) Specimen test in progress . . . . . . . . . . . . . .
16
4.1
Chauvenet’s criterion for rejecting a reading obtained from the book Experimentation, Validation and Uncertainty Analysis for Engineers by Coleman and
Steele [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
4.2
Tensile strength vs specimen gauge cross section . . . . . . . . . . . . . . . . .
24
4.3
Yield strength vs specimen gauge cross section . . . . . . . . . . . . . . . . . .
25
4.4
Engineering stress vs strain curves for specimen gauge cross section (1x3) . . .
26
5.1
Factors for two-sided tolerance interval [1] . . . . . . . . . . . . . . . . . . . . .
30
5.2
Schematic for MCM for uncertainty propagation when random standard uncertainties for individual variables are used [1] . . . . . . . . . . . . . . . . . . . .
33
Distribution of MCM results for yield strength of carbon steel SA516 grade 70.
Expanded uncertainties for each variable being calculated at 95% confidence . .
43
Convergence study for MCM value of UF for 95% combined uncertainty for each
variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
5.3
5.4
x
Acronyms
ASTM
american standards of testing and materials
SEM
scanning electron microscope
EBSD
electron backscatter diffraction
MEMS
microelectromechanical systems
MMST
modified miniature specimen test
FE
finite element
EDM
electric discharge machining
FIB
focused ion beam
LVDT
linear variable differential transformer
MCM
monte carlo method
DRE
data reduction equation
SS
stainless steel
YS
yield strength
UTS
ultimate tensile strength
1
Chapter 1
Introduction
Steels and alloys experience material degradation when encountered with accidents or
exposed to elevated services, especially after long-term exposure. Depending on the types of
materials, the degradation could include embrittlement, creep, sensitization, phase precipitation, other phase transformation/formation, corrosion, oxidation, etcetra. When materials
experience degradation, it is important to evaluate material properties for repair, extended
use, or life prediction. For the material evaluation, it is often to prefer to obtain the actual
material mechanical properties. However, for many important components, such as vessels in
refinery industry, it is not realistic to obtain sufficient material to perform standard mechanical
testing due to the relative large sample size requirement, which typically needs window cut
from vessels. Window cut from service vessels will involve significant challenges of its replacement from cost and code requirements, and thus, it is typically not feasible to obtain materials
for standard mechanical testing. Nevertheless, it is more feasible to obtain boat samples, such
as about 2 in. (50mm) long, from serviced vessels, and it would be highly valuable for material
evaluation if mechanical testing could be performed on boat samples, which requires miniature
mechanical testing method.
Miniature mechanical testing has become an important aspect of industrial and research
labs, especially for analyzing the mechanical properties. The requirement of miniature testing depends on many factors such as availability. A number of tensile testers have been
employed for miniature testing with samples of different sizes and geometries. For example,
MTI SEMTester can test miniature specimens. Universal Tensile Testing machines can also
be employed for miniature testing depending on the minimum cross head travel as miniature
specimens have very short gauge length. However, there is no study to verify the miniature
mechanical testing results against the standard mechanical testing results.
2
The current study will verfiy the miniature mechanical testing for stainless steel Type 304,
sensitized Type 304 and SA516 Grade 70 carbon steel as a viable replacement for standard
sized mechancial testing. Miniature tensile specimen designs are developed in a combination of
varying gauge cross-sectional area, and then tested on two different tensile testers, Tinius Olsen
H50KS and MTI SEMTester 1000 EBSD. It is shown in this study that miniature specimens of
a particular gauge cross section exhibit similar mechanical properties to that of its equivalent
conventional size.
Since only a minimum number of specimens are tested, there is a possibility of variation
from the expected conventional results from miniature specimens. This uncertainty in the results could be due to a number of factors, either machine oriented, which classify as systematic
uncertainties, or procedure oriented, which classify as random uncertainties, in a broad sense.
Also, the variation in miniature measurements reflects local variations in the measurements
that are not measured with full sized specimens. This is due to lesser presence of voids and defects in miniature specimens when compared to standard full sized specimens. An estimation
of uncertainty analysis will help generalize the mechanical properties of the materials being
reviewed, and hence can be compared to conventional testing on a general scale which has not
been covered by any study so far.
The objectives for this study are as follows:
• To identify an optimum procedure for extraction of miniature tensile specimens of high
strength steels
• To obtain a suitable specimen geomtery relevant to that obtained from a boat sample,
and the miniature tensile tester utilized for the purpose
• To analyze the data obtained with miniature specimens and compare the results with
the tensile tests of equivalent standard specimen results
• To propagate the mechanical properties obtained after tensile testing of miniature specimens by conducting a Monte Carlo uncertainty estimation
3
Chapter 2
Literature Review
2.1
Overview of the field of inquiry
The origin of miniaturization of specimens is in the nuclear industry because of expensive
and limited irradiation space, specimen sizes had to be miniaturized for experimental purposes
of irradiation programs in reactors. However, in some cases there is a need to keep the test
specimen similar in size to structural components [2]. The miniature specimen test techniques
are broadly classified as follows [3]:
• Tests that are based on miniaturization of conventional specimen sizes such as miniature
tensile, fatigue, impact and fracture toughness tests
• Tests based on novel techniques using disk sized specimens such as disk bend tests, shear
punch/small punch tests as shown in Figure 2.1. The small specimen test techniques for
disk bend tests and small punch tests do not have conventional counter parts. These
techniques therefore need to be validated before being effectively used
• Tests based on ball indentation techniques as shown in Figure 2.2
Fig. 2.1: Shear punch test fixture
4
Fig. 2.2: Indigenously developed table top ball-indentation test system
A few methodologies that evolved in the past on miniature testing and its requirement in
various fields are briefly described as follows:
1. Service life of a component is greatly dependent on the extent of monitoring the material degradation of the component when in service. In order to monitor the service
life, nondestructive testing is required to evaluate material properties while keeping the
component in service. For this purpose miniature mechanical test named small punch
test has been used [4]. Sample punch test is claimed to be the most effective test as the
specimens can be extracted from the highly stressed zones of the components and also,
the samples can be extracted as many times as required allowing continuous monitoring
of the component while in service.
2. Micro-scale testing has also been developed to characterize the performance and reliability of Micro-Electro-Mechanical Systems (MEMS). Various testing techniques have been
developed to measure the mechanical response of small specimens. Different methodologies aiming at determining the mechanical properties of small scale samples have
evolved in the recent past. A survey describes that a technique involving instrumented
5
indentation, most specifically, nano-indentation, coupled with careful loading and unloading response of the material make it possible to measure both the hardness and
elastic modulus of material [5]. However, strength and strain hardening effects cannot be measured with indentation techniques. For manufacture of MEMS, a number
of machining processes such as photolithography, vapor deposition and electro plating
are available in exclusive foundries of MEMS. For machining miniature specimens from
bulk specimens, traditional machining processes are applicable such as wire EDM, laser
machining, chemical mechanical polishing, and FIB milling may be used [5].
3. Recent development in nanomaterials and metallic glasses are facing challenges with
preparation, handling and testing of small volumes of materials and also lack clear understanding of test results. Hence miniaturization has become a general trend [2]. In
the past, Modified Miniature Specimen Test (MMST) was employed to determine yield
strength, tensile strength and uniform elongation of unirradiated and irradiated reactor
pressure vessel steels [6].
