SQ

GRAIN SIZE EFFECTS IN MICRO-TENSILE TESTING

OF AUSTENITIC STAINLESS STEEL

by

Whitney A. Poling

A thesis submitted to the Faculty and Board of Trustees of the Colorado School of Mines in partial

fulfillment of the requirements for the degree of Master of Science (Metallurgical and Materials Engineering).

Golden, Colorado

Date:

Signed:

Whitney A. Poling

Signed:

Kip O. FindleyThesis Advisor

Signed:

Nicholas Barbosa IIIThesis Advisor

Golden, Colorado

Date:

Signed:

Professor Michael J. Kaufman,

Professor and HeadDepartment of Metallurgical and Materials Engineering

ii

ABSTRACT

A micro-tensile test system is under development at the National Institute for Standards and Technol-

ogy (NIST) and is based on tensile test techniques developed for thin films, but it provides the capability

of testing samples from bulk material. Example specimen dimensions are a gage section length of 342 m,

width of 75 m, and thickness of 80 m. The micron size scale of the specimens presents the potential to an-

alyze local deformation of a material within a specific region of the microstructure. The micro-tensile system

could also be useful in material modeling, because samples can be constructed to match micromechanical

modeling simulations of specific microstructures and microstructure size scales. The primary objectives of

this research are to fundamentally understand the grain size effect on micro-scale tensile testing results and

to further the goal of using micro-tensile technology as a tool to study mechanical behavior of small volumes

of material with minimal size effects.

316L stainless steel was selected as the candidate material because of its austenite stability. Both

ASTM E-8 rectangular sheet-type tensile specimens and micro-tensile specimens were tested for mean grain

sizes of 3.1 m, 9.5 m, and 17.5 m, which were obtained through heat treatments of 40% and 85% cold

reduced sheet. The grain sizes were selected such that there was wide range in the number of grains in the

micro-tensile specimen cross-section. For all three grain sizes, the yield stresses, flow stresses, and strain

hardening rates of the micro-tensile specimens were less than those of the conventional tensile specimens. A

specimen size effect scaling model suggests that the lower flow stresses and lower strain hardening rates are

attributable to the high grain size to specimen size ratios of the micro-tensile specimens. The scatter in the

yield stress and flow stress data was higher for the micro-tensile specimens, but the amount of scatter did not

show a strong dependence on grain size. It was expected that the scatter would increase with an increasing

grain size to specimen size ratio, but high uncertainty in the calculated stress from the micro-tensile tests

could also contribute to the observed scatter. Electron back scatter diffraction (EBSD) was used to calculate

the possible standard deviation in the minimum average Taylor factor in the cross-section of the micro-tensile

specimens. The increase in the standard deviation of the minimum average Taylor factor with increasing

grain size to specimen size ratio could contribute to the observed scatter in the yield stress and possibly

the flow stress of the micro-tensile specimens. Recommendations for improving precision and accuracy of

measurements associated with the micro-tensile test system are also discussed.

iii

TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

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

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Industrial Relevance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

CHAPTER 2 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Small-Scale Tensile Testing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Specimen Size Effects in Small-Scale Tensile Testing . . . . . . . . . . . . . . . . . . . . . . . 3

2.3 Grain Size Strengthening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

CHAPTER 3 EXPERIMENTAL METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 Grain Growth Heat Treatments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.4 Microstructural Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4.1 Metallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4.2 Electron Backscatter Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.5 Conventional Tensile Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.6 Micro-Tensile Specimen Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.6.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.6.2 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.6.3 Cross-Sectional Area Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.7 Set-Up of Micro-Tensile Test System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.8 Micro-Tensile Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.9 Digital Image Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.10 Assessment of Measurement Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

iv

CHAPTER 4 RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1 Heat Treatments and Grain Size Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.1 Grain Growth Heat Treatment Experiments . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.2 Sheet Grain Sizes for Tensile Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Conventional Tensile and Micro-Tensile Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.1 Grain Size Strengthening - Hall-Petch Behavior . . . . . . . . . . . . . . . . . . . . . . 48

4.2.2 Grain Size Strengthening - Flow Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3 Specimen Size Scaling Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.4 Taylor Factor Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

CHAPTER 5 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

CHAPTER 6 FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

APPENDIX A METHOD OF GRAIN SIZE MEASUREMENTS . . . . . . . . . . . . . . . . . . . . 62

APPENDIX B SELECTION OF EBSD SCAN SIZE FOR TAYLOR FACTOR ANALYSIS . . . . . . 64

APPENDIX C COMMERCIAL SOFTWARE DIGITAL IMAGE CORRELATION EXAMPLE . . . 65

v

LIST OF FIGURES

2.1 Tensile strength and yield strength from tensile tests of 1747 aluminum sheet with varying thicknessand a constant grain size of 0.126 mm. The approximate number of grains in the specimen cross-section was calculated as (sheet thickness)/(grain size). The tensile strength and yield strengthdecrease as the grain size to sheet thickness ratio increases. . . . . . . . . . . . . . . . . . . . . . 4

2.2 (a) Henning and Vehoffs model of the gage section of a small-scale tensile specimen. The grainsare represented by equal size cubes. This model was used in a Taylor factor analysis to predict thescatter in the initial yield stress of small-scale tensile specimens. Two different relative lengths ofthe gage section were investigated. Lrel = 4 is more representative of a typical tensile specimenwith a gage length that is several times longer than the specimen width or thickness. (b) Therelative standard deviation of the initial yield stress is plotted versus the number of grains in thespecimen cross-section. The bars on the experimental data points represent the 90% confidenceinterval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 True stress-strain curves for aluminum multicrystals with differing number of grains (n) in thespecimen cross-section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Tensile and yield strength as a function of proportion of surface grains relative to interior grainsin the tensile specimen cross-section (). The tensile tests were performed for a constant grainsize and varying aluminum sheet thickness. The equation for calculating and a schematic of thetensile specimen cross-section are shown to the right. . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 Schematic of a single crystal under uniaxial tension. The angles and are used to calculate theSchmid factor m. The applied stress necessary to plastically deform the crystal depends on m. . 9

2.6 (a) Scaling parameters and were calculated by fitting Equation 2.6 to experimental data withM = 2.6. At approximately n = 15, substantial specimen size effects may be observed as themechanical behavior of the specimen transitions from polycrystal (n > 15) to multicrystal (n 15) to multicrystal (n< 15) behavior. Plot from [23]. (b) The experimentally determined flow stress data fromHansen for aluminum [8] is compared to predicted flow stresses from the scaling model for thesame specimen size to grain size ratios (n). Plot from [23].

10

and calculated data occurs for n = 3.2. The large difference in flow stresses between the n = 3.2 and n =

3.9 specimens from the experiment was speculated to be related to the orientation of the individual grains

in the specimen [8]. The discrepancy between the experiment and the model for n = 3.2 could suggest that

the limit of the model was reached due to too few grains in the specimen cross-section.

11

CHAPTER 3

EXPERIMENTAL METHODS

Heat treatment experiments were performed on 316L stainless steel sheet with two different amounts

of cold reduction to obtain a range of mean grain diameters for the micro-tensile and conventional tensile

tests. Micro-tensile and conventional tensile testing were performed on sheets with mean grain diameters of

3.1 m, 9.5 m, and 17.5 m. Fabrication of the micro-tensile specimens is described. A detailed description

of the MEMS-based micro-tensile test system, micro-tensile testing, and data analysis is also provided.

3.1 Experimental Design

To study grain size effects with the micro-tensile specimens, three primary target grain sizes were

selected: 5, 10, and 20 m. The grain sizes and specimen cross-section dimensions were chosen such that

there was a wide range in the number of grains in the micro-tensile specimen cross-section. The size scale of

the micro-tensile specimens was expected to potentially result in lower flow stresses due to the high surface

area to volume ratio. Further, the number of grains in the micro-tensile specimen cross-section was expected

to directly affect the scatter in the mechanical property data. Conventional tensile tests were performed for

the same grain sizes as the micro-tensile specimens for comparison of the mechanical behavior. Additionally,

the conventional tensile tests were performed on two secondary target grain sizes: 7 and 50 m. The five

different grain sizes for the conventional tensile tests were used to determine the Hall-Petch relationship

for the material. The Hall-Petch data from the conventional tensile tests were used to evaluate grain size

strengthening in the micro-tensile specimens.

