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THESIS

This is to certify that the

thesis entitled

EXPERIMENTAL VERIFICATION OF THE
NEUBER RELATION AT ROOM AND
ELEVATED TEMPERATURES

presented by
Lonnie J. Lucas

has been accepted towards fulfillment
of the requirements for

Master's degreein Mechanics

 

 

 

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6 #74 ‘07

EXPERIMENTAL VERIFICATION OF THE NEUBER RELATION
AT ROOM AND ELEVATED TEMPERATURES

BY

Lonnie J. Lucas
A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Metallurgy, Mechanics and Materials Science

1982

ABSTRACT

EXPERIMENTAL VERIFICATION OF THE NEUBER RELATION
AT ROOM AND ELEVATED TEMPERATURES

BY

Lonnie J. Lucas

The accuracy of the Neuber equation at room temperature
and l,200°F was experimentally determined under cyclic load
conditions with hold times. All strains were measured with
an interferometric technique at both the local and remote
regions of notched specimens. At room temperature, strains
were obtained for the initial response at one load level and
for cyclically stable conditions at four load.Levels. Stress-
es in notched members were simulated by subjecting smooth
specimens to the same strains as were recorded on the notch-
ed specimen. Local stress-strain response was then predict-
ed with excellent accuracy by subjecting a smooth specimen
to limits established by the Neuber Equation. Data at
l,200°F were obtained with the same experimental techniques
but only in the cyclically stable conditions. The Neuber
prediction at this temperature gave relatively accurate re-
sults in terms of predicting stress and strain points.
However, predicted interaction of the creep and stress
relaxation behavior differed from experimentally measured

values.

ACKNOWLEDGMENTS

This thesis project was funded by the National
Aeronautics and Space Administration.

I would like to thank my wife, Beth, for her help
during the period of this work. I would also like to
thank Shari Sawdey and Michelle Ward for their help in
preparing the figures and typing. A special thank you
goes to Dr. John Martin who is an excellent teacher,

advisor and friend.

TABLE OF CONTENTS

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

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

Chapter 1
Chapter 2
2.1
2.2
2.3
2.4
Chapter 3
Chapter 4
4.1

4.2

4.3

4.4
Chapter 5
5.1
5.2
5.3
5.4

INTRODUCTION . . . . . . . . . . . . .
INTERFEROMETRIC STRAIN GAGE . . . . . .
Fundamentals of the I.S.G. . . . . . . .
Hardware and Software for the I.S.G. . .
The I.S.G. at Elevated Temperatures . .
Comparison Test . . . . . . . . . . . .
SAMPLES AND MATERIALS . . . . . . . . .
EXPERIMENTAL METHODS . . . . . . . . .
Strain Measurement in Notched Specimens

Simulation of Stresses in Notched
SpeCimns O O O O O O O O O O O O O O 0

Elevated Temperature Tests . . . . . . .

4.3.1 Elevated Temperature Stress
Simulation . . . . . . . . . . .

Neuber Prediction of Notch Root Behavior
EXPERIMENTAL RESULTS AND DISCUSSION . .
Room Temperature I.S.G. Measurements . .
Room Temperature Stress Simulation . . .
Room Temperature Neuber Prediction . . .

High Temperature I.S.G. Strain
Measurements . . . . . . . . . . . . .

ii

Page

iv

14
23
25
33
33

37
39

41
45
52
52
60

66

70

5.5 High Temperature Stress Simulation

5.6 High Temperature Neuber Prediction

Chapter 6 CONCLUSIONS .

LIST OF REFERENCES

iii

Page
78
84
89

90

LIST OF TABLES
Table Page

1 Material Properties . . . . . . . . . . . . . 26

iv

Figure

10

11

12
l3

14

15
16
17
18
19
20

LIST OF FIGURES

Fringe Pattern Generation Principles . . . .
Interference Patterns . . . . . . . . . . .
Orientation of Fringe Patterns . . . . . . .
Vickers Hardness Tester . . . . . . . . . .
Negative Replica of Indentations . . . . . .
Schematic of I.S.G. . . . . . . . . . . . .
Interferometric Strain Gage . . . . . . . .
Interferometric Technique at 1,200°F . . . .
Induction Heating Coils . . . . . . . . . .
Black-body Radiation Versus Wavelength . . .