4. Quasi static tensile tests were performed for 1.2mm gauge TRIP800 steel sheets with
miniature tensile sample to determine the effects of sample geometry and loading rate
on tensile ductility [7]. The samples with smaller gauge length will yield higher level of
ultimate ductility since the post uniform elongation is achieved mainly in very narrow
neck region. This is because in the miniature sample geometry, there is the localized
nature of the neck during the post-uniform elongation just before fracture [7]. Similar
increase in ductility was observed in miniature samples of AA5182 samples under quasi
static loading in comparison with full size samples. Also, higher loading rate yielded
higher upper yield strength compared to the results for TRIP steels reported by other
authors. This is different from the reported strain rate sensitivity results for mild steel
where a typical reduction of ductility is observed at high strain rates. However these
discussions are valid for quasi static conditions of the sample being tested.
6
5. In one of the miniature tensile behavioral studies, the samples were made of ultrafinegrained Cu having the same width of 1mm and varying gauge length while keeping
thickness constant and vice versa were studied. The results showed that thinner samples
are susceptible to shear failure, resulting in smaller reduction in area [8]. However, the
gauge length has no evident influence on the failure mode or area reduction. This article
further summarizes that shorter and thicker specimens tend to be more ductile. The
thickness effect is mainly seen during necking/fracture modes. The gauge length effect
originates from strain.
6. In another study it is shown that tensile testing of very thin sheet metal, usually specimens lesser than 10µm (in this case Cu 99.9%), show trend in decreasing mechanical
properties [9]. However, many authors have mentioned the need for efficient modelling
of the constitutive behavior and the use of simulation and FE analysis [10].
2.2
Parameters effecting the mechanical properties of miniature specimens
1. Miniaturization causes scaling effect, which leads to a different material behavior in the
microscale compared to the macroscale. In general, this effect is pronounced largely and
relates to specimen size and geometry and other factors such as microstructural constraints (grain size through specimen thickness, microstructural anisotropy, microstructural inhomogeneity, etc.), surface effect, residual stress [2].
2. Non uniform stress distributions during testing provide an additional length parameter
that affects the size-effect. The tensile specimen size-effect should be considered when
comparing mechanical properties, such as tensile ductility measured on non-standardized
dog-bone specimens [8]. To minimize any undue stress, the sample must be aligned with
the center line of the two test machine grips [11]. Furthermore, with an ever increasing
popularity of miniaturized tensile specimens in research of materials, there is a need for
a standardized protocol to be adopted.
7
3. Specimen dimensions and strain measurement methods largely influence the tensile
stress-strain curves. Miniature specimens are too small for use with conventional extensometers so that the strains are usually calculated from the crosshead displacements.
Uniform elongation and post necking elongation increase with decreasing gauge length
and increasing specimen thickness [12]. With a decrease in thickness, the gauge part is
effectively transformed from a bulk to sheet geometry and the stress state within the
gauge changes from a more or less biaxial condition to uniaxial stress state condition
under tension, thereby resulting in a change from diffuse necking to localized necking.
A reduction in gauge length prolongs the stress-strain curves to higher fracture strains
or higher apparent ductility primarily by prolonging the uniform elongation. However,
in [13], the yield stress is known to be thickness independent for thickness larger than
the critical thickness, usually 6 to 10 times the average grain size for ferrous materials.
4. The variation in micro-sample measurements reflects local variations in the material that
are not measured with full sized specimens. This is due to lesser presence of voids and
defects in miniature samples when compared to the standard full size samples [14].
All the above listed effects are not limited to strength dependence on area, many times it
relates to various microstructural constrains [2, 8–10, 12–27].
2.3
Advances to miniature testing techniques
For tensile specimens of most components in service, the dimensions are larger than the
internal microstructural features, and thus the importance of external size-effects on mechanical properties can be a minor issue and hence can be neglected. However, with the advent
of MEMS, and other miniature testing requirement module samples with microstructural dimensions, external size effects have become a more prominent feature to look at. Thus many
small scale devices have been designed to get reliable results from these miniature samples [5].
Nevertheless, there is a need for a handbook for collecting accurate and reliable values of
elastic, plastic, fracture, and fatigue properties of materials at different gage lengths.
8
Optical techniques have emerged replacing the grip system to hold the specimen in position
as the grip displacement do not give correct strain measurements most of the time [5]. Results
from these miniature tests give an insight into size effects such as elastic, plastic brittle and
ductile behavior at miniature level. Also, the influence of voids and boundary cracks and
surface roughness may also have an effect on the mechanical properties that are being tested
for. With the need of requirement of understanding the mechanical behavior at small scale
level, the trend towards miniature testing will increasingly continue.
9
Chapter 3
Experimental Procedure
3.1
Component Element Properties
The as-received samples of stainless steel Type 304 and SA516 Grade 70 carbon steel
are initially tested for actual composition. The component element properties are as shown
in Table 3.1. One of the sample slabs of stainless steel Type 304 is furnace heat treated to
promote sensitization.
Table 3.1: Component element properties
A: SA516 Steel, 1/2” Plate
C
Mn
P
S
Si
Cr
Ni
Mo Cu
0.14-0.20 0.70-1.00 0.040 max 0.050 max
0.17
0.98
0.015
0.007
0.18
0.11
0.11
0.02 0.26
B: 304/304L. Steel, 1/2” Plate
0.03 max 2.00 max 0.045 max 0.03 max
1.00 max 18.0-20.0 8.0-20.0
0.02
1.62
0.029
0.02
0.40
17.8
8.5
0.02 0.26
C: 304/304L. Steel, 1/2” Plate Sensitized
Same as plate B but furnace heat treated (CRESS electric furnace, model C162012/SD)
at 650◦ C(1200◦ F) for 7 days (168 hours)
Microstructures of the as-received samples are captured as shown in Figure 3.1, 3.2 and
3.3. For the preparation of material sample, small portion (about an inch) of the samples are
cut with a band saw. These sample pieces are placed in a mold on to which resin and hardener
mixture is poured and is allowed to solidify. The casting on the sample piece aided in grinding
and polishing for the preparation of flat polished sample. After etching on the flat polished
samples, traces of its microstructure are revealed and the image is captured using an optical
microscope.
10
Fig. 3.1: Microstructure of SA516 steel etched with 4% Nital
Fig. 3.2: Microstructure of 304/304L steel etched with Vilella’s reagent
11
Fig. 3.3: Microstructure of sensitized 304/304L steel etched with Vilella’s reagent
3.2
Sensitization
Sensitization occurs when stainless steel is exposed to very high temperatures (425◦ C to
815◦ C). Sensitization effect causes precipitation of chromium carbides at grain boundaries as
chromium carbides are insoluble at high temperatures. For carbide to precipitate, it must
get chromium from the surrounding material causing chromium depleted zone around the
grain boundaries. This chromium depleted zone will be less corrosion resistant, specifically
to intergranular corrosion. Sensitization is important at welded joints. This is because the
welded zones experience very high temperatures causing sensitization [28].
The difference in the grain boundaries is evident in the microstructures. The Grain
boundaries in the sensitized microstructure are darker due to the precipitation of carbides.
The microstructures of stainless steel as-received vs sensitized stainless steel taken at 20x
magnification is shown in Figure 3.4 & 3.5.