The coarsest grain size micro-tensile specimens were predicted to have a large amount of scatter in

mechanical properties due to very few grains in the specimen cross-section. With less than 20 grains in

the specimen cross-section, the orientation of each grain becomes an important factor in the mechanical

behavior of the specimen. Specifically, the scatter in the initial yield stress was expected to be proportional

to the scatter in the minimum average Taylor factor in the specimen cross-sections based on the following

yielding criterion: the tensile specimen will initially yield at the location where the average Taylor factor

in the specimen cross-section is a minimum [14]. To perform an analysis of the scatter in the minimum

average Taylor factor for the micro-tensile specimens, grain orientation data in the specimen cross-section

were necessary. A possible non-destructive method to obtain these data would be to construct a three-

dimensional rendering of the microstructure from surface scans of the gage section, but the surface quality

of the micro-tensile specimens was not high enough to perform electron backscatter detection (EBSD) scans

on the gage section. Instead, the heat treated materials were well characterized using EBSD, and the

microstructures of the cross-sections were simulated from the data. The gage sections of several specimens

from each grain size were simulated with sets of randomly selected EBSD scans. The average Taylor factor

was calculated for each scan, and the standard deviation in the minimum average Taylor factor was estimated

for the micro-tensile specimens of each grain size allowing for a comparison of the expected scatter in the

initial yield stress with the experimental data.

12

3.2 Material

To isolate the grain size effect on measured mechanical properties from other potential microstructure

effects, a single phase material is required. Austenitic stainless steels are good candidates. 316 stainless

steel was selected to ensure that stress or strain-induced austenite to martensite transformation does not

affect the mechanical behavior. More specifically, 316L stainless steel was chosen because it has a low carbon

content, which helps to prevent carbide formation during heat treatments. Such carbides could interfere

with isolating the grain size effect on measured mechanical properties.

In order to obtain the desired range of grain sizes, sheets with two different amounts of cold reduction

were used. A sheet of 40% cold-reduced 316L stainless steel, provided by AK Steel, was used to obtain

grain sizes of approximately 10 m and larger. The thickness of the sheet was approximately 0.75 mm. For

grain sizes below 10 m, a sheet of 85% cold-reduced 316L stainless steel, obtained from Hamilton Precision

Metals, was used. The thickness of the sheet was 0.13 mm. The sheet chemistries provided by the suppliers

are given in Table 3.1.

Table 3.1 Sheet chemistries of 316L stainless steels in weight percent.

wt pct C Mn Si Ni Cr Mo

40% CR 0.022 1.38 0.54 10.10 17.14 2.1085% CR 0.016 1.44 0.50 10.12 16.48 2.07

wt pct Ti Nb V Al N S

40% CR 0.002 0.013 0.044 0.005 0.034 0.01485% CR N/R N/R N/R N/R 0.028 0.001

wt pct P Cu B Sn Co W

40% CR 0.033 0.26 0.0019 0.009 0.16 0.03085% CR 0.036 0.33 N/R N/R 0.30 N/R

The steel sheets were cut into approximately 130 mm by 250 mm pieces for heat treating and specimen

fabrication. The conventional tensile specimens, micro-tensile specimens, and metallography and EBSD

specimens for each of the three primary target grain sizes of 5, 10, and 20 m were all obtained from the

same 130 mm by 250 mm sheet sample. Conventional tensile specimens and metallography specimens were

also taken from sheet samples for the target grain sizes of 5 and 70 m. The schematic in Figure 3.1 shows

where the different types of specimens were sectioned from the sheet. One of the metallography specimens

and the EBSD specimen came from the front edge of the sheet that was sheared off post-heat treating to

prepare foils for the micro-tensile specimens. The sheared off material was approximately 130 mm by 25 mm.

Another metallography sample was taken from the opposite end of the sheet for comparison to the front

edge sample.

3.3 Grain Growth Heat Treatments

Based on recrystallization experiments, grain growth curves, and kinetic analyses of grain growth in

316L stainless steel from literature [2527], heat treatment screening experiments were performed on test

coupons to identify the specific conditions to achieve a wide range of grain sizes in the 316L stainless steel

sheet with 40% and 85% cold reduction. The material heat treatments were designed to avoid sensitization

13

Figure 3.1 Schematic of a sheet sample with dashed lines indicating the sectioning of the sheet for ma-chining tensile specimens. Material for the micro-tensile specimens and EBSD specimen camefrom the section that was sheared off post-heat treating. The two boxes indicate the approx-imate locations from which metallography specimens were taken, and the oval indicates theapproximate location from which the EBSD specimen was taken. The back edge was near theback of the furnace, and the front edge was near the furnace door.

of the stainless steel, which can occur at temperatures below 950 C [28]. Rapid quenching is required toavoid the most detrimental and fastest carbide precipitation in the temperature range of 800-900 C [28, 29].Water quenching was found to severely distort the material; therefore, air cooling was implemented instead.

Since the sheets were quite thin, it was assumed that the sheets would rapidly cool in ambient air. All heat

treatments were performed in open air at either 950 C or 1000 C.

The target grain diameters for the conventional tensile tests were 5, 7, 10, 20 and 50 m. This range

of grain sizes was selected to develop the Hall-Petch behavior of bulk 316L stainless steel as a reference for

comparison with the yield strengths from the micro-tensile specimens. The target grain diameters for the

micro-tensile specimens were 5, 10 and 20 m. Three grain sizes were used in the micro-tensile experiments

due to challenges associated with preparation and cost constraints. These target grain diameters were selected

to obtain a large variation in the number of grains in the cross-section of the micro-tensile specimens. The

design of the micro-tensile specimens is described in more detail in Section 3.6.1. The heat treatments that

produced mean grain diameters close to the target grain diameters were used to heat treat the large sheet

samples. The selected heat treatments and sheet mean grain diameters are given in Table 3.2. More detailed

results and statistical analysis of the grain size data are provided in Section 4.1.

14

Table 3.2 Summary of sheet heat treatments. The sheet mean grain diameter is the average of the meangrain diameter of the sheet from two different locations as indicated in Figure 3.1.

Cold Reduction Temperature ( C) Time (min) Sheet Mean GrainDiameter (m)

85% 950 3 3.185% 950 10 4.840% 950 20 9.540% 950 110 17.540% 1000 190 54.2

3.4 Microstructural Characterization

The grain size analysis of the heat treated experimental coupons and the sheet samples was conducted

using optical micrographs of etched metallography specimens. EBSD scans were performed on each of the

three primary grain sizes to simulate the texture in cross-sections of the micro-tensile specimens for Taylor

factor analysis.

3.4.1 Metallography

Metallography specimens were prepared from the grain growth experiment coupons and from unused

portions of the heat treated sheets. Electrolytic etching was performed with 10% (w/v) oxalic acid at 1.8 V

for approximately 2 minutes.

The mean grain diameters were measured using a software program called analySIS. The software

is capable of detecting the location of a grain boundary based on the change in contrast in the micrograph,

and it draws a line along all of the detected grain boundaries. The micrograph often requires further manual

editing to remove erroneous boundaries identified by the program and to add boundaries where they are

obviously missing. Once the boundaries are overlaid on the micrograph, the software measures the area

within the boundaries, which it outputs as individual grain areas. The grain diameters were calculated in a

similar manner to the manual planimetric measurement technique described in ASTM E112-96 (2004) [30].

The square root of the individual grain area was taken to approximate the diameter of the grain, and the

average of all the grain diameters gave the mean grain diameter of the sample. A detailed description of the

method used for making the grain size measurements is provided in Appendix A.

3.4.2 Electron Backscatter Diffraction

The objective of the EBSD experiments was to estimate the standard deviation of the minimum average

Taylor factor in the cross-sections of the micro-tensile specimens. The standard deviation of the minimum

average Taylor factor was predicted to be proportional to the standard deviation in the initial yield strength

of the micro-tensile specimens. The experimental set-up for the Taylor factor analysis was based on the

small-scale tensile specimen model and initial yielding criterion of Henning and Vehoff [14].

A LEO 1525 Gemini Schottky field-emission scanning electron microscope with a 20 kV accelerating

voltage was used for the EBSD experiments. The EBSD patterns were collected with TSL OIM Data

Collection 5 and TSL OIM Analysis 5 was used to analyze the patterns. Step sizes were selected to

15

minimize scan time without losing accuracy in the average Taylor factor. A comparison of average Taylor

factors obtained using different step sizes on the same scan area for the three different grain sizes are provided

in Appendix B. A step size of 3 m was used for the fine and medium grain sizes, and a 4 m step size was

used for the coarse grain size.

Henning and Vehoffs tensile specimen model in Figure 2.2a was used to develop an experimental

design. The gage section of the micro-tensile specimen was represented by a set of EBSD scans. The

average thickness and width of the micro-tensile specimens from micrograph measurements were used as the

dimensions of the scan size. The number of scans necessary to represent the entire gage section came from

the assumption that each scan represented a slice of the gage section that was one grain thick. Thus, the

number of scans was given by the length of the reduced section, 342 m, divided by the mean grain size of

the sheet. The EBSD experimental matrix is given in Table 3.3.

Table 3.3 EBSD experimental matrix for simulating the average Taylor factors along the micro-tensilegage section.