Fringe Pattern Intensity Signals at 70°F
and l'OOOOF o o o o o o o o o o o o o o o o

Fringe Pattern Intensity Signals at l,lOO°F .

Fringe Intensity Before and After Increasing
PMT Gain 0 O O O O I O O O O O O O O O O 0

Comparison of I.S.G. and Clip-on Gage Strain
output 0 O O O O I O O O O O O O O O O O 0

Smooth Axial Specimen . . . . . . . . . . .
Hourglass Specimen . . . . . . . . . . . . .
Notched Specimen Geometry . . . . . . . . .
Notched Specimen . . . . . . . . . . . . . .
Determination of Strain Profile . . . . . .

Test set Up 0 O O O O I O O O O O O O O O O

Page

10
11
12
13‘
16
17

19

20

21

22

24
27
28
29
30
32

34

Figure
21
22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

Load Pattern and Recording Technique . . .
Spotwelding Thermocouple to Specimen . . .

Diametral Extensometer and Hourglass
Specimen . . . . . . . . . . . . . . . .

Analog Circuit for Transverse to Axial
Strain Conversion . . . . . . . . . . . .

Construction of Neuber Versus Time Graph .

The Neuber Relation is Defined for Each
Seperate Reversal . . . . . . . . . . . .

Analog Circuit for Neuber Prediction . . .

Room Temperature I.S.G. Strain Measurement
versus Load 0 O O O O C I O O O O O O O 0

Load, Notch Root Strain, and Remote Strain
Versus Time . . . . . . . . . . . . . . .

Load Level 1 Versus Strain Across Notched
SpeCimen O O O O O O O O O O O O O O O 0

Load Level 2 Versus Strain Across Notched
SpeCimen O 0 O O O O O O O O O O O O O 0

Load Level 3 Versus Strain Across Notched
SpeCimen O I C O O O O I O C O O O O O 0

Load Level 4 Versus Strain Across Notched
Specimen . . . . . . . . . . . . . . . .

Smooth Specimen Simulation of Initial
Stresses . . . . . . . . . . . . . . . .

Strain and Simulated Stress for Remote
Region 0 O O O O O O O I O O O O O O O 0

Smooth Specimen Simulation of Stabilized
Stresses . . . . . . . . . . . . . . . .

Strain and Simulated Stress for Local
Region I O O O O O O I I O O O O O O O O

Neuber Prediction Curves for Initial Local
BehaVior O O I I O O O O O O O O O O O 0

vi

Page
36

40

42

44

46

48

51

54_

55

56

57

58

59

61

63

64

65

67

Figure

39

4O

41
42
43
44
45
46
47
48

49

50

51

52

53
54

55

Neuber Prediction and Stress Simulation of
Initial Local Behavior . . . . . . . . . . .

Neuber Prediction and Stress Simulation of
Stabilized Local Behavior . . . . . . . . .

Load Versus Local Strain at 1,200°F . . . . .
Load Versus Local Strain at 1,200°F . . . . .
Load and Local Strain Versus Time at 1,200°F.
Load and Local Strain Versus Time at 1,200°F.
Load and Local Strain Versus Time at 1,200°F.
Load and Local Strain Versus Time at 1,200°F.
Load Versus Remote Strain at 1,200°F . . . .

Local Simulated Stress Versus Strain at
1,200°F O O O O O O O O O O O O O O O O O 0

Local Simulated Stress Versus Strain at
1,200°F O O O C I O O O O O O O O O O O O 0

Local Strain and Simulated Stress at 1,200°F
Level 3 O O O O C O O C O O O O O O O O O 0

Remote Simulated Stress Versus Strain at
1,200°F O O O O O O O O O O O O O O O O O 0

Remote Strain and Simulated Stress at 1,200°F
Level 3 O O C O I O O O O O O O O O O O I O

Neuber Prediction Curves for Local Level 3 .