12
Fig. 3.4: Microstructure of 304/304L steel
Fig. 3.5: Microstructure of sensitized 304/304L steel
13
3.3
Tensile Specimen Preparation
The miniature specimens in this study are sheet specimens unlike in most previous research
where only nano-scale or thin films have been studied. An important factor to be considered in
specimen design for tensile tests is the aspect ratio. The ASTM standard specimen size has an
aspect ratio (gauge length/diameter or in this case thickness) of 4:1. In order to get comparable
results to that of the standard specimens, the miniature specimens were designed to have the
same and multiples of the standard aspect ratio to study the effect of varied cross-sectional
area. The gauge length is considered constant in all specimens to have uniform elongation.
The first batch of specimens is tested on the SEMtester 1000 EBSD Figure 3.6 (designed to
be fit inside a scanning electron microscope (SEM)). The minimum sample size (l × w) is
(43 × 10) mm. Maximum displacement travel is 10mm with maximum load capacity of 1000lb
(4500N). Suitable specimen design is created in Solid Edge software after careful examination
of the grip area and the crosshead capacity of the miniature tensile tester. All specimens had
constant gauge length of 18.21 mm. The dimensions were (1 to 2) mm (0.5 to 1) mm. Each
material condition was tested for three specimen dimensions and 2 repeats in each combination.
For machining miniature specimens, traditional machining processes such as wire EDM, laser
machining, and chemical mechanical polishing, and FIB milling maybe used [5, 29, 30]. In this
study, wire EDM and water jet machining were used to cut out miniature specimens. Wire
EDM process is advantageous mainly because it produces a stress and burr-free cutting, has
efficient production capabilities and is cost effective with an excellent finishing.
The cutting mechanism in wire EDM is by bombarding the work piece with intense short
pulses of electricity and each pulse leaves a tiny pit on the work piece [30]. This causes
surface roughness on the specimen. Studies have proven that, the surface roughness that is
induced during specimen preparation causes reduction in fatigue strength and areas of stress
concentration. For example, fatigue strength of the SiC/Al metal matrix composite suffers
due to the presence of surface roughness which is an aftermath of wire EDM machining.
Therefore, the specimens were sanded before testing. Most of the specimens in the first
batch did not fracture upon reaching maximum strain travel resulting in incomplete stress
14
Fig. 3.6: SEMTester 1000 EBSD instrument for miniature testing
strain curves. Hence repeatability was very poor. Therefore, a second batch of specimens was
designed with reduced gauge length and cross sectional dimensions to see complete failure of
specimens and improve on the repeatability of the results. Keeping the aspect ratio comparable
to that of the standard, the gauge length and width were maintained constant at 8 mm and
1 mm respectively. The thickness varied from 0.1 mm to 0.2 mm Figure 3.7. The sanding of
the specimens causes slight decrease in thickness.
From each material condition and 2 specimen designs, 2 repeats were conducted. Another
batch of specimens was designed to be tested on Tinius Olsen machine Figure 3.9 in order to
have a comparison of mechanical properties from two different tensile testers. The gauge length
and width were maintained constant at 12 mm and 1 mm, respectively. The thickness varied
from 0.5, 1, and 2 to 3 mm Figure 3.8.
15
Fig. 3.7: Specimen cross section at gage length is (1x0.2) mm
Fig. 3.8: Variation in specimen gage thickness from 3 mm to 0.5 mm
16
(a)
(b)
Fig. 3.9: (a) Tinius Olsen H50KS (b) Specimen test in progress
From each material condition and 4 specimen designs, 4 repeats were conducted to ensure
repeatability. All the tests were conducted at a strain rate of
0.001s−1 .
The Tinius Olsen H50KS is a 50KN capacity model having maximum crosshead travel of
1100 mm. Load measurement accuracy is ±0.5% of indicated load from 2% to 100% capacity
and position measurement accuracy of ±0.01% of reading or 0.001 mm, whichever is greater.
17
Chapter 4
Experimental Results and Discussion
4.1
Conventional (Macro-sized specimen) Tensile Testing
The same materials under study are tested on a conventional scale and the following results
(from engineering stress-strain curves) are obtained from AZZ WRI facility. The specimens are
designed with respect to ASTM A370-10 standard with a displacement rate of 0.075 in/min
(1.9 mm/min) until 0.4% offset and post yield loading rate at 0.7 in/min (17.78 mm/min)
until failure. Two tests per material is experimented on a Tinius Olsen tester.
Table 4.1: Conventional (Macro-sized) results
Specimen
Material
Stainless Steel
(SS)
grade 304/304L
Sensitized (SS)
grade 304/304L
Carbon Steel
4.2
Average
Yield Stress (YS)
(MPa)
Average
Tensile Stress (UTS)
(MPa)
Young’s
Modulus
(GPa)
Elongation
(%)
276.48
610.19
187.88
61
269.59
617.08
165.82
57
344.39
529.17
204.08
35
Miniature Tensile Testing
The miniature specimens designed for micro tensile tester are tested, and the Yield
Strength (YS) and Ultimate Tensile Strength (UTS) obtained are as shown in Table 4.2.
All specimens are flat and have a gauge length of 8 mm. The SEMTester 1000 EBSD is usually used for testing textiles, foods, biomaterials to paper and polymers. Hence, the results
obtained for such high strength metals are not that accurate as that obtained from Tinius
Olsen. Prestress values are taken care of during data analysis. All specimens are tested at a
strain rate of 0.001s−1 . The results obtained from Tinius Olsen are as shown in Tables 4.3
18
Table 4.2: Results obtained from SEMTester 1000 EBSD
Cross-sectional
Yield Strength (YS) MPa
Gauge Area(mm2 ) (Average value)
Stainless Steel (SS) grade 304/304L
1x0.1
129.00
1x0.2
92.19
Sensitized (SS) grade 304/304L
1x0.1
575.00
1x0.2
172.02
Carbon Steel
1x0.1
47.57
1x0.2
48.34
Ultimate Tensile Strength (UTS) MPa
(Average value)
781.84
1174.57
794.21
806.55
377.32
445.50
through 4.5. All specimens are flat and have a constant gauge length of 12 mm. Prestress
values are subtracted (zeroed) during data analysis. All specimens are tested at a strain rate
of 0.001s−1 .
Application of Chauvenet’s Criterion:
The experiment is affected in numerous ways such as human error while mounting the
specimen and taking the readings, vibrations from surrounding instruments etc. All of these
qualify as random errors. Certain errors like load cell defect and LVDT offsets qualify as
systematic errors as the same error value is affecting every test. Also, in sample to sample
experiments, the variability inherent in the samples themselves causes variations in measured
values in addition to the random errors in the measurement system. A combination of these
errors influence the results obtained from the instrument.
Chauvenet’s criterion is applied to exclude the values which lie outside a certain range as
dictated by the ratio of maximum acceptable deviation to standard deviation
∆Xmax
Sx
(4.1)
From the table in Figure 4.1, Chauvenet’s criterion for rejecting a reading [1], for a set of
four specimens in each cross-sectional gauge area, the absolute value of the ratio of Maximum
Acceptable Deviation to Standard Deviation is 1.54. Hence, any ratio greater than 1.54 is
considered as an outlier. The Chauvenet’s criterion can be applied on any stress value because
19
if one value is considered an outlier, the entire test on that specimen is discarded on the whole.
Fig. 4.1: Chauvenet’s criterion for rejecting a reading obtained from the book Experimentation,
Validation and Uncertainty Analysis for Engineers by Coleman and Steele [1]
From the calculations in Tables 4.3 through 4.5, none of the tests had to be discarded.
Thus, the experiment procedure was correct and all the results are valid.