Grain Size (m) Scan Size (m) Scans/Specimen

3 78 x 76 1149 83 x 71 3817 65 x 75 20

A set of large scans, approximately 480 x 1200 m in size, was performed for each grain size. Each

large scan was then cropped down into a set of smaller scans of the size specified in Table 3.3 for a particular

grain size. The confidence indices of the data points in the scan were used to assess the quality of the

scan. A confidence index is a measure of the confidence of the TSL OIM Data Collection 5 software that

a pattern was correctly indexed by the indexing algorithm [31]. The range of the confidence index is zero to

one with one being the highest confidence value. Any points with a confidence index of less than 0.1 were

considered to be poor quality and were filtered out. The scans were assumed to have good quality and were

used for the Taylor factor analysis if they had 85% or more data points with confidence indices greater than

0.1 and had an average confidence index of 0.5 or greater. The Taylor factor for each point in the scan can

be calculated by EBSD analysis software based on the crystallographic orientation indexed for the point and

the primary slip systems and deformation gradient for the sample. A definition of the Taylor factor and

the relationship between the average Taylor factor and applied tensile stress for a specimen are provided

in Section 2.3. The average Taylor factor was computed by the TSL OIM Data Collection 5 software for

all of the scans for each grain size with the face-centered cubic primary slip systems({111} 110) and the

following deformation gradient: xx = 0.5, yy = 0.5, zz = 1, xy = yz = xz = 0. The selecteddeformation gradient represents loading in uniaxial tension with the tensile axis perpendicular to the scan

area. The EBSD scans were performed parallel to the rolling direction of the sheet, which means that the

specimen cross-section simulated by the scan is actually rotated 90 with respect to the real micro-tensilespecimen cross-section. Only the standard deviation in the minimum average Taylor factor was of interest,

and the standard deviation should not be affected by a 90 difference in orientation. The set of average Taylorfactors were entered into the statistical program R [32], and each set of average Taylor factors was divided

into ten random samples to represent ten micro-tensile specimens. The minimum average Taylor factor

from each sample was found, and the standard deviation of the ten minimum average Taylor factors was

16

calculated. The number of scans necessary to simulate the gage section of the finest grain size micro-tensile

specimens was quite large; therefore, only six specimens were simulated.

An average Taylor factor for the heat treated material was necessary for the scaling model described

in Section 2.3. For each of the three grain size conditions, the average Taylor factor was computed for a

scan size of approximately 480 x 550 m with the face-centered cubic primary slip systems({111} 110)

and the following deformation gradient: xx = 1, yy = 0.5, zz = 0.5, xy = yz = xz = 0. The scanwas performed parallel to the rolling direction of the sheet, and the selected deformation gradient represents

loading in uniaxial tension in the rolling direction.

3.5 Conventional Tensile Testing

All of the conventional tensile specimens were machined to the dimensions set forth by ASTM E8/E8M-

09 for a rectangular sheet-type tension specimen [33]. Four specimens were tested for each mean grain diam-

eter. The tensile tests of the specimens from the 9.5 m, 17.5 m, and 54.2 m mean grain diameter sheets

prepared from the 0.75 mm thick 40% cold-reduced sheet were performed using a MTS Alliance RT/100

test system with TestWorks software and a 20,000 lbf load cell. A Shepic 50.8 mm (2 in) extensometer

with 50% extension was used to measure displacement. In each of the tests, the test was briefly stopped and

the extensometer was re-positioned to capture the full elongation of the tensile specimen, which exceeded

the 25.4 mm (1 in) extension limit of the extensometer. The displacement rate was set at 2.54 mm/min (0.1

in/min), and the strain rate was approximately 6 104 s1. All of the specimens failed within the gagelength and within the extensometer.

The 3.1 m and 4.8 m mean grain diameter specimens were prepared from the 0.13 mm thick 85%

cold-reduced sheet, and the test set-up was altered to accommodate the thinner specimens. A 1,000 lbf load

cell and Instron low load grips with parallel grip faces were used to test the thin sheet. The tests were

performed using an Instru-Met test system with TestWorks software. The displacement rate was set at

2.54 mm/min (0.1 in/min), and the strain rate was approximately 6 104 s1. An Instron 25.4 mm (1 in)blade-type extensometer with 100% extension was used to measure displacement. Two of the 4.8 m mean

grain diameter specimens were tested with wedge-shaped grips and warped in the grip sections during the

tests. Further, for one of the two specimens, an Instron 50.8 mm (2 in) blade-type extensometer with 50%

extension was used and was re-positioned after reaching its displacement limit. The shorter extensometer

and the parallel grips were used for a subsequent test such that the tensile specimen was gripped as close

to the reduced section as possible in order to prevent warping. Four specimens were tested for each mean

grain diameter, but only three of the 4.8 m tests were successful. All of the specimens failed outside of the

extensometer.

3.6 Micro-Tensile Specimen Preparation

The micro-tensile specimen dimensions were based on the scaled-down dimensions of a conventional

ASTM sheet-type tensile specimen [33]. The specific dimensions and selected grain sizes were chosen to

achieve a target number of grains in the micro-tensile specimen cross-section. The specimens were machined

from thin foils with micro-electrical discharge machining. The cross-sectional areas of the micro-tensile

17

specimens were calculated using electrical resistance measurements, and the thickness and width of each

specimen were also approximately measured from optical and SEM micrographs.

3.6.1 Design

The micro-tensile specimens were designed to contain a specific average number of grains per area in

the specimen cross-section. Based on the work of Henning and Vehoff [14] discussed in Section 2.2, the

number of grains in the specimen cross-section can be divided into three different ranges of interest as shown

in Figure 3.2. The following results were predicted for the three different conditions identified for study:

Figure 3.2 The relative standard deviation of the initial yield stress is plotted versus the number of grainsin the specimen cross-section for small-scale tensile specimens. The plot is demarcated intothree different ranges of the number of grains in the specimen cross-section: less than 20 grains,20-100 grains, and more than 100 grains. The impact of size effects on mechanical behavior isexpected to differ between the three ranges. Plot from [14].

1. A micro-tensile specimen condition with greater than 100 grains in the gage cross-section was expected

to correspond well with bulk mechanical behavior and have little scatter in the mechanical property

data.

2. A micro-tensile specimen condition with approximately 50 grains in the gage cross-section was expected

to behave similarly to the bulk material, but there was expected to be more scatter in the mechanical

property data between micro-tensile samples and the bulk samples.

3. A micro-tensile specimen condition with fewer than 20 grains in the gage cross-section was expected

to exhibit significantly different mechanical behavior when compared to the bulk mechanical behavior

and was expected to show considerable scatter in the mechanical property data between specimens.

Table 3.4 shows the analysis performed to estimate the average number of grains per micro-tensile cross-

sectional area for the three target grain sizes for the micro-tensile specimens. The grains were modeled as

18

equal size squares, and the method of analysis came from the planimetric method for grain size measurement

according to ASTM E112-96 (2004) [30]. The average grain area, Agrain, is given by

Agrain = (d)2 (3.1)

where d is the theoretical mean grain diameter. The number of grains per area, NA, and the theoretical

number of grains in the specimen cross-section, Nspecimen, are given by

NA =1

Agrain(3.2)

Nspecimen = NA A (3.3)

where A is the specimen cross-sectional area. The specimen width, w, was selected to be similar to the

scaled-down dimensions of the conventional tensile specimens, and the specimen thickness, t, was chosen

to meet the three conditions described above. In Table 3.4, thicknesses 5 m above and below the target

thickness of 80 m are also included to account for some variability in the exact thickness between specimens.

The foils for the micro-tensile specimen fabrication are prepared by thinning down a sample of bulk material,

and there is some variability in the thickness throughout the foil.

As shown in the analysis, a slight difference in thickness between samples of the same grain size will

not have a significant impact on the micro-tensile test results compared to the large differences between

grain sizes. For example, a 10 m thickness difference between samples of a target grain size of 10 m would

potentially result in only a difference of 7 grains in the cross-sectional area between the two samples; whereas,

an 85 m thick tensile specimen with a grain size of 20 m compared to a 75 m thick tensile specimen

with a grain size of 10 m would potentially have a difference of 38 grains in the gage cross-sectional areas.

The methodology for the specimen design was used to make slight adjustments to the nominal width of the

micro-tensile specimens based on the actual mean grain diameter achieved through heat treatments and the

final thickness after thinning the material down to a foil, as described in Section 3.6.2.