Neuber Prediction and Stress Simulation at
1,200°F I O I O O O O O O O I O O O O O O O

Neuber Prediction and Stress Simulation at
l'zoooF O O C O O O O O 0 O O O O O O O O 0

vii

Page

68

69
71
72
73
74
75
76
77

79

80

81

82

83

87

88

CHAPTER 1

INTRODUCTION

There has been a demand in recent years for the air-
craft industry to provide a more energy efficient turbine
propulsion system. Part of this task involves trying to
understand the limitations of the current materials and
structures being used, especially in the 'hot section' of
the engine (l)*. The hot section components include the
turbine blades, vanes, and combustors which operate under
severe stresses and temperatures. To make improvements in
these parts it is first necessary to compile test data which
describe the events leading up to failure. Theoretical mod-
els can then be developed and compared with experimental
data until the failure modes and component lives may be
predicted.

The combustor, fabricated from the alloy Hastelloy X,
is one component which has gone through the initial testing
phase and is now being examined from a theoretical stand-
point. Failures in the combustor linem'Ihave been attribut-

ed to thermal-mechanical fatigue which causes cracking and

 

*Numbers in parenthesis refer to references listed in the
reference table. Numbers in brackets refer to equations.

buckling (2). A number of constitutive theories have been
proposed for predicting the nonlinear stress-strain behavior
near holes which serve as cracking sites in the liner (3).
When these theories are incorporated into finite element
codes, the final package becomes very complex and requires
a large computer facility.

The purpose of this study is; to examine a more basic
theory, namely the Neuber relation, to see how well it
can predict local stress-strain behavior in notched speci-
mens of Hastelloy X. For cyclic loading the Neuber equation

is written,
(Ac) (As) = (Kt')2 (AS) (Ae) [11

where: A0 and A5 are the notch root stress and strain ranges,
respectively;
AS and Ae are the remote stress and strain ranges,
respectively;
th.is the elastic stress concentration factor.

Much of the work involving Neuber's relation has fo-
cused on stress redistribution near a notch (4) and the
accompanying variation in the stress and strain concentra-
tion factors throughout fatigue life (5,6,7). One of these
researchers, Guillot (6), evaluated Neuber's equation at
moderately elevated temperatures (500°F) and found that
conservative results were obtained for life predictions in

1018 steel and 7475 aluminum. Both Bofferding (5) and

Guillot (6) used an Interferometric Strain Gage (I.S.G.)
(8-11) to measure notch root strains.

Equation [1] by itself is indeterminate. Knowing the
remote stress or strain range leaves three unknowns. The
relationship between stress and strain at both the remote
and local locations is needed. Crews and Hardrath (12)
assumed that the notch stress could be found by reproduc-
ing measured notch strains in smooth samples. This assump-
tion was upheld by Stadnick (13) and other researchers
(14,15) who showed that the smooth specimen simulation
gave good results in predicting fatigue lives of notched
specimens. For this study it was assumed that smooth speci-
mens could be used to supply the needed stress-strain re-
lationship.

Stadnick and Morrow (16) worked on automating the
techniques for performing tests on smooth specimens that
were controlled according to the Neuber Equation. They
evaluated various approaches for subjecting a smooth speci-
men to the same stresses and strains which would exist at
a notch. These methods consisted of manual control, and
analog or digital computer control of the Neuber parameters.

Separate research efforts have been devoted to using
smooth specimens to simulate notch root response, develop-
ing laser based measurement devices and establishing high
temperature testing techniques. This study utilized all
of these tools to determine the accuracy of Neuber's equa-
tion for cyclic loading of notched specimens at temperatures

up to 1,200°F.

CHAPTER 2

INTERFEROMETRIC STRAIN GAGE

2.1 Fundamentals of the I.S.G.

 

The Interferometric Strain Gage is a noncontacting
laser device capable of measuring strains Over a very short
gage length (SO-100 microns). Figure 1 illustrates the
fundamental principles upon which the I.S.G. is designed.
This cut—away view shows two surface indentations which
form the gage length on a specimen. Parallel rays reflect-
ing off the indentations have a path difference of d sina,
where a is the angle between the normal incident laser beam
and the light rays of interest. When the following relation

is satisfied,

d sino = mA (m = 0, i l, i 2,...) [l]
where: A = wavelength of laser light

the laser rays will interfere constructively to form bright
interference fringes such as those in Figure 2. Each bright
fringe is defined by an integer, m, from Equation [1]. The
orientation of the fringe patterns with respect to the laser

is shown in Figure 3.