4.3
Optimum Specimen Size
The optimum miniature specimen size is the one whose results correspond to that of
conventional tensile test results. From the comparison, the miniature stainless steel (SS)
and Sensitized SS grade 304 specimens tested on Tinius Olsen give comparable yield and
engineering tensile strength results. However, one can observe that there is a slight increase
in yield strength (YS) values for miniature SS and Sensitized SS specimens when compared to
its equivalent conventional sized specimens.
20
Table 4.3: Mechanical properties for Stainless Steel Type 304 miniature specimens
Sl. No.
Yield Strength
(MPa)
Specimen
1
2
3
4
X̄ 1
Sx 2
Interval
Specimen
1
2
3
4
X̄
Sx
Interval
Specimen
1
2
3
4
X̄
Sx
Interval
Specimen
1
2
3
4
X̄
Sx
Interval
Cross-Section
275.45
306.70
303.95
283.82
292.48
15.26
292.48±15.26
Cross-Section
296.41
301.92
306.97
285.99
297.82
8.99
297.82±8.99
Cross-Section
191.72
229.31
212.73
217.82
212.90
15.73
212.90±15.73
Cross-Section
157.69
147.38
160.38
173.45
159.73
10.73
159.73±10.73
1
2
Average Value
Standard Deviation
Xi − X̄ /Sx (1x3)mm2
1.12
0.93
0.75
0.57
(1x2)mm2
0.16
0.46
1.02
1.32
(1x1)mm2
1.35
1.04
0.01
0.31
(1x0.5)mm2
0.19
1.15
0.06
1.28
Ultimate Tensile
Strength (MPa)
Xi − X̄ /Sx Elongation (%)
691.00
640.85
640.42
611.00
645.81
33.20
645.81±33.20
1.36
0.15
0.16
1.05
58
66
63
66
63.25
3.77
63.25±3.77
690.07
620.00
639.10
592.10
635.32
41.29
635.32±41.29
1.33
0.37
0.09
1.05
50
58
42
46
49
6.83
49±6.83
508.00
579.06
524.00
588.04
549.76
39.71
549.76±39.71
1.05
0.74
0.65
0.96
50
46
50
42
47
3.83
47±3.83
543.00
435.00
500.00
511.00
497.25
45.33
497.25±45.33
1.01
1.37
0.06
0.30
25
25
46
25
30.25
10.50
30.25±10.50
21
Table 4.4: Mechanical properties for Sensitized Stainless Steel Type 304 miniature specimens
Sl. No.
Yield Strength
(MPa)
Specimen
1
2
3
4
X̄
Sx
Interval
Specimen
1
2
3
4
X̄
Sx
Interval
Specimen
1
2
3
4
X̄
Sx
Interval
Specimen
1
2
3
4
X̄
Sx
Interval
Cross-Section
301.38
262.31
267.01
305.85
284.14
22.65
284.14±22.65
Cross-Section
263.26
236.75
174.29
266.47
235.19
42.73
235.19±42.73
Cross-Section
149.71
197.64
105.84
156.14
152.33
37.58
152.33±37.58
Cross-Section
79.74
86.84
140.15
103.81
102.64
26.97
102.64±26.97
Xi − X̄ /Sx (1x3)mm2
0.76
0.96
0.76
0.96
(1x2)mm2
0.66
0.37
1.43
0.73
(1x1)mm2
0.07
1.21
1.24
0.10
(1x0.5)mm2
0.85
0.59
1.39
0.04
Ultimate Tensile
Strength (MPa)
Xi − X̄ /Sx Elongation (%)
740.00
683.42
548.39
612.00
645.95
83.51
645.95±83.51
1.13
0.45
1.17
0.41
66
58
66
42
58
11.31
58±11.31
546.00
526.00
443.60
595.00
527.65
63.09
527.65±63.09
0.29
0.03
1.33
1.07
50
42
58
58
52
7.66
52±7.66
472.00
485.95
407.00
455.00
454.99
34.40
454.99±34.40
0.49
0.90
1.40
0.00
58
58
42
46
51
8.25
51±8.25
299.00
341.00
398.00
373.00
352.75
42.76
352.75±42.76
1.26
0.27
1.06
0.47
38
50
33
42
40.75
7.18
30.25±10.50
22
Table 4.5: Mechanical properties for SA516 grade 70 carbon steel miniature specimens
Sl. No.
Yield Strength
(MPa)
Specimen
1
2
3
4
X̄
Sx
Interval
Specimen
1
2
3
4
X̄
Sx
Interval
Specimen
1
2
3
4
X̄
Sx
Interval
Specimen
1
2
3
4
X̄
Sx
Interval
Cross-Section
301.99
291.92
282.37
287.34
290.91
8.36
290.91±8.36
Cross-Section
318.37
233.47
291.50
219.83
265.79
46.84
265.79±46.84
Cross-Section
180.81
208.90
106.05
161.29
164.26
43.45
164.26±43.45
Cross-Section
70.76
153.42
145.70
175.63
136.38
45.55
136.38±45.55
Xi − X̄ /Sx (1x3)mm2
1.33
0.12
1.02
0.43
(1x2)mm2
1.12
0.69
0.55
0.98
(1x1)mm2
0.38
1.03
1.34
0.07
(1x0.5)mm2
1.44
0.37
0.20
0.86
Ultimate Tensile
Strength (MPa)
Xi − X̄ /Sx Elongation(%)
465.44
440.00
432.00
433.00
442.61
15.63
442.61±15.63
1.46
0.17
0.68
0.61
25
33
25
25
27
4
27±4
461.00
362.00
465.00
494.00
445.50
57.58
445.50±57.58
0.27
1.45
0.34
0.84
25
33
21
25
26
5.03
26±5.03
328.00
350.00
296.01
307.00
320.25
23.86
320.25±23.86
0.32
1.25
1.02
0.56
42
21
17
25
26.25
10.99
26.25±10.99
203.00
240.00
250.00
352.00
261.25
63.79
261.25±63.79
0.91
0.33
0.18
1.42
17
25
25
13
20
6
20±6
23
This can be explained due to lesser presence of anomalies in the microstructure when
compared to the macro-sized specimens. Also, lesser number of voids is present in a miniature
specimen. Further, sensitized stainless steel is heat treated for several hours (168) leading to
softening of the metal. This is explained due to the diffusion of Cr, C and N towards the
grain boundaries from the grain body, thus indicating a fall in hardness of stainless steel.
This is usually seen after 1 hour of sensitization. When the duration is longer than 1 hour,
the hardness marginally increases due to the precipitation at the grain boundaries and also
formation of martensitic microstructure [31].
The carbon steel specimens show more ductility owing to lesser yield strength (YS) values
at miniature level. Such a ductile nature is seen only in carbon steel specimens unlike the
stainless steel (SS) specimens hence carbon specimens were more sensitive to miniaturization.
Also, the tensile strength values are lower than the conventional results. A plausible reasoning
is due to the brittle nature of carbon steel and at miniature level the volume is much lesser
than the conventional owing to lesser strength and higher ductility. Very thin cross sections
that were tested on microtensile tester gave considerably high results for SS specimens and
slightly comparable results for carbon steel specimens as shown in Figure 4.2 and 4.3.
Considering all the above reasoning, the recommended miniature size for all three materials is a gauge area of (1 × 3) with an aspect ratio 4:1 which is the same as that of the
standard ASTM testing for a constant gauge length of 12 mm. The overall specimen size was
approximately 50 mm (2 in.). The engineering stress versus strain curve for four specimens of
gauge dimension (1 × 3) from each material is as shown in Figure 4.4.