Table 3.4 Analysis to estimate the average number of grains per cross-sectional area of the micro-tensilespecimens based on modeling the grains as equal size squares. The width of the specimens was kept

constant, and the thickness was varied.

w (mm) t (mm) A (mm2) d (mm)Agrain(mm2)

NA(grains/mm2)

Nspecimen

0.070 0.075 0.0053 0.005 0.000025 40000 2120.070 0.075 0.0053 0.01 0.0001 10000 530.070 0.075 0.0053 0.02 0.0004 2500 13

0.070 0.080 0.0056 0.005 0.000025 40000 2240.070 0.080 0.0056 0.01 0.0001 10000 560.070 0.080 0.0056 0.02 0.0004 2500 14

0.070 0.085 0.0060 0.005 0.000025 40000 2400.070 0.085 0.0060 0.01 0.0001 10000 600.070 0.085 0.0060 0.02 0.0004 2500 15

19

3.6.2 Fabrication

The design of the micro-tensile specimens came from a scaled-down version of the conventional sheet-

type tensile specimen described in ASTM E8/E8M-09 [33]. The micro-tensile design is given in Figure 3.3a,

and for reference, the ASTM E8/E8M-09 sheet-type tensile specimen dimensions are provided in Figure

3.3b. Slight adjustments were made to some of the micro-tensile dimensions, such as the width and fillet

radius. The final thickness of the micro-tensile specimens was dependent on the thinning process; thus, the

nominal width of the gage section was adjusted to 75 m for the 17.5 m mean grain diameter material and

70 m for the 3.1 m and 9.5 m mean grain diameter materials to meet the target number of grains in the

specimen cross-section.

(a)

(b)

Figure 3.3 (a) Schematic of the basic micro-tensile specimen design. The width was adjusted to 70 mfor the finest and medium grain size foils to compensate for the foil thickness. (b) Schematic ofthe basic ASTM E8/E8M-09 sheet-type tensile specimen design. Specimen design from [33].

The first step of the fabrication process was to thin the stainless steel sheet to the desired thickness

of the micro-tensile specimens. The thinning process involved grinding and polishing a small sample of

material from the sheet to achieve a foil with planar surfaces. The foil was carefully prepared to minimize

mechanical defects such as scratches on the surface, which could significantly impact the performance of the

micro-tensile specimens. The material was polished with 1 m diamond polishing media and finished with

20

0.05 m alumina. The thinning process was repeated with several samples from the same sheet to obtain

a foil with a machinable area within a thickness tolerance of 5 m. An area of 2.5 mm by 3.7 mm wasnecessary to machine ten micro-tensile specimens from a foil, and a foil was only selected for machining if

it had a usable area with a maximum thickness variation of 10 m across the entire machinable area. A

Brunswick Instrument MP1-VF thickness probe with a resolution of 0.0002 mm was used to measure the

thickness across the sample throughout the thinning process. Post-machining, the thicknesses of the micro-

tensile specimens were measured from SEM micrographs. The thickness of the micro-tensile specimen gage

section varied by less than 10 m between all of the specimens from the same foil, and the thickness of each

individual specimen appeared to vary by less than 1 m along the gage section.

All three sets of the micro-tensile specimens were machined using micro-electrical discharge machining

(micro-EDM). EDM is a process in which material is removed from an electrically conductive workpiece

through controlled sparks, which vaporize the workpiece material [34]. This process results in a shallow heat

affected zone and re-cast layer at the machined surface. An attempt to remove the damage layer on the

micro-tensile specimens was done through micro-milling. Examples of as-machined specimens are shown in

the SEM image in Figure 3.4.

Figure 3.4 SEM micrograph showing two machined micro-tensile specimens. The specimens remain at-tached to the foil by tabs. Prior to testing, notches are cut in the tabs with a focused ionbeam (FIB), and the specimens are removed from the foil with fine-tipped tweezers.

Micro-milling was incomplete on most of the micro-tensile specimens from the 17.5 m mean grain

diameter material. The foil appeared to have shifted during machining, and one of the gage section sidewalls

on most of the specimens received little or no micro-milling. Example SEM images of the sidewalls are

provided in Figure 3.5. The thickness of the total damage layer from micro-EDM is strongly related to the

pulse energy [35]. The nominal energy that was used to machine the micro-tensile specimens was 350 nJ;

the damage layer thickness from this very low pulse energy was estimated to be 0.5-1.0 m [36]. Therefore,

21

the partial damage layer that was left on the micro-tensile specimens from the 17.5 m mean grain diameter

material was estimated to account for approximately 1% of the micro-tensile specimen width. The impact of

the partial damage layer on the micro-tensile specimen mechanical behavior was considered to be low, since

the volume of un-affected, ductile material in the gage section was comparatively very large. A custom-

made clamping system was designed and machined to prevent future foil samples from moving during the

machining process.

(a) (b)

(c) (d)

Figure 3.5 SEM micrographs of one of the sidewalls of three micro-tensile specimens machined from the17.5 m mean grain diameter foil. The specimens were imaged while still attached to the foil;thus, the orientation of each specimen is the same. There are varying degrees of micro-millingon the sidewall in the micrographs. For example, the surface of the sidewall in (a) appears tobe completely milled; whereas, the sidewall in (c) shows no signs of milling and the sidewallin (b) shows partial milling. The curl of material visible in the micrographs is a small amountof material that was pushed up by the micro-milling process.(d) Schematic of a micro-tensilespecimen showing the orientation of the specimens in the SEM micrographs.

22

3.6.3 Cross-Sectional Area Measurements

The width and thickness of each micro-tensile specimen were measured from optical and SEM micro-

graphs, respectively. However, the excess material that remained on the edge of the micro-tensile specimens

after micro-milling obscured the exact location of the edge in the micrographs. The location of the edge

had to be estimated for the width and thickness measurements, which could reduce the accuracy of the

calculated cross-sectional area. Thus, the cross-sectional area of each micro-tensile specimen was calculated

from electrical resistance measurements.

The cross-sectional area of a long conductor with constant cross-section dimensions and constant re-

sistivity can be calculated using

R =l

A(3.4)

where R is resistance, is electrical resistivity, l is the length of the conductor, and A is the cross-sectional

area of the conductor. Equation 3.4 is a solution to Maxwells equations for a simple geometry. It is applicable

to the micro-tensile specimens, since the gage section of each micro-tensile specimen has an approximately

constant width and thickness.

The electrical resistance measurements were performed at an electrical probe station with manual three-

axis screw-driven micro-manipulators with tungsten probes, which are needle-like electrodes for contacting

the specimen. A four-wire (Kelvin) configuration was used. Current probes were placed on the grips near

the ends of the specimen. The voltage probes were placed close to the gage section in the tapered regions

where the width of the specimen narrowed from the grip width to the gage width. Figure 3.6 shows a micro-

tensile specimen with the current and voltage probes in contact with the specimen. The current probes were

spaced apart from the voltage probes to allow the current to spread across the entire width of the specimen.

Current was applied with a power source, and the voltage between the voltage probe tips was measured

with a multimeter. A program written by David T. Read at NIST controlled the application of current, the

measurement of voltage, and the calculation of the specimen resistance.

The current applied to the specimen was ramped between -100 mA and 100 mA. A full ramping cycle

consisted of 20 steps starting at 0 mA, progressing to a maximum negative current, stepping back to 0 mA,

progressing to a maximum positive current, and returning to 0 mA. At each step the current and voltage

were measured. Three full ramping cycles were performed for each specimen. The voltage versus current

data were plotted as they were acquired, and the software calculated the specimen resistance at each step

based on a linear fit to the voltage versus current data. The calculated resistance based on 60 sets of data

was used for the cross-sectional area calculation. The resistance was calculated using Ohms Law

V = IR (3.5)

where V is the voltage, I is the current, and R is the resistance. The value of l in Equation 3.4 was measured

as the length between the voltage probe tips from an optical microscope image.

The voltage probes were placed in the tapered region outside of the gage section to avoid deeply

scratching the gage section surface with the probe tips; therefore, Equation 3.4 cannot be applied directly.

A correction factor for the resistance measurement was calculated to account for the increase in width

of the specimen beyond the gage section where the voltage field becomes non-uniform. Two-dimensional

23

Figure 3.6 Optical microscope image of a micro-tensile specimen with current probe needles placed onthe grips and voltage probe needles placed near the gage section.

simulations, based on Maxwells equations, of the voltage field in the micro-tensile specimen geometry were

performed with ANSYS finite element analysis (FEA) software by David T. Read. The FEA mesh was

generated from computer-aided design (CAD) drawings that matched the actual specimen geometry. It was

assumed that the thickness of any individual micro-tensile specimen was constant throughout the specimen;

therefore, a two-dimensional simulation was sufficient to predict how the voltage field was altered due to the

specimen geometry. A plot of voltage as a function of position along the specimen axis from the ANSYS

simulation is shown in Figure 3.7. The linear fit within the position range corresponding to the gage section

shows that Equation 3.4 is satisfied within this region. The second linear fit on the plot is between the two

positions at which the voltage was actually read on the specimen. The second linear fit has a shallower

slope than the linear fit within the gage section. Thus, if the experimentally determined resistance, based on

the measurement points shown in Figure 3.7, was used in Equation 3.4 under the assumption of a uniform

voltage field, then the calculated area would be too high. To correct for the non-uniformity of the cross-

sectional area beyond the gage section, a correction factor for the resistance measurement was determined.