Incident Laser Beam

 

 

 

 

 

 

 

 

 

FIGURE 1 FRINGE PATTERN GENERATION
PRINCIPLES

 

 

FIGURE 2 INTERFERENCE PATTERNS

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When a tensile load is applied to the specimen, the
indentations will move apart a distance 6d. Since mA is a
constant for a given fringe, a must decrease for tensile
loads which means that the fringes will move toward the
laser beam. For compressive loads, the spacing d decreases,
causing an increase in a and a movement of the fringes away
from the incident beam. To monitor this motion, a fixed
observation point at angle a0 is chosen. The relationship

between fringe motion and displacement is defined as,

A

sin a
o

 

5d = ( ) 6m [2]

Where: 5m is the fraction of fringes passing the

observation point.

Letting 6m = l and substituting typical values into Equation
[2].

6328 x 10"10 6

_ m 4 - _ .
6d - ( sin 12v. ) - l x 10 m - l m1cron

 

By knowing the initial spacing between indentations, do,

the strain can be calculated,

1 micron
100 microns

 

Ez-grg-z =1% [3]
O

Fringe motion can also be caused by rigid body motion which
occurs in most loading schemes. However, these effects are
cancelled out since one pattern of fringes moves toward the
incident laser while the other pattern moves away. By using

the following equation,

€=§—g= A (l 2)
do (dOT Sln a0 2 [4]

 

 

Where: Sml and 6m2 are the upper and lower fringe

patterns.

the relative displacement of the specimen can be averaged
out.

A Vicker's hardness tester with a pyramidal diamond
(Figure 4) was used to form the indentations. Figure 5
shows scanning electron micrographs of a negative replica
from a notched specimen. The upper photo shows two inden-
tations which are 50 microns from the edge of a 0.2 inch
diameter hole. The lower photo shows an enlarged view of
the replica. The indentations are generally 25 microns on

a side and 5 microns deep.

2.2 Hardware and Software for the ISG

 

A schematic diagram of the ISG is shown in Figure 6.
Figure 7 shows the actual components of the ISG. Fringe
patterns impinge upon the two servo controlled mirrors posi-
tioned at angle a0 and are then reflected onto a pair of
photomultiplier tubes (PMT's). A cover with a narrow slit
is fitted over the face of each PMT so that only part of
one bright fringe may shine through. A bright band of con-
structive interference is accompanied by a high voltage out-
put from the PMT.

During operation of the system a D/A converter outputs

a ramp function to the servo mirrors. This causes the

10

 

FIGURE 4 VICKERS HARDNESS TESTER

11

Edge of Hole
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FIGURE 5 NEGATIVE REPLICA OF INDENTATIONS

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mirrors to rotate through a small angle, sweeping several
fringes across the slits of the PMT's. A minicomputer di-
vides each sweep into 256 mirror positions and stores an
accompanying PMT voltage for each position. In this manner,
the computer can locate and store the mirror positions which
accompany each maximum (bright) and minimum (dark) fringe
band. Applied loads to the specimen cause the fringes to
move with respect to do and therefore the mirror positions
recorded for maximum and minimum fringe bands will also
change.

Every 100 milliseconds the mirrors complete a sweep and
the change in fringe position is calculated. This allows
the analog strain voltage to be determined using Equation
[4]. The strain output is continuously available as in
standard extensometers but because of the calculation time
involved, there are limitations on the strain rates for
which data may be obtained. For strain ranges near 0.5% the
limit on cyclic frequency is about 0.1 Hz. This speed is
quite sufficient for most applications. A more detailed

discussion of the I.S.G. may be found in Reference (5).

2.3 The I.S.G. at Elevated Temperatures

The interferometric strain gage offers several advan-
tages over conventional techniques. It operates over a
small gage length to obtain strain information at critical
locations. It is also noncontacting, which allows it to

hue used in hostile environments such as those found in high

15

temperature fatigue applications. Three requirements must
be met in order to utilize this gage. One is that the path
of the incoming laser beam and rays reflected back by the
specimen must not be obstructed. Another is that the re-
flective ability of the specimen must be maintained. The
final requirement is that blackbody radiation emitted from
the specimen not overshadow the interference fringes.