24
800
700
Tensile Strength (MPa)
600
500
400
300
200
100
0
0
0.5
1
1.5
2
2.5
3
3.5
3
3.5
3
3.5
Cross-Sectional Area (mm2)
(a) Stainless Steel grade 304
800
700
Tensile Strength (MPa)
600
500
400
300
200
100
0
0
0.5
1
1.5
2
2.5
Cross-Sectional Area (mm2)
(b) Sensitized Stainless Steel grade 304
600
Tensile Strength (MPa)
500
400
300
200
100
0
0
0.5
1
1.5
2
2.5
Cross-Sectional Area, (mm2)
(c) Carbon Steel(SA5 16) grade 70
Fig. 4.2: Tensile strength vs specimen gauge cross section
25
350
Yield Strength (MPa)
300
250
200
150
100
50
0
0
0.5
1
1.5
2
2.5
3
3.5
Cross-Sectional Area (mm2)
(a) Stainless Steel grade 304
350
Yield Strength (MPa)
300
250
200
150
100
50
0
0
0.5
1
1.5
2
2.5
Cross-Sectional Area (mm2)
3
3.5
(b) Sensitized Stainless Steel grade 304
350
Yield Strength (MPa)
300
250
200
150
100
50
0
0
0.5
1
1.5
2
2.5
3
3.5
Cross Sectional Area (mm2)
(c) Carbon Steel(SA5 16) grade 70
Fig. 4.3: Yield strength vs specimen gauge cross section
26
800
700
Stress (MPa)
600
500
400
300
200
Specimen
Specimen
Specimen
Specimen
100
0
0
0.1
0.2
0.3
0.4
0.5
Strain (mm/mm)
0.6
0.7
1
2
3
4
0.8
0.9
(a) Stainless Steel grade 304
800
700
Stress (MPa)
600
500
400
300
200
Specimen
Specimen
Specimen
Specimen
100
0
0
0.1
0.2
0.3
0.4
0.5
Strain (mm/mm)
0.6
0.7
0.8
1
2
3
4
0.9
(b) Sensitized Stainless Steel grade 304
500
450
400
Stress (MPa)
350
300
250
200
150
Specimen
Specimen
Specimen
Specimen
100
50
0
0
0.05
0.1
0.15
0.2
Strain (mm/mm)
0.25
0.3
1
2
3
4
0.35
(c) Carbon Steel(SA5 16) grade 70
Fig. 4.4: Engineering stress vs strain curves for specimen gauge cross section (1x3)
27
When gauge length is maintained constant, with decrease in thickness, the gauge part is
effectively transformed from a bulk to sheet geometry and the stress state within the gauge
changes from a more or less biaxial to a uniaxial stress state condition, thereby resulting in
a change from diffuse necking to localized necking. In a tension test on a ductile material, a
diffuse necking - so called because its spatial extension is much larger than the sheet thickness
- begins to develop in the sample when the strain hardening is no longer able to compensate
for the weakening due to the reduction of the cross-section. After some elongation under
decreasing load, a localized neck usually appears in the region of the diffuse neck. In the
localized neck, severe thinning occurs leading to ultimate failure. The transition from diffuse
to localized neck happens much faster in a sheet specimen (in this case miniature specimen)
when compared to a bulk specimen (conventional specimen). This causes reduction in yield
strength since elastic zone is shortened and hence, material exhibits more ductility as a sheet
miniature specimen than a conventional specimen.
Miniature specimens have smaller volumes compared to conventional specimens. Therefore, it might be because of smaller volume, the interatomic forces give way too soon under
tension (as they have small number of grains) causing decreased strain hardening effect.Yield
stress is more sensitive to increase in strain rate than the tensile strength. High rates of strain
cause a yield point to appear in specimens of low carbon steel which otherwise do not appear
when tested under ordinary rates of loading [32]. All specimens were tested with a strain rate
of 0.001s−1 . However, miniature carbon steel specimens show yield point as seen in Figure
4.4(c).
The Modulus of elasticity in all three material specimens was found to be as low as about
6895 MPa (1000 ksi) in all the three specimens. There are various reasoning for this behavior of
miniature specimens. The literature shows that with the increase of gauge length, the Youngs
Modulus values also increase with different strain rates [2]. A strong dependence of elastic
modulus on gauge length suggests that some shear might occur causing a deviation from the
linear stress-strain relationship [2].
It is reported in many studies that the smaller the gauge length, the lower the slope of the
28
linear stress/strain relationship is detected for samples with shorter gauge length (with respect
to the volume of the conventional specimens). However, shorter linear portion of stress strain
graph does not influence the overall elongation of the material until maturation. Also, steel
crystal is highly anisotropic; grain orientation anisotropy can be a valuable explanation for
low Modulus values [33]. Youngs Modulus is the materials resistance to elastic deformation.
The greater the modulus, the stiffer the material or smaller the elastic strain that results from
application of a given stress. On atomic scale, macroscopic elastic strain is manifested as small
changes in the interatomic spacing and the stretching of interatomic bonds.
As a consequence, the magnitude of the modulus is a measure of resistance to separation
of adjacent atoms which is interatomic bonding forces [34].
29
Chapter 5
Uncertainty analysis
5.1
Tolerance intervals in Sample Populations
In addition to the confidence interval for a sample population, there is a useful statistical
interval called the tolerance interval. The tolerance interval gives information about the parent
population that the specimens came from [1]. The Gaussian interval that contains 95% of the
parent population is simply the X̄ ± 1.96σ, i.e the average value of the specimens being tested
± 1.96 times the random standard deviation. However, when we are dealing with specimens
and the statistical quantities X̄ and Sx , the determination of the range that contains a portion
of the parent population is not as straightforward. What one can estimate is the range that
has a certain probability of containing a specified percentage of the parent population.
Considering a set of four specimens of (1 × 3) cross-sectional area, and to find an estimate
of 99% of the parent population with 95% confidence is
X ± CT 95(99) (4) Sx
(5.1)
where CT is a factor which is given as a function of the number of readings in Table A.3 as
in Figure 5.1 [1], for four specimens CT 95(99) factor is 8.299. (Here the number of specimens
is four but note that if the number of specimens tends to infinity, the concept of a confidence
level for the tolerance interval does not apply because the tolerance interval approaches the
Gaussian interval.) For YS readings of SS grade 304 specimens with gauge length 12 mm,
using the mean X 292.48 MPa and Sx 15.26 MPa from Table 4.3 in Eq.5.1
292.48 ± [8.299] [15.26] M P a
(5.2)
30
Fig. 5.1: Factors for two-sided tolerance interval [1]
prob (165.80 ≤ µ ≤ 419.17) M P a
(5.3)
For UTS readings, using the mean X 645.82 MPa and Sx 33.21 MPa in Eq.5.1
645.82 ± [8.299] [33.21] M P a
(5.4)
prob (370.25 ≤ µ ≤ 921.39) M P a
(5.5)
For YS readings of Sensitized SS grade 304 specimens with gauge length 12 mm, using
the mean X 284.14 MPa and Sx 22.65 MPa from Table 4.4 in Eq.5.1
31
284.14 ± [8.299] [22.65] M P a
(5.6)
prob (96.17 ≤ µ ≤ 472.10) M P a
(5.7)
For UTS readings, using the mean X 645.95 MPa and Sx 83.50 MPa in Eq.5.1
645.95 ± [8.299] [83.50] M P a
(5.8)
prob (−47.04 ≤ µ ≤ 1338.93) M P a
(5.9)
For YS readings of carbon steel (SA5 16) grade 70 specimens with gauge length 12 mm,
using the mean X 290.91 MPa and Sx 8.36 MPa from Table 4.5 in Eq.5.1
290.91 ± [8.299] [8.36] M P a
(5.10)
prob (221.56 ≤ µ ≤ 360.26) M P a
(5.11)
For UTS readings, using the mean X 442.61 MPa and Sx 15.63 MPa in Eq.5.1
442.61 ± [8.299] [15.63] M P a
(5.12)
prob (312.89 ≤ µ ≤ 572.33) M P a
(5.13)
Eq.5.3,5.7 and 5.11 gives the tolerance interval for 99% of the parent population with
95% confidence for YS. From Table 4.1, the conventional YS of SS specimens are 276.48 MPa,
Sensitized SS specimens are 269.59 MPa and carbon steel specimens are 344.39 MPa. Eq.5.5,5.9
and 5.13 gives the tolerance interval for UTS. The conventional UTS of SS specimens are 610.19
MPa, Sensitized SS specimens are 617.08 MPa and carbon steel specimens are 529.17 MPa.