From the ANSYS simulation, a ratio of the resistance for the actual specimen geometry to the resistance

for a constant cross-section was calculated for varying distance between the probe tips. An example of a

set of correction factors is provided in Figure 3.8. Two sets of correction factors were calculated for each

of the three different sets of micro-tensile specimens based on computer-aided design (CAD) drawings of

two specimens from each condition. Since the widths of the specimens varied slightly within each set of

specimens, the two specimens selected for analysis had the smallest and largest widths as determined from

optical micrographs. As shown in Figure 3.8, the correction factors for the two extremes in widths of the

24

Figure 3.7 At the top is an optical microscope image of a micro-tensile specimen with voltage probeneedles placed near the gage section. The image was used to construct a CAD drawing ofthe specimen geometry for the ANSYS simulation. An ANSYS simulation of steady stateelectrical conduction through the micro-tensile specimen geometry was performed to analyzethe voltage field across the specimen. The plot shows the voltage versus position results forthe specimen geometry. One of the fitted lines corresponds to the uniform voltage field withinthe gage section and the other line corresponds to the two positions at which the voltage wasactually measured on the specimen in the image. The difference in the two lines is due to thenon-uniformity of the voltage field beyond the gage section. ANSYS simulation provided byDavid T. Read, NIST.

25

Figure 3.8 An example plot of the correction factors for the electrical resistance measurements of the ma-terial with a mean grain diameter of 9.5 m. The ANSYS simulation was performed on thelargest and smallest optically-measured gage widths. Since the correction factors were fairlyclose for the specimens with the extremes in width, one set of correction factors, calculatedfrom a representative micro-tensile specimen geometry, was used to adjust all of the electricalresistance measurements of a particular grain size. ANSYS simulation and correction factorsprovided by David T. Read, NIST.

9.5 m mean grain diameter material were very close. The percent difference in the correction factors due to

variation in specimen widths for each grain size is given in Table 3.5. For example, the 9.5 m mean grain

diameter material had the largest percent difference in correction factors, and the difference was only 0.65%.

Thus, one set of correction factors was used for all of the micro-tensile specimens of a particular grain size.

The resistivity value for Equation 3.4 was determined experimentally for each of the three grain size

conditions. A long, narrow reference strip was cut from the foil near the location where the micro-tensile

specimens were extracted. The electrical resistance of the strip was measured as described for the micro-

tensile specimens. The dimensions of the reference strip were measured at the locations of the probes, and

an average cross-sectional area was calculated. The thickness was measured with the Brunswick Instrument

thickness probe, and the width and length between probe tips were measured with a stereo microscope

equipped with digital stage micrometers that have a resolution of 0.001 mm. Equation 3.4 was used to

calculate the electrical resistivity based on the electrical resistance and cross-sectional area measurements of

the reference strip.

26

Table 3.5 Percent difference in electrical resistance correction factors from the specimens with extremesin gage width for the fine, medium, and coarse grain sizes. The difference in correction factors was

calculated for the mean probe separation. Note that the difference in correction factors increases withincreasing probe separation, as evident in Figure 3.8.

Grain Size Maximum Minimum Mean Probe Difference in(m) Width (m) Width (m) Separation (m) Ractual/Rstraight

3.1 82.0 72.1 414 0.37 %9.5 70.9 73.2 474 0.65 %17.5 78.8 70.7 518 0.53 %

3.7 Set-Up of Micro-Tensile Test System

The MEMS-based micro-tensile test system under development at NIST by Nicholas Barbosa III and

David T. Read is based on tensile test techniques developed for thin films [37, 38]. The overall set-up of the

micro-tensile test system is shown in Figure 3.9a, and a detailed schematic of the silicon frame is provided in

Figure 3.9b. The micro-tensile test apparatus is constructed around a probe station and consists of a silicon

frame mounted on an aluminum collet, a microscope above the silicon frame, and an aluminum support

structure with three inchworm motors, a pull rod, a load cell and a displacement sensor. The entire test

apparatus is attached to an optical table to minimize vibrations and other disturbances to the micro-tensile

test.

The silicon frame is clamped to a mounting stub, which is attached to a collet fixture with a set screw.

The collet fixture attaches to a plate on the probe station with a set screw. Figure 3.10 is a schematic of

the collet assembly. Two collet designs were used. The old collet design had one collet set screw and one

mounting stub set screw. Further, the collet was constructed of two intersecting parts. A new collet was

designed to increase stability in the system. The new design consists of one solid piece with three collet set

screws and two mounting stub set screws. The collet is aligned underneath the microscope such that the

specimen gage section can be imaged during the test.

The silicon test frame is fabricated through bulk micromachining of a silicon wafer. In Figure 3.9b,

the dark areas are the silicon frame and the light gray areas are where material was removed through bulk

micromachining. A suspended plate in the center of the frame is supported by four slender silicon beams

connected to the fixed outer portion of the frame. The pull rod fits in the opening of the plate to apply

force to the tensile specimen. The frame contains recesses that accept the tapered grips of the micro-tensile

specimens.

The load cell is constructed from two fixed 1.8 mm (0.05 in) spring steel beams and an inductive eddy

current displacement sensor with a range of 1 V and a resolution of 0.0001 V. The leaf spring fixtureconsisting of the spring steel beams separated by ceramic blocks can be interchanged depending on the force

capacity necessary for the tensile test. The pull rod is attached to the bottom of the leaf spring fixture with

two screws, and the load cell and eddy current displacement sensor are secured to an aluminum support

structure. The support structure actuated along three axes by inchworm motors with 0.05 m resolution.

The support structure with the load cell, pull rod, and displacement sensor is secured to a larger aluminum

support structure containing the inchworm motors that is aligned and screwed to the collet plate. As the

pull rod applies a force to the suspended plate in the frame, the spring steel beams in the load cell deflect

and the gap between the ceramic block and the eddy current displacement sensor changes. The change in

27

(a)

(b)

Figure 3.9 (a) A schematic illustrating the overall set-up of the micro-tensile testing system. Schematicprovided by Nicholas Barbosa III, NIST. (b) A schematic of the silicon micro-tensile test frame.The central plate is suspended by four slender silicon beams. There are fixed and mobile gripsetched in the silicon. The black circle indicates the place on the beam used as a starting pointfor the frame alignment.

28

Figure 3.10 A schematic of the collet assembly. The collet is mounted on a stub on the probe stationplate, and the mounting stub with the clamped silicon frame is attached to the collet.

the size of the gap causes a change in the voltage reading from the eddy current displacement sensor. The

load cell is calibrated by removing the support structure containing the load cell, pull rod, and displacement

sensor and mounting the support structure vertically on the optical table. Known masses are hung from the

pull rod and the corresponding voltages are recorded. The load cell calibration constant comes from a linear

fit to the voltage versus mass plot.

Alignment of the micro-tensile test system was performed in three steps, and it involved aligning the

microscope, camera, tensile frame and actuator to one another. Each alignment was performed using the

live view of the image capture software. In each step, a reference object was brought into focus and either

the microscope or the object was moved such that the reference object traveled in the field of view of the

image capture software. A box drawn onto the field of view was used to assess whether the reference object

was moving along either a horizontal or vertical axis. If the reference object moved off-axis, an alignment

adjustment was made to the system.

First, the camera was aligned to the microscope. The camera was screwed into a collet that attached

to the microscope, and the camera was turned 90 with respect to the microscope. A small round spot onthe frame was used as the reference object. The microscope was moved forward and back such that the small

round spot traveled left and right across the field of view. The camera collet was rotated until the small

round spot traveled straight along a horizontal line. The camera collet was tightened down, and the same

camera-to-microscope alignment was used for a set of sequential tests.

Second, the microscope-to-actuator alignment was checked. The actuator was precisely aligned with

the microscope during the construction of the micro-tensile system; thus, no adjustment should be necessary

29

for the microscope-to-actuator alignment prior to any of the tests. The bottom edge of the pull rod, close

to where the pull rod contacts the silicon frame, was brought into focus. The pull rod was moved in the

directions of tension and compression, which correspond to the forward and backward movement of the

microscope. The back edge of the pull rod was compared to a horizontal box drawn on the live camera

image to insure that it did not travel off-axis. The actuator and probe station positions were not altered,

but the alignment correction, if necessary, would be to rotate the entire probe station.