The problem of obstruction of light rays depends upon
the method of heating the specimen. If an enclosed furnace
is used, optical ports for the incident beam and exiting
fringes must be provided. Quartz windows with melting
temperatures near 4,000°F have been used successfully (l7) .
For this experiment, high frequency induction heating was
employed. The coils were designed to allow fringes to exit
the specimen surface. Figure 8 shows the I.S.G. during a
test conducted at14200°Fh The plexiglass enclosure has a
2 inch wide slit for passing of light rays to and from the
specimen. Figure 9 is a closer View of the heating coil and
notched specimen.

Since the indentations essentially form the "gage" for
the system, their sides must be smooth and reflective. For
most metals at high temperature, oxidation begins to break
down the integrity of the surface finish. There are three
‘ways of solving the oxidation problem:

1. Protect the specimen in an inert atmosphere.

2. Attach non-oxidizing tabs to the specimen and place

the indentations on them.

l6

 

FIGURE 8 INTERFEROMETRIC TECHNIQUE AT 1,200°F

l7

 

FIGURE 9 INDUCTION HEATING COILs

18

3. Coat the specimen with a non-oxidizing material.

The latter method was used to retard oxidation in
Hastelloy X specimens. A vacuum evaporator was used to de-
posit 0.14 microns of 40% gold — 60% paladium at 2 x 10-5
torr. For purposes of this study it was concluded that the
thin coating would have no significant effect on stress~
strain behavior. It should be noted that the coating was
applied after indenting the specimen.

Figure 10 is a plot of black-body radiation at various
temperatures (17). As temperatures increase, the radiation
intensity increases while the wavelength decreases into the
region of visible light (.4 to .7 microns). Wavelengths
of two common lasers are also plotted in Figure 10. The
He-Ne laser (used in this experiment) has a wavelength of
0.633 microns while the argon lasers have more powerful
lines at 0.514 microns and 0.488 microns. The best choice
for high temperature applications of the I.S.G. would clearly
be the argon type laser.

Figures 11 through 13 show the relative intensity of fringe
patterns being reflected by a Hastelloy X specimen at var-
ious temperatures. The PMT output for a bright fringe is
represented as an upper peak of the sine wave type pattern.
In Figure 11, the intensity peaks dropped very little as the
temperature of the specimen was raised to 1,000°F. Figure
12 shows the effects of oxidation over a ten minute period
at 1,100°F. The upper photo in Figure 13 was taken at

1,200°F. The bright fringe bands have been reduced to

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0.2 0.5 I 2 5 IO 20 so IOO

Wavelength- microns

FIGURE 10 BLACK-BODY RADIATION VERsus
WAVELENGTH

20

ROOM TEMPE RATU RE

 

1,000 F

 

FIGURE 11 FRINGE PATTERN INTENSITY SIGNALS
AT 70°F AND 1,200°F

21

1,100 F

03723“

 

AFTER 10 MINUTES AT 1,100 F

03725.!

 

FIGURE 12 FRINGE PATTERN INTENSITY
SIGNALS AT 1,100°F

22

1,200 F

057.13.!

 

AFTER INCREASING GAIN AT 1,200 F

 

FIGURE 13 FRINGE INTENSITY BEFORE AND
AFTER INCREASING PMT GAIN

23

lower levels due to oxidation and black-body radiation which
begins to occur. The bright bands were again made distin-
guishable in the lower photo of Figure 13 by increasing the

gain of the PMT circuit.

2.4 Comparison Test

 

A test was conducted at room temperature to determine
the accuracy of the I.S.G. for the material and strain ranges
to be used in the experiment. An MTS extensometer (0.3 inch
gage length) and the I.S.G. device were both used to simul-
taneously monitor the strain in an axial specimen of
Hastelloy X. The specimen was cyclically loaded under
strain control by using the extensometer to generate a feed-
back signal. Hysteresis 100ps were then recorded for var-
ious strain ranges. Figure 14 shows the results at four
different strain levels. The results for the two measure-
ment devices were superimposed for comparison.

This test showed that the I.S.G. strain values were
within 5-10% of the extensometer output. The total strain
ranges were predicted exceptionally well. The comparison
test allowed confident strain measurements to be made during

the remainder of the test program.