Hence our miniature specimen tensile testing experiments give approximate values as that of
the conventional testing as per ASTM A370-10 standard.
32
5.2
Propagation of Mechanical Properties by Monte Carlo Method (MCM)
Monte Carlo simulation provides a distribution of errors for a result that is a function
of multiple variables. In most cases, the result of our experiment or simulation will depend
on several variables through a data reduction equation (DRE) or a simulation solution. This
method is not limited to simple expressions but can also be used for highly complicated
equations or for numerical solutions of advanced simulation equations.
In this section, a general approach for the MCM for uncertainty propagation is discussed
and this technique is applied to the data reduction equations of the current study. The
mechanical properties of the three materials are propagated to parent population. It is seen
that our experimental results, obtained by a set of four specimens in each gauge cross section,
fall in the same interval as that of the propagated values of the parent population. The
section concludes with a discussion for the convergence of the combined standard uncertainty
(or relative uncertainty with respect to each iteration represnting a specimen) in this MCM
analysis.
5.2.1
General Approach for MCM
Figure 5.2 presents a flowchart that shows the steps involved in performing an uncertainty
analysis by the MCM. The figure shows the sampling techniques of two variables, but the
methodology is general for DREs or simulations that are functions of multiple variables. First,
the assumed true value for each variable (load, area, stress etc.) in the result is the input.
These would be the Xbest values that we have for each variable. Then, the estimates of the
random standard uncertainty, s and the elemental systematic standard uncertainties bk for
each variable are input. An appropriate probability distribution function is assumed for each
error source. The random errors are usually assumed to come from a Gaussian distribution.
The systematic errors are chosen based on the analyst’s judgment as it is upto the analyst to
use the best information.
33
Fig. 5.2: Schematic for MCM for uncertainty propagation when random standard uncertainties
for individual variables are used [1]
For the flowchart in Figure 5.2, the random standard uncertainties for X and Y are
assumed to come from Gaussian distributions and that each variable has three elemental
systematic standard uncertainties, one Gaussian, one triangular, and one rectangular.
For each variable, random values for the random errors and each elemental systematic error
are found using an appropriate random number generator (Gaussian, triangular, rectngular,
etc.). The individual error values are then summed and added to the true values of the
variables to obtain ”measured” values. Using these measured values, the result is calculated.
This process corresponds to running the simulation once.
34
5.2.2
Propagation of Mechanical Properties
The Data Reduction Equations (DRE) for the study are the equations for YS, UTS and
elongation (el) given by,
σy (M P a) =
load
area
(5.14)
σ (M P a) =
load
area
(5.15)
el =
lf − l0
l0
(5.16)
where σy is YS, σ is UTS, lf is final gauge length after fracture and l0 is the original gauge
length. The unit of load is in KN and that of area is mm2 . All lengths are in mm. However,
elongation or strain is dimensionless.
Before testing, the thickness and width of the specimens are carefully measured at three
different locations in the gauge length. The average of the readings is considered as the
thickness and the width values which are subsequently used for calculation of area. Random
standard uncertainty for (95%) confidence interval (Eq.5.17) for area is calculated using Table
5.1.
U ncertainty (95%) = 1.96 × Sx
(5.17)
35
The uncertainty analysis is conducted only on optimum specimen size as it yields almost
the same mechanical properties as that of a conventional specimen.
Table 5.1: Specimen area
Specimen No.
Area (mm2 )
SS grade 304 - (1x3) specimens
1
2.90
2
2.89
3
2.88
4
2.88
Average
2.89
Standard Deviation 0.00957
Sensitized SS grade 304 - (1x3) specimens
1
4.26
2
2.84
3
2.74
4
2.83
Average
3.17
Standard Deviation 0.07297
Carbon Steel (SA5 16) grade 70 - (1x3) specimens
1
2.97
2
2.95
3
2.88
4
2.90
Average
2.93
Standard Deviation 0.04203
For SS 304 = 1.96 × 0.00957
∴Uncertainty for area = 0.0188mm2 or 1.8%
For Sensitized SS 304 = 1.96 × 0.07297.
∴Uncertainty for area = 0.14303mm2 or 14.30%
For Carbon Steel 70 = 1.96 × 0.04203
∴Uncertainty for area = 0.0824mm2 or 8.24%
Below is the table for force applied by the load cell in each test for yielding to take place.
Yielding point is crucial in every material testing as the specimen enters the plastic zone.
Tensile strength does not depend on when the yielding takes place for a given material. The
36
load cell has a value of +/ − 0.5% of the applied load value. This specification is in the Tinius
Olsen manual.
Table 5.2: Load values for (YS) and (UTS) in specimens of gauge area (1x3) mm2
Specimen No. (YS) Load (KN) (UTS) Load (KN)
SS grade 304 - (1x3) specimens
1
0.80
2.01
2
0.89
1.85
3
0.87
1.84
4
0.82
1.86
Average
0.85
1.89
Sensitized SS grade 304 - (1x3) specimens
1
1.28
3.15
2
0.74
1.94
3
0.73
1.50
4
0.86
1.73
Average
0.90
2.08
Carbon Steel (SA5 16) grade 70 - (1x3) specimens
1
0.90
1.38
2
0.86
1.30
3
0.81
1.24
4
0.83
1.25
Average
0.85
1.29
Stainless steel grade 304/304L
Table 5.3: Variables and their uncertainties - SS grade 304
Variables
Area (A)mm2
YS Load (F )KN
UTS Load (Fu )KN
Nominal Value
2.89
0.85
1.89
Uncertainty (95%)
1.8%
0.5%
0.5%
Calculations for finding the uncertainty of YS using Eq.5.14
Uσy
2
∂Uσy 2
∂Uσy 2
2
=
(UF ) +
(UA )2
∂F
∂A
2
−F 2
1
2
=
(UF ) +
(UA )2
A
A2
(5.18)
37
The MCM nominal value (true value) for load and area (variables) are obtained from running
the MCM algorithm for 10000 (M times) iterations in MATLAB. However, since the nominal
value for area is very less, no significant changes were seen after running 10000 iterations,
although the load values changed every single time. 10000 iterations is not just a random
number chosen. It is later seen in this section that this number generates a converged value
for relative uncertainty of the results.