Third, the silicon frame was aligned to the microscope. One of the thin silicon beams was used as the

reference object. Starting at the mobile plate end of the beam as indicated by the black circle in Figure

3.9b, a rectangular box was drawn flush against one edge of the beam. Then, the microscope was moved

right or left such that the length of the beam traveled through the field of view to the end attached to the

fixed portion of the frame. The collet was rotated to bring the beam edge flush against the rectangular box.

The adjustment steps were repeated until the edge of the beam traveled flush against the drawn box for

the entire length of the beam, and then the collet set screw was tightened down. The frame alignment was

performed before every tensile test.

During the micro-tensile test, data were collected through two computers. One computer controlled the

camera attached to the microscope, and the other computer controlled the inchworm motors and recorded

the force and displacement measurements. The two computers were synchronized during the test, and the

image file name and corresponding data point number were recorded throughout the test.

3.8 Micro-Tensile Testing

After electrical resistance measurements were performed on a set of micro-tensile specimens, a notch was

cut with a focused ion beam (FIB) in each of the tabs attaching the micro-tensile specimens to the foil. The

specimens were carefully removed from the foil with fine-tipped tweezers. Then, the micro-tensile specimens

were spray-coated with carbon black to provide a contrast pattern for digital image correlation (DIC). The

carbon black was suspended in methanol for spray coating, and the particle size was approximately 1-5 m.

Once a specimen was ready for testing, it was placed into the grips of the silicon frame using fine-tipped

tweezers and a manual three-axis screw-driven micro-manipulator with a tungsten needle.

A custom-built load cell capable of reaching 8 N was used for all of the micro-tensile tests. During

the test, the actuator was ramped at a rate of 0.3 m/s, and the strain rate was approximately 4 104s1. Images of the gage section were acquired every three seconds. The 9.5 m and 17.5 m mean grainsize micro-tensile specimens were tested with an old collet design, and all of the specimens were pulled to

failure. The 3.1 m mean grain size micro-tensile specimens were tested with an improved collet design and

were pulled to approximately 10-30% strain. The collet design is discussed in Section 3.7. Multiple silicon

frames were used for the tests due to frame fracture. Most of the medium and coarse grain size specimens

were tested with a single frame, and the calibration from that frame was used for the data analysis of all

the tests of medium and coarse grain size micro-tensile specimens. Similarly, four out of five tests of the fine

grain size micro-tensile specimens were performed with the same frame, and the calibration for that frame

was used for the data analysis of all five tests.

The load cell calibration used for the data analysis of the 9.5 m and 17.5 m mean grain size micro-

tensile specimens was 4.364 N/V. The load cell calibration was performed after the leaf spring fixture was

30

constructed and was not performed again when the leaf spring fixture was re-installed on the test apparatus

for the 9.5 m and 17.5 m mean grain size micro-tensile tests. Prior to the 3.1 m mean grain size micro-

tensile tests the pull rod was removed from the load cell and re-attached. Post-testing, the load cell was

re-calibrated, and the new calibration value was 4.817 N/V. The impact of the two different calibration

values on the micro-tensile data is discussed in Section 4.2.1. The tightness of the screws in the load cell

and pull rod assembly could potentially have a significant impact on the load cell calibration due to the high

sensitivity of the inductive eddy current displacement sensor.

The force value from the micro-tensile test raw data is the total force applied to the frame and the

specimen. The design of the frame causes parallel loading of the frame and the specimen when force is applied

with the pull rod. The slender beams that suspend the mobile plate in the center of the silicon frame are

fixed at both ends. When the pull rod applies force to the mobile plate, the beams bend and allow the plate

to move. As the plate moves, force is applied to the micro-tensile specimen. After the specimen fractures,

there is load that remains applied to the frame that keeps the mobile plate in a displaced position. At the

end of the test, the frame must be unloaded. Thus, the difference between the total force and the force on

the frame gives the force on the specimen. The force on the frame was calculated with a frame calibration

and the displacement of the frame. The frame calibration was determined from frame unloading force versus

displacement curves. Since the frame was designed to bend under loading along the micro-tensile specimen

axis, a cubic expression was the best fit to the force versus displacement curve. The constants in the three

expressions were averaged for the frame calibration.

3.9 Digital Image Correlation

Displacement of the frame and the specimen during the micro-tensile test were measured with sub-

pixel DIC. The DIC software was written by David T. Read at NIST based on previous DIC development

work [39]. Each set of images to be analyzed was defined as a sequence. A subset image of size 80 x 80

pixels was selected in a region of interest on the first image of the sequence. The subset image was defined

such that high contrast and non-periodic shapes were captured in it. Such features were necessary for the

software to correctly determine motion vectors to track the movement of the subset image between images

in the sequence [39]. The reference subset image and a sequential subset image were matched at a sub-pixel

level using the cubic B-splines interpolation method. The fit that provided the minimum average difference

squared in gray values between the two subset images, as determined using the Newton-Raphson method,

was used to estimate the horizontal and vertical movement of the subset image. The horizontal movement of

the subset image parallel to the tensile direction was defined as the value u; similarly, the vertical movement

of the subset image perpendicular to the tensile direction was defined as the value v. Four subset images

were used to track the frame, and two were used to track the specimen. Specifically, subset images were

placed on the fixed upper grip, fixed lower grip and near the fixed end of the gage section. Corresponding

subset images were placed on the mobile upper grip, mobile lower grip and near the mobile end of the gage

section. Figure 3.11 shows an example of the placement of the subset images. In the image of the specimen,

the upper grips are the set of fixed and mobile grips that are above the specimen. Since the contrast of

the specimen changed during the test, the subset image was re-referenced every ten images in the image

sequence.

31

Figure 3.11 Optical micrograph of a micro-tensile specimen under elastic loading. The specimen wascoated with carbon black to create an irregular, high contrast pattern for the DIC. Theboxes show the approximate placement of the subset images.

The u and v displacements of the upper and lower grips of the frame and the gage section of the micro-

tensile specimen were calculated as the difference in the u and v values of the two corresponding subset

images for every image in the sequence:

du = ufixed umobile (3.6)

dv = vfixed vmobile (3.7)

To account for any slight movement in v or slight offset of the two corresponding subset images in v, the

displacement, dr, was calculated as

dr =R2x +R

2y + 2Rxdu+ 2Rydv + du

2 + dv2 |l| (3.8)

where l is the distance between the centers of the two subsets and Rx and Ry are the initial offsets of the

center of the mobile subset with respect to the center of the fixed subset. The schematic in Figure 3.12

shows how Rx and Ry were calculated. The frame displacement was calculated as the average of the upper

grips dr and the lower grips dr. The specimen strain was calculated as dr/l.

Commercial DIC software can efficiently analyze thousands of subset images. Cheng Liu at Los Alamos

National Laboratory performed DIC on a set of images from one of the 9.5 m mean grain diameter micro-

tensile tests with commercially available software from Correlated Solutions, Inc. The results are provided

in Appendix C.

3.10 Assessment of Measurement Uncertainty

Expressions for the combined standard uncertainties were derived using the law of propagation of

uncertainty with the assumption that all of the uncertainty components were uncorrelated. The terms in

the uncertainty expressions are defined for the micro-tensile tests. If the uncertainty expression also applies

to the conventional tensile tests, then the terms are defined again.

The combined standard error of the resistivity measured from the reference strips is given by

=

(A

l

)22R +

(R

l

)22A +

(RA

l2

)22l (3.9)

32

Figure 3.12 The values of Rx and Ry are equal to the offset of the mobile subset image from the fixedsubset image in the horizontal and vertical directions, respectively. Rx1, Rx2, Ry1 and Ry2are the pixel coordinates of the subset image at the fixed end of the image. Similarly, Rx3,Rx4, Ry3 and Ry4 are the pixel coordinates of the subset image at the mobile end of theimage.

where l is the average length between probe tips, R is the measured resistance, A is the average cross-sectional

area of the strip, l is the standard deviation of the length measurements, R is the standard error of the

slope fit to the voltage and current data, and A is the standard deviation of the area measurements.

The combined standard error of the micro-tensile cross-sectional area based on electrical resistance is

given by

A,er =

(l

R

)22 +

( R

)22l +

(l

R2

)22R (3.10)

where l is the average length between probe tips, R is the measured resistance, is the resistivity from the

reference strip, l is the standard deviation of the length measurements, R is the standard error of the

slope fit to the voltage and current data, and is the combined standard error of the resistivity.

The combined standard error of the micro-tensile cross-sectional area from micrograph measurements

is given by

A,m =

(tSEM )22w,optical + (woptical)

22t,SEM (3.11)

where tSEM is the average thickness measured from an SEM micrograph, woptical is the average width

measured from an optical micrograph, w,optical is a combination of the standard deviation of the width

measurements and the resolution of the micrograph, and t,SEM is a combination of the standard deviation

of the thickness measurements and the resolution of the micrograph. The form of equation 3.11 can also be

used to determine the combined standard error of the conventional tensile cross-sectional area. The thickness

and width terms are the average thickness and width measurements, and w and t are a combination of

the standard deviation of the measurements and the resolution of the measuring instrument.