24

 

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CHAPTER 3

SAMPLES AND MATERIALS

All of the specimens used in this program were supplied
by NASA. The material chosen was Hastelloy X which is an
austenitic, nickel-base superalloy used in components re-
quiring oxidation resistance up to 2,200°F (18). Some of
the pertinent material properties for this alloy are given
in Table 1. Reference (19) gives a more complete listing
of properties for Hastelloy X. Three types of specimens
were used in this experiment. These consisted of axial,
hourglass, and notched configurations. Specimen dimensions
and photographs are shown in Figures 15 through 18.

An elastic stress concentration factor for the cir-
cular notches specimens was found experimentally using the
I.S.G. From Peterson (20),the stress concentration factor,
Kt, was given as 2.37. The experimentally determined stress
concentration factor which is defined here as Kg‘was found
to be equal to 2.27. Both the theoretical and experimental
stress concentrations were determined for a spot which was
50 microns from the edge of the notch. This was as near
to the notch as the small indentations could be made. Figure

19 shows where five sets of indentations for the I.S.G. were

25

26

Table 1. MATERIAL PROPERTIES

 

 

 

 

 

 

 

 

Test Dynamic Modulus Ultimate 0.2% Offset
Temperature of Elasticity Tensile Yield
°C (°F) MPa (Ksi x 103) MPa (ksi) MPA (ksi)
21 (70) 206,850 (30.0) 785.3 (119.9) 360 (52.2)

650 (1,200) 154,448 (22.4) 572.3 (83.0) 272 (39.5)

 

3.85 REF.

   

 

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T

(ALL DIMENSIONS IN INCHES)

 

FIGURE 15 SMOOTH AXIAL SPECIMEN

 

 

 

 

 

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FIGURE 16

__._ 1.2 .220 DIA.

28

1.5R.

5/8 - 18 NF

 

 

4.00 REF.

 

 

 

 

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I

(ALL DIMENSIONS IN INCHES)

 

HOURGLASS SPECIMEN

29

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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FIGURE 18 NOTCHED SPECIMEN

31

placed across the width of a notched specimen. Room tem-
perature strain measurements were made at each of these
locations while cycling well below the proportional limit.
The actual strain data and the calculated strain profile
are both shown. By taking the ratio of strains at location
#5 and location #1, the strain concentration factor was
determined. For elastic strains, the stress and strain
concentration factors are equal (21). Using this informa-
tion the experimental stress concentration factor, Kt' , was
determined 0§:'= 2.27). This experimentally determined
value oflq:'as well as the designations for the remote and
local areas (locations #5 and #1) were used throughout the
test program.

It should be noted that the I.S.G. was used to measure
both local and remote strains for evaluating the Neuber
equation. Other investigators (5,6) have restricted load
levels to insure that the remote region remained linearly
elastic. This allowed the remote strain to be calculated
by knowing the stress in the net section and the modulus of
elasticity. In this experiment, the complications of de-
fining a net section stress were avoided since the remote
strain was measured directly. There were also no limitations
on plasticity in the remote region. This allowed the Neuber

relation to be evaluated over a greater range of loading

conditions.

132

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FIGURE 19 DETERMINATION OF STRAIN PROFILE

CHAPTER 4

EXPERIMENTAL METHODS

4.1 Strain Measurement in Notched Specimens

 

This chapter describes the methods which were used to
determine the stresses and strains in notched specimens.
SectionsI4.land 4.2 refer to room temperature testing
techniques while Section 4.3 gives a description of experi-
mental methods used for running tests at 1,200°F. The
Neuber prediction tests at room temperature and 1,200°F
are explained in Section 4.4.

A computer controlled, MTS closed-loop testing machine
was used to perform all of the tests. The loading capacity
of the system was 12 Kips. Figure 20 shows the test set up.
A woods metal gripping arrangement was employed so that
specimens could be mounted in a stress free condition.