If we assume that all distributions are Gaussian, then the standard uncertainty inputs for
the MCM will be previous percent uncertainties divided by 2. Plugging the nominal values
obtained for the variables from the program, Eq.5.18 becomes
Uσy
2
=
1
2.89
2 0.8452 × 103 × 0.005
1.96
2
+
−0.85 × 103
2.892
2 2.9 × 1.8 × 10−2
1.96
2
= 0.5565 ± 7.3464
= 7.90
(5.19)
In the above equation, (UF ) is 0.8452 × 103 (N ) × 0.005/1.96 where 0.8452 is the MCM
nominal value for load to cause yielding and multiplied by one standard uncdertainty, i.e the
standard uncertainty inputs for the MCM will be previous percent uncertainties divided by
2 or 1.96 for considering only the random uncertainties at 95% confidence. Similarly, (UA ) is
(2.9 × 1.8 × 10−2 )/1.96 where 2.9 mm2 is the MCM nominal value for area.
∴ Uσy = 2.81 MPa
Therefore, the uncertainty in the YS is the average value obtained from our experiment,
292.48 ± 2.81 MPa.
Calculations for finding uncertainty in UTS using Eq.5.15
∂Uσ 2
∂Uσ 2
(UFu )2 +
(UA )2
∂Fu
∂A
2
−Fu 2
1
2
=
(UFu ) +
(UA )2
A
A2
(Uσ )2 =
(5.20)
38
The nominal value of the load changes as the load applied at breaking is different from the
load applied at yielding.
2
(Uσ ) =
1
2.89
2 1.8942 × 103 × 0.005
1.96
2
+
−1.89 × 103
2.892
2 2.9 × 1.8 × 10−2
1.96
2
(5.21)
= 2.7949 ± 36.3211
= 39.12
∴ (Uσ ) = 6.25 MPa
Therefore, the uncertainty in the UTS is the average value obatined from our experiment,
645.82 ± 6.25 MPa.
For finding the uncertainty in elongation, the final length of each specimen needs to be
calculated after testing. The final specimen length was measured from the neck of the gauge
length of the fractured specimen.
Table 5.4: Final length of specimens of gauge cross section (1x3) after tensile testing - SS
grade 304
Specimen No.
1
2
3
4
Average
Standard
Deviation
Final Length lf mm
19.00
20.00
19.50
20.00
19.63
0.4787
From Eq.5.17, uncertainty for lf is 0.9383 mm. Therefore, the final length is 19.63 ±
0.9383 mm. Using Eq.5.16, the mean of the final length as 19.63 and the original gauge length
as 12 mm, elongation E is 0.636 or 64%. Calculations for finding uncertainty in E using
Eq.5.16 are
2
∂Uel 2
∂Uel 2
Ulf +
(Ul0 )2
∂lf
∂l0
2
2
−lf 2
1
=
Ulf +
(Ul0 )2
2
l0
l0
(Uel )2 =
(5.22)
39
The original length for all specimens is 12 mm nevertheless the least count of the measuring
caliper is taken as the uncertainty. In this case the uncertainty value is 0.001. Upon substitution of nominal values for final and original length randomly generated by MATLAB for MCM
analysis,
(Uel )2 =
1
12
2 20.468 × 0.9383
1.96
2
+
−19.63
122
2 11.99 × 0.001
1.96
2
(5.23)
= 0.6667
∴ (Uel ) = 0.8165
Therefore, the total uncertainty in the elongation is within the range of 0.636 ± 0.8165 or
64% ± 81.6% of the final length. This elongation can be really high if the material is ductile.
A ductile material has low yielding point and hence undergoes yielding for a long period before
failure. Another reason for high elongation is inclusions in the alloy material. Note that when
10000 iterations represents 10000 experimental trials that have random outcomes [35]. When
applied to uncertainty estimation, random numbers are used to ramdomly sample parameters’
uncertainty space instead of point calculation carried out by a small test of experiments.
Hence it is a method where propagation of uncertainty is achieved. Therefore, the possibility
of elongation being greater than the value itself is almost insignificant. The range is a number
interval within which elongation may lie without reaching to the extremes.
Sensitized SS grade 304/304L
Table 5.5: Variables and their uncertainties - Sensitized SS grade 304
Variables
Area (A)mm2
YS Load (F )KN
UTS Load (Fu )KN
Nominal Value
3.17
0.90
2.08
Uncertainty (95%)
14.30%
0.5%
0.5%
40
Uncertainty calculation in YS is obtained by using Eq.5.14 and 5.18. The nominal value
for load and area are obtained from running the MCM algorithm for 10000 iterations in
MATLAB.
Uσy
2
=
1
3.17
2 0.9105 × 103 × 0.005
1.96
2
+
−0.90 × 103
3.172
2 2
3.16 × 14.30 × 10−2
1.96
(5.24)
∴ Uσy = 21.38 MPa
Therefore, the uncertainty in the YS is the average value obtained in our experiment,
284.14 ± 21.38 MPa.
Calculations for finding uncertainty in UTS using Eq.5.15 and 5.20
2
(Uσ ) =
1
3.17
2 2.0756 × 103 × 0.005
1.96
2
+
−2.08 × 103
3.172
2 3.16 × 14.30 × 10−2
1.96
2
(5.25)
∴ Uσ = 47.77 MPa
Therefore, the uncertainty in the UTS is the average value obtained in our experiment,
645.95 ± 47.77 MPa.
Uncertainty in elongation is calculated based on the following table.
Table 5.6: Final length of specimens of gauge cross section (1x3) after tensile testing - Sensitized SS grade 304
Specimen No.
1
2
3
4
Mean
Standard
Deviation
Final Length lf mm
20.00
19.00
20.00
17.00
19.00
1.414
From Eq.5.17, uncertainty for lf is 2.7714 mm. Therefore, the final length is 19.00 ±
2.7714 mm. Using Eq.5.16, the mean of the final length as 19.00 and the original gauge length
41
as 12 mm, elongation E is 0.583 or 58%. Calculations for finding uncertainty in E using
Eq.5.16 and Eq.5.22
2
(Uel ) =
1
12
2 18.11 × 2.7714
1.96
2
+
−19
122
2 11.99 × 0.001
1.96
2
(5.26)
∴ Uel = 2.1339
Therefore, the uncertainty in the elongation is 0.5383 ± 2.1339.
Carbon steel SA516 grade 70
Table 5.7: Variables and their uncertainties - carbon steel SA516 grade 70
Variables
Area (A)mm2
YS Load (F )KN
UTS Load (Fu )KN
Nominal Value
2.93
0.85
1.29
Uncertainty (95%)
8.24%
0.5%
0.5%
Uncertainty calculation in YS is obtained by using Eq.5.14 and 5.18. The nominal value
for load and area are obtained from running the MCM algorithm for 10000 iterations in
MATLAB.
Uσy
2
=
1
2.93
2 0.8459 × 103 × 0.005
1.96
2
+
−0.85 × 103
2.932
2 2.90 × 8.24 × 10−2
1.96
2
(5.27)
∴ Uσy = 12.09 MPa
Therefore, the uncertainty in the YS is the average value obtained in our experiment,
290.91 ± 12.09 MPa.