The combined standard error of the load applied to the specimen is given by

P,specimen =2P,raw +

2P,frame (3.12)

33

where P,raw is the standard error of the slope fit to force and voltage calibration data and P,frame is the

sum of the standard deviations of the constants from cubic fit curves to the frame calibration data.

The combined standard error of stress is given by

=

(1

A

)22P +

(P

A2

)22A (3.13)

where A is the specimen cross-sectional area, P is the load on the specimen, P is the combined standard

error of the load applied to the specimen, and A is the combined standard error of the micro-tensile cross-

sectional area. P was selected as the load corresponding to the 0.2% offset yield strength. The uncertainty

in stress was also calculated for the 20% offset flow stress. The 0.2% offset yield strength and flow stresses

up to a 20% strain offset were used to compare the mechanical behavior between the micro-tensile and

conventional tensile specimens. All of the micro-tensile specimens with the exception of one of the finest

grain size specimens were pulled to approximately 30% strain or to fracture. For applying Equation 3.13 to

the conventional tensile tests, P is the uncertainty in the load cell calibration and A, P , and A are defined

the same as for the micro-tensile tests.

The combined standard error of strain is given by

=

(1

l

)22dr +

(dr

l 1l

)22l (3.14)

where l is the gage length in pixels, dr/l is the strain, and l is the length in microns of one pixel. The value

of dr was determined from images taken of a static specimen under no load, and the images were acquired

over a similar amount of time and at the same rate as images taken during the micro-tensile test from initial

loading of the specimen up to the initial yield point. The uncertainty in strain was calculated for the strains

corresponding to the 0.2% offset yield strength and the 20% offset flow stress. For applying Equation 3.14

to the conventional tensile tests, l is the gage length in millimeters and l is the resolution of the calipers

used to measure the gage length. The term dr for the micro-tensile specimens is equivalent to l for the

conventional tensile specimens. Therefore, l/l is the strain and l is the resolution of the extensometer.

Table 3.6 summarizes the calculated uncertainties from all fifteen micro-tensile specimens. The uncer-

tainty in stress for each specimen was calculated as ,er for the area from electrical resistance measurements

and as ,m for the area from micrograph measurements. The high ,er values are the result of the high

A,er values, which are in turn due to the high values of . A large variation in the cross-sectional area of

the reference strip is responsible for the high uncertainty in the resistivity. A precisely machined reference

strip could reduce the resistivity uncertainty in future work. By comparison, the ,m values are low because

the low standard deviations of the width and thickness measurements cause low A,m values. Comparing

the uncertainty in stress at the 0.2% offset yield strength and at the 20% offset flow stress, the high values

of A,er cause a substantial increase in ,er with increasing load, whereas, ,m is much more consistent

between the two loads due to the low values of A,m. Thus, low uncertainty in the area measurement is

critical for reducing the uncertainty in stress. The low uncertainty associated with the cross-sectional area

calculated from micrograph measurements reflects the uniform appearance of the micro-tensile specimens,

but the accuracy of the area calculation is limited by the obscured edges of the specimens. Since it was

difficult to judge the location of the edges in the micrographs, the measurements could be off by several

34

microns. For example, if a specimen had real gage section dimensions of 75 x 80 m and the dimensions

were measured as 77 x 83 m, then the percent difference in the cross-sectional area would be 6.5%. This

difference would translate directly to a 6.5% difference in the calculated stress values from the tensile test.

If the micro-tensile specimens had sharp, visible edges and 90 corners post-machining, then the micrographthickness and width measurements would have much higher accuracy.

The calculated uncertainties from all nineteen conventional tensile specimens are also provided in

Table 3.6. The uncertainty in the stress is low for the thicker tensile specimens, but it appears to be

relatively high for the thinner tensile specimens. The uncertainty in the cross-sectional area is low for both

sheet thicknesses, but the cross-sectional area of the thin tensile specimen is substantially smaller than that

of the thicker specimen. A small cross-sectional area results in a high uncertainty in the stress according to

Equation 3.13. The cross-sectional area term in Equation 3.13 has the greatest effect on the stress uncertainty

for the conventional tensile specimens.

Table 3.6 Summary of the combined standard uncertainties for the conventional and micro-tensile tests.The uncertainties in stress and strain were calculated for the 0.2% offset yield strength (0.2% YS) and the20% offset flow stress (20% FS). The uncertainties for the conventional tensile tests are provided separatelyfor the thin material used to obtain the 3.1 and 4.8 m mean grain diameters and the thicker material used

to obtain the 9.5, 17.5, and 54.2 m mean grain diameters.

Combined Standard Uncertainty Mean Range

Mic

ro-T

ensi

leT

ests

0.0782 m 0.0640 - 0.0875 mA,er 574 m

2 512 - 635 m2

A,m 44 m2 22 - 68 m2

P,specimen 0.0182 N 0.0143 - 0.0202 N,er at 0.2% YS 25.7 MPa 22.3 - 31.1 MPa,m at 0.2% YS 4.2 MPa 3.4 - 5.6 MPa,er at 20% FS 49.5 MPa 45.7 - 54.5 MPa,m at 20% FS 5.9 MPa 4.2 - 8.3 MPa at 0.2% YS 8.1 105 6.8 105 - 9.9 105 at 20% FS 2.5 104 2.1 104 - 3.0 104

Con

venti

onal

Ten

sile

Tes

ts

Th

ick

(0.7

5m

m)

A 0.0638 mm2 0.0637 - 0.0639 mm2

at 0.2% YS 5.0 MPa 4.9 - 5.2 MPa at 20% FS 5.9 MPa 5.8 - 6.2 MPa at 0.2% YS 2.0 105 - at 20% FS 2.0 105 -

Th

in(0

.13

mm

)

A 0.0634 mm2 -

at 0.2% YS 30.6 MPa 29.8 - 32.3 MPa at 20% FS 36.4 MPa 35.3 - 38.3 MPa at 0.2% YS 3.7 105 2.0 105 - 4.0 105 at 20% FS 3.7 105 2.0 105 - 4.0 105

35

CHAPTER 4

RESULTS AND DISCUSSION

4.1 Heat Treatments and Grain Size Analysis

Heat treatment experiments were performed on coupons of 316L stainless steel sheet with 40% and 80%

cold reduction to obtain a wide range of grain sizes for the tensile tests. The potential impact of the grain

size distribution on the micro-tensile specimen results was examined using the spread and the upper tail of

the distribution. From the experiments, heat treatments were selected for the conventional and micro-tensile

specimen conditions.

4.1.1 Grain Growth Heat Treatment Experiments

The mean grain diameter (d) results from the grain growth heat treatment experiments are summarized

in Figure 4.1a, and example micrographs from three of the heat treatments are provided in Figures 4.1b,

4.1c, and 4.1d. For all of the heat treatments, the grains appeared to be equiaxed and to have coarsened

with a unimodal distribution.

Histograms of the individual grain diameter measurements for the three heat treatments selected for

the micro-tensile specimen materials are given Figure 4.2. The distribution of individual grain diameters of

a single phase material is approximately log-normal [30]. The distributions of the log transformed individual

grain diameter data are also provided in Figure 4.2. The right tail of the untransformed distribution provides

an estimate of the largest grains. For example, only 0.6% of the grains are larger than 10 m in the

distribution in Figure 4.2a for the 3.5 m mean grain diameter material. The largest grains from this heat

treatment should not have an adverse effect on the micro-tensile specimen behavior, because the largest

grains are a very small percentage of the sample and are several times smaller than the specimen width or

thickness. In the distribution in Figure 4.2b for the 9.1 m mean grain diameter, 1.8% of the grains are

larger than 20 m. The largest measured grains were 20-25 m in diameter, which is approximately three to

four times smaller than the micro-tensile specimen width and thickness. While these grains are closer to the

size of the specimen dimensions, they are expected to be a small percentage of the grains that comprise the

micro-tensile specimen. They may have a slight impact on the scatter in the mechanical properties depending

on their crystallographic orientation and the number of large grains in each micro-tensile specimen. In the

distribution in Figure 4.2c for the 16.8 m mean grain diameter, 5.8% of the grains are larger than 30 m

with a maximum measured grain diameter of 54.5 m. It may be possible that one or more grains in the

gage section of a micro-tensile specimen could span at least half of the width or thickness of the specimen.

If a single large grain comprises a substantial portion of the micro-tensile specimen cross section, it could

have a significant impact on the overall mechanical behavior of the micro-tensile specimen. The presence of

very large grains in these specimens could contribute to the amount of scatter in the mechanical properties.