Forone of the room temperature tests, five sets of indenta-
tions were placed on a notched specimen at the locations
described in Figure 19. After the specimen was mounted,
the laser of the I.S.G. was focused on the set of indenta-
tions nearest the edge of the notch. Several parameters
were then adjusted so that the I.S.G. would function pro-

perly. The gain and offset controls for the servo mirrors

33

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were set so that the fringe patterns were sweeping symmetri-
cally across the photomultiplier tubes. The electronic
analog of the fringe patterns was viewed on an oscillo-
scope until all adjustments had been properly made.

A function generator was then programmed to output a
symmetric ramp loading pattern voltage. The loading pattern
and the recording set up is illustrated in Figure 21. The
time for loading between tensile and compressive peaks was
set at 20 seconds with a 100 second hold time between peaks.
A dual pen recorder was used to plot both the load pattern
and the resulting strain response versus time. Another
plotted X-Y data of load versus strain.

Load and notch root strain were recorded for the first
six reversals of loading in order to obtain the initial
response of the material. Hastelloy X is a cyclically
hardening alloy which reaches a stable condition quite
rapidly. Taking this information into account, the speci-
men was run for fifty additional reversals of loading until
the strain responseluni stabilized. Load and strain values
were then recorded for several reversals of loading to des-
cribe the stable condition of the material.

With the material fully stabilized, the I.S.G. was
focused on the set of indentations which was second farth-
est from the notch. (See Figure 19 for location #2).

Load and strain data were again obtained for several re-
versals of loading. Similar measurements were performed

at locations #3, #4, and #5. At that point, the level of

36

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37

the peaks in the loading pattern was increased. The period
of loading remained constant and the 100 second hold times
were still imposed. The I.S.G. was again used to obtain
strain data at locations #1 through #5, The procedure was
repeated for two more increased levels of loading. For
the remainder of this report, the lowest load level is
referred to as Level 1 with the higher loads being Level
2, Level 3 and Level 4.

The next portion of the experiment involved setting
up a new notched specimen so that initial remote strain at
location #5 could be recorded. With the loading pattern
set for Level #1, the first six reversals of loading were
recorded in terms of load and strain data. The stable be-
havior at this location had already been established in the
previous test.

It should be noted that the initial response was re-
corded for locations #1 and #5 because these are defined
as the local and remote locations for the Neuber analysis.
The information from intermediate locations #2, #3, and #4
was recorded for observation rather than to be used in

the Neuber predictions.

4.2 Simulation of Stresses in Notched Specimens

The strain data from the notch root and remote loca—
tions were imposed upon smooth specimens to obtain the
stresses. This was the only practical method known for

extracting stress information at a notch.

38

A smooth specimen was set up under strain control with
a 0.3 in. gage length extensometer attached. A graph show-
ing the previously obtained strain versus time data at the
notch (location #1) was placed on the dual-pen recorder.
The strain voltage from the extensometer was fed into the
recorder.

As the pen swept across the graph, the operator was
required to manually control the test system so that the
strain level would exactly duplicate the original I.S.G.
data. During this time the stress data were plotted on the
other channel of the dual-pen recorder. Stress versus
strain plots were also recorded. This method of simulating
stresses proved to be very satisfactory from an experimental
viewpoint. All parameters such as strain rate, creep, and
total strain were reproducediJIthe smooth specimen as they
occurred in the notched plate. The X-Y plots of notch
root stress versus strain were considered direct experimen-
tal data to which the Neuber predictions could later be
compared.

To characterize the overall behavior (and for the
eventual Neuber prediction) the remote stresses in the
notched member were also determined. To accomplish this,
the above procedure was repeated on another smooth specimen
using the remote strain plots. Therefore, the stress-
strain behavior of a notched specimen had been established
for the initial response at Load Level #1 and the stable

condition at Load Levels 1 through 4.

39

4.3 Elevated Temperature Tests

 

A set of tests similar to those described in Sections
4.1 and 4.2 were conducted at 1,200°F so that Neuber's
rule could be evaluated under more adverse circumstances.
There were several reasons for choosing this temperature.
Although the I.S.G. had produced strain measurements at
temperatures up to 1,500°F, the data usually became dis-
torted after a short time. This was due to a breakdown of
the Gold-Palladium coating which caused a loss of reflec-
tive ability. At 1,200°F the I.S.G. would respond more
consistently for a long period of time. This was important
since the loading scheme was lengthy and involved 100
second hold times. Hastelloy X also began to experience
time dependent effects at 1,200°F.