Calculations for finding uncertainty in UTS using Eq.5.15 and 5.20
2
(Uσ ) =
1
2.93
2 1.2923 × 103 × 0.005
1.96
2
+
−1.29 × 103
2.932
2 2.90 × 8.24 × 10−2
1.96
2
(5.28)
∴ Uσ = 18.56 MPa
42
Therefore, the uncertainty in the UTS is the average value obtained in our experiment,
442.61 ± 18.56 MPa. Uncertainty in elongation is calculated based on the following table.
43
Table 5.8: Final length of specimens of gauge cross section (1x3) after tensile testing - carbon
steel SA516 grade 70
Specimen No.
1
2
3
4
Mean
Standard
Deviation
Final Length lf mm
15.00
16.00
15.00
15.00
15.25
0.5
From Eq.5.17, uncertainty for lf is 0.98 mm. Therefore, the final length is 15.25 ± 0.98
mm. Using Eq.5.16, the mean of the final length as 15.25 and the original gauge length as 12
mm, elongation E is 0.271 or 27%. Calculations for finding uncertainty in E using Eq.5.16
and 5.22
2
(Uel ) =
1
12
2 15.17 × 0.98
1.96
2
+
−15.25
122
2 11.99 × 0.001
1.96
2
(5.29)
∴ Uel = 0.6321
Therefore, the uncertainty in the elongation is 0.271 ± 0.6321.
5.3
Convergence study
The sampling process is repeated M times to obtain a distribution for the possible result
values. The primary goal of the MCM propagation technique is to estimate a converged value
for the standard deviation SM CM , of this distribution. 2s of this distribution is the resultant
uncertainty at 95% confidence (assuming the distribution is Gaussian). An appropriate value
for M is determined by periodically calculating SM CM during the MCM process and stopping
the process when a converged value of SM CM is obtained. The SM CM is the combined standard
uncertainty of the result (uncertainty in each reading/iteration). The number of iterations, in
this case is 10000 (M ).
We do not need to have a perfectly converged value of SM CM to have reasonable estimate
of uncertainty. Once the SM CM values are converged to within 1-5%, then the value of SM CM
44
is a good approximation of the combined standard uncertainty of the result.
Figure 5.3 gives the distribution for uncertainties calulated in Eq.5.27 for YS in carbon
steel specimens. The resultant uncertainty is calulated using Eq.5.30
2UF
F
(5.30)
M CM
where, UF is the uncertainty calculated using the true value generated for a variable (load)
in the Eq.5.27 and F is the true value generated by MCM process. This value of expanded
uncertainty is stored with every iteration. In Figure 5.3, load and area uncertainties are
included. The distribution is Gaussian, hence our assumption is valid.
350
300
No. of Specimens
250
200
150
100
50
0
2.5
2.6
2.7
2.8
2.9
3
3.1
Uncertainty, %
3.2
3.3
3.4
3.5
Fig. 5.3: Distribution of MCM results for yield strength of carbon steel SA516 grade 70.
Expanded uncertainties for each variable being calculated at 95% confidence
The averge uncertainty is 3.0344% and the standard deviation is 0.1200% for the entire
range of specimens.
The convergence plot is as shown in Figure 5.4. The plot describes the convergence of
relative uncertainty of the specimens. The value of standard deviation, s of the resultant
45
uncertainty as in Eq.5.30 was calculated after every iteration, starting from the first iteration.
The plot of these s values which represent combined standard uncertainty, SM CM is plotted
in Figure 5.4. The value of relative uncertainty or the combined standard uncertainty was a
fully converged value from a large number of iterations as seen.
0.08
0.07
Relative uncertainty, %
0.06
0.05
0.04
0.03
0.02
0.01
0
0
1000
2000 3000 4000 5000 6000 7000 8000
Monte Carlo Iterations, M (No. of specimens)
9000 10000
Fig. 5.4: Convergence study for MCM value of UF for 95% combined uncertainty for each
variable
We see that after 1000 iterations, the value has converged to within 3% and by about 2000
iterations to within less than 1% of the fully converged value. About 6.3% is the combined
standard uncertainty for 10000 specimens. Since these are characteristic plots, similar trend
is seen with mechanical properties of stainless steel and sensitized steel as well. Any further
addition to 10000 would still give us a converged value and hence our selection of 10000
iterations is reasonable.
46
Chapter 6
Conclusion and Future Work
6.1
Conclusions
This study clearly shows that the tensile testing results of miniature specimens are de-
pendent not only on the property of the material itself and testing conditions (strain rate,
tensile machine employed, etc.), but also on specimen size and geometry. Miniature specimens
can be of any cross-section but to choose the best cross-section for a particular application is
challenging. Four different specimen gauge cross-sections having the same or multiple of the
ASTM standard aspect ratio is designed with a constant gauge length of 12mm. With the
current facilities, the specimens were tested upon two different machines, Tinuis Olsen and
SEMTester 1000 EBSD.
From the results obtained, larger consistency in results was obtained from Tinius Olsen
machine. The specimens with gauge cross- section (1 × 3) shows repeatability in results which
is considerably agreeable to that of the ASME- SA 240 standard and the macro sized testing
performed by AZZ WSI, hence considered an optimum specimen cross-section. The variation
in results is present in all specimen lots but more pronounced in the smaller cross-section
specimens. Variations are attributed to a number of factors such as stress concentration
effects while machining the specimen, especially near the fillet area and voids present within.
To minimize any undue stress, the specimen must be aligned with the center line of the two
test machine grips [11]. Therefore, Chauvenets criterion is applied to remove the results that
are most effected from the above mentioned factors so that more reliable mean and standard
deviation is obtained. However, in the optimum specimen size considered, there was no big
variation that was observed after the application of Chauvenets criterion. Tolerance interval
for the parent population is estimated within 95% confidence and the method assumes to have
a Gaussian parent population. To exclude the assumption made previously, MCM analysis for
47
uncertainty propagation is employed as the inputs to MCM analysis need not be Gaussian.
The mean of the UTS for SS and Sensitized SS grade 304/304L specimens may lie within
370.25 to 921.39 MPa and 1338.93 MPa, respectively. The mean for the experimented lot in
this study for SS specimens is 645.82 ± 6.25 MPa and for Sensitized SS specimens is 645.95
± 47.77 MPa, which lies well within the range. The range of UTS within which Carbon Steel
specimens may lie is 312.89 to 572.33 MPa and the mean for the experimented lot from this
study is 442.61 ± 18.56 MPa which is well within the expected range.
A conclusion can be drawn from this study that uncertainty analysis must be conducted in
sample to sample experiments to know whether the range of the results obtained are agreeable
with that of the standard results. In most studies, we see that a certain experiment is simply
rejected for giving inconsistent results failing to understand the random and systematic errors
that are prevalent in experimentation. Propagation of uncertainty is required in every research
field to know whether the experimental results observe the same trend as that of the parent
population.
6.2
Future Work
The mechanical properties obatined with the set of specimens, categorized as optimum
specimen design in this thesis gives a good approximation to that of the conventional specimens. The uncertainty analysis further ensures our study and validates the properties obtained on a large scale. However, the miniature specimens in this study was focussed on high
strength steels. Many types of metals can be studied at miniature level, specimens obtained
from boat samples to understand its behavior. Many a time, structures are subjected to high
temperatures and is difficult to conduct a conventional test of an equivalent material at that
temperature. Miniature mechanical properties should be determined at varied temperature
levels. Further, SEM should be employed along with the SEMTester, miniature testing instrument to find out the physics at miniature level as many properties depend on a large number
of variables.
48
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