36

(a) (b)

(c) (d)

Figure 4.1 (a) Log-log plot of the mean grain diameters obtained from heat treatments of 40% and 85%cold rolled (CR) 316L stainless steel sheet. The time represents the total time for recrystalliza-tion and grain growth. Example micrographs are provided for the following heat treatments:(b) 85% CR, 950 C, 3 min, (c) 40% CR, 950 C, 20 min, and (d) 40% CR, 1000 C, 190min. The mean grain diameter, d, is labeled on the micrograph. The plane of the micrographswith respect to the rolling direction is indicated with the coordinates above (b). Electrolyticetch with 10% (w/v) oxalic acid at 1.8 V for approximately 2 minutes. The dark spots in themicrographs are light pitting from the electrolytic etch.

37

(a) (b) (c)

(d) (e) (f)

Figure 4.2 Distribution of grain diameters from analySIS software measurements of individual grain areas from micrographs of heat treated 316Lstainless steel. Mean grain diameters for the heat treatments are (a) 3.5 m, (b) 9.1 m, and (c) 16.8 m. These heat treatments wereselected for the three micro-tensile specimen grain size conditions. Distributions of the log transformed grain diameter data for thesame heat treatments are given in (d), (e), and (f).

38

The spread in the grain size distribution was further quantified to assess its possible influence on

micro-tensile mechanical behavior. Simply taking the standard deviation of the data does not appropriately

describe the spread in a log-normal distribution. The log standard deviation can be calculated, but it is

not straightforward to interpret this value in terms of actual grain sizes because it is the standard deviation

of a transformed distribution. The log mean and log standard deviation from a transformation of the

individual grain size distribution for each heat treatment are provided in Table 4.1. The log standard

deviation is the same for nearly all of the heat treatments, which indicates that the apparent spread in the

untransformed distributions plotted in Figures 4.2a, 4.2b, and 4.2c scales with mean grain size. The spread

in the untransformed data sets was examined through relative comparisons of the interquartile ranges. The

interquartile range (IQR) encompasses the middle 50% of the data and is given by

IQR = Q3 Q1 (4.1)

where Q3 is the 75th quantile or third quartile of the data and Q1 is the 25th quantile or first quartile of

the data. The IQR values for the experimental heat treatments are given in Table 4.1. The magnitude of

the IQR approximately scales with the mean grain diameter, which suggests that the apparent spread in the

grain size distribution increases with grain growth. In terms of the micro-tensile specimens, it is important

that Q1 and Q3 are close to the mean grain diameter, because the number of grains in the specimen cross-

section was estimated based on a simple grain model of equal size squares. Although the values of dQ1 andQ3 d increase with increasing d, the values of Q1/d and Q3/d are very similar for the three heat treatmentsselected for the micro-tensile specimen materials in agreement with the log standard deviations. The value

of Q1/d is in the range of 0.6-0.7, and the value of Q3/d is approximately 1.3. Since the relative values of

Q1 and Q3 are approximately the same for the three heat treatment conditions, the number of grains in the

specimen cross-section should scale proportionally with the average grain size.

Table 4.1 Summary of the grain sizes obtained through different heat treatments of cold rolled (CR) 316Lstainless steel sheet. The heat treatments selected for the fine, medium, and coarse micro-tensile grain sizesare in bold. The term Ntotal is the total number of measured grains. Log mean and log standard deviation(sd) are the mean and standard deviation of the log-transformed grain size data. The individual grain sizes

measured from micrographs follow an approximately log-normal distribution.

CR T ( C) t (min) d (m) Log Mean( m)

Log Sd( m)

Q1(m)

Q3(m)

IQR(m)

Ntotal

85% 950 3 3.5 1.1 0.5 2.2 4.4 2.2 101385% 950 10 6.1 1.7 0.5 3.9 7.8 3.9 161640% 950 20 9.1 2.1 0.5 5.8 11.6 5.8 75940% 950 40 12.1 2.4 0.5 7.4 15.9 8.5 39240% 950 70 14.0 2.5 0.5 9.3 18.3 9.0 31340% 950 110 16.8 2.7 0.5 11.3 21.5 10.2 122140% 1000 70 32.3 3.3 0.6 20.1 41.8 21.7 27140% 1000 190 53.1 3.9 0.5 34.6 67.4 32.9 1285

4.1.2 Sheet Grain Sizes for Tensile Testing

The mean grain diameters measured from the two sheet metallography specimens for each of the five

conditions for conventional tensile testing are provided in Table 4.2. The three conditions for which micro-

39

Table 4.2 Summary of 316L stainless steel sheet heat treatments. Conventional tensile testing wasperformed for all five conditions. The three conditions for which micro-tensile testing was performed are in

bold. The sheet mean grain diameter is the average of the mean grain diameter of the sheet from twodifferent locations as indicated in Figure 3.1.

CR T ( C) t (min) Back Edged (m)

Front Edge d(m)

PercentDifference

Sheet d(m)

85% 950 3 3.5 2.8 22% 3.185% 950 10 5.8 3.8 40% 4.840% 950 20 10.0 8.9 11% 9.540% 950 110 18.2 16.8 8% 17.540% 1000 190 55.3 53.1 4% 54.2

tensile testing was also performed are in bold in Table 4.2. The label front edge refers to the metallography

sample taken from the sheared off material and the label back edge refers to the metallography sample

taken from the opposite end of the sheet as indicated in Figure 3.1. The mean grain diameter of the entire

sheet was taken as the average of the mean grain diameters from the two metallography specimens. The

measured grain size from the back edge is systematically greater than the grain size measured from the

front edge, as shown by the percent difference in Table 4.2. The percent difference between the two mean

grain diameters was calculated as ((d, back d, front)/((d, back + d, front) 0.5)) 100. The back edge ofthe sheet was near the back of the furnace and the front edge was near the front close to the door, so a small

temperature gradient could be responsible for the difference in mean grain diameters. The percent difference

generally decreases with increasing hold time, which could be the result of the temperature gradient in the

furnace decreasing over time and the hold temperature stabilizing. The micro-tensile specimens came from a

sample of material close to the front edge metallography specimen. Thus, the mean grain diameter from the

front edge specimen is a more accurate estimate of the mean grain diameter in the micro-tensile specimens

than the sheet mean grain diameter. Since the gage sections of the conventional tensile specimens correspond

to the center of the sheet, the sheet mean grain diameter is a reasonable estimate of the mean grain diameter

in the conventional tensile specimens. In the grain size strengthening analysis presented in Sections 4.2.1

and 4.2.2, the front edge d will be used for the micro-tensile specimens, and the sheet d will be used for the

conventional tensile specimens.

4.2 Conventional Tensile and Micro-Tensile Tests

The stress-strain curves from the micro-tensile and conventional tensile tests for the 3.1 m, 9.5 m,

and 17.5 m mean grain diameter conditions are provided in Figures 4.3, 4.4, and 4.5. The stress-strain

curves for the 9.5 m and 17.5 m mean grain diameter conditions were constructed with an old load cell

calibration and the curves for the 3.1 m mean grain diameter condition were constructed from a load cell

calibration performed after the completion of the micro-tensile tests as discussed in Section 3.8. The yielding

behavior and strain hardening rate differ between the micro-tensile and conventional tensile specimens for

all three grain sizes that were tested. A plot of strain hardening rate versus true strain is also included in

Figures 4.3, 4.4, and 4.5. The stress-strain curves from the conventional tensile tests for the 4.8 m and

54.2 m mean grain diameter conditions are provided in Figure 4.6.

Many of the conventional tensile tests display yield points in the stress-strain curves, which is not

typically observed with 316L stainless steel [4045]. A yield point did occur in some 316L stainless steel

40

(a) (b)

(c) (d)

Figure 4.3 (a) Engineering stress-strain curves focused on yielding behavior from the four conventionaland five micro-tensile tests for the sheet d = 3.1 m. The micro-tensile specimen d may becloser to 2.8 m due to a slight variation in the mean grain diameter of the sheet. (b) The entireengineering stress-strain curves. Four of the micro-tensile tests were stopped at approximately30% strain, and one test was stopped prior to 10% strain. (c) True stress-strain curves. (d)Instantaneous strain hardening rate as a function of true strain. The strain hardening rateswere calculated from the true stress-strain data.

41

(a) (b)

(c) (d)

Figure 4.4 (a) Engineering stress-strain curves focused on yielding behavior from the four conventionaland five micro-tensile tests for the sheet d = 9.5 m. The micro-tensile specimen d may becloser to 8.9 m due to a slight variation in the mean grain diameter of the sheet. (b) The entireengineering stress-strain curves. The conventional tensile tests were stopped at approximately30% strain, and the extensometer was re-positioned to capture the full elongation of thespecimen. (c) True stress-strain curves. (d) Instantaneous strain hardening rate as a functionof true strain. The strain hardening rates we