The philosophy behind the room temperature and the
high temperature experiments was essentially the same.
However, the preparation and testing techniques used at
high temperature were more complex. After a notched speci—
men was indented and coated with Gold-Palladium, it had to
be carefully mounted so that the induction heating coils
would not disrupt the reflected fringe patterns. The
specimen was then brought up to temperature using a 5KW
induction heater. A chromel-alumel type thermocouple spot
welded to the reverse side of the specimen provided tem-
perature feedback (see Figure 22). Cooling coils both
above and below the specimen were used to isolate the heat

from the load cell and the woods metal pot. The loading

40

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41

pattern was similar to the one shown in Figure 21. The ten-
sile and compressive peaks were lower than those used at
room temperature to allow for the larger amounts of strain
at 1,200°F.

Some difficulties were encountered when trying to ob-
tain initial cyclic data with the I.S.G. at high temperature.
Bofferding (5) had experienced similar problems during his
room temperature experiments. Improvements in the system
have since allowed room temperature data to be obtained
very consistantly during all phases of testing. However,
at elevated temperatures the I.S.G. must sometimes be
readjusted after the test has begun. Because of this,
the specimen was cycled until it exhibited stable strain
response. Then the load pattern and strain values at the
notch root (location #1) were recorded for several rever-
sals. The I.S.G. was next focused on the remote location
(location #5) and strain data were recorded. Measurements
were obtained at the notch root and remote locations for

four different load levels.

4.3.1 Elevated Temperature Stress Simulation

 

The strains measured at the local and remote locations
in the notched plate were replayed onto hourglass specimens
at 1,200°F. A diametral extensometer, shown in Figure 23
was used to measure transverse strain. A computed axial

strain was obtained by combining the load and transverse

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strain voltages in an analog computer circuit. A schematic
diagram of the circuit is shown in Figure 24.

The proper output voltage representing total strain
was obtained by adjusting three potentiometers. Pot #00
was set so that the transverse strain was increased by a
factor of two. Pot #01 was used to dial in the elastic
strain voltage, knowing the applied stress and the modulus
of elasticity. The last step involved setting pot #02 to
obtain the elastic Poisson's ratio. This was actually done
by turning pot #02 until the plastic strain was equal to
zero for very small load levels. The final axial strain
output was used as the feedback signal for direct control.
The switches (81, $2, $3) allow the option of recording
transverse, elastic, plastic, or total strain. For a
detailed discussion of this strain measurement technique,
see Reference (22).

High frequency induction coils were used to heat the
hourglass specimens. A thermocouple spot welded at the
minimum diameter of the specimen provided a feedback signal
for temperature control. Care was taken to insure that the
thermocouple would not interfere with the sensing arms of
the extensometer. The strains which had been measured in
the notched specimen at 1,200°F were replayed onto the
hourglass specimen to obtain the stresses. The techniques
were the same as those for the room temperature simulation

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4.4 Neuber Prediction of Notch Root Behavior

 

The Neuber equation allows local behavior in a notched
plate to be determined as a function of remote stress and
strain. In this experiment, a smooth specimen was manually
controlled according to remote stress-strain information
in order to establish local behavior.

The measured strains and simulated stresses at the
remote location in the notched specimen had been recorded
on a time scale (Section 4.2 & 4.3.1). Figure 25 shows how
the remote data werereplotted to be used in the Neuber
prediction. The values ASl and Ael were obtained for the
first reversal of loading. (The 100 second hold period at
the tensile peak was considered part of the first reversalJ
At each five second interval during the first reversal, AS
and Ae were multiplied together and their product was multi-
plied by the notch factor squared, (Kt'F. This procedure
allowed one point to be plotted on the Neuber versus time
graph at each five second interval. The value of C1 shown
in Figure 25 would be the last point plotted for the first
reversal. This point defined a new origin for constructing
the Neuber plot of the second reversal. Values of A52 and
482 were taken with respect to the ending point of
the first reversal. The changes in stress and strain were
again multiplied by(Kt')2 until the second reversal was
completed. This plotting procedure was repeated for every

subsequent reversal. The Neuber versus time plot was then

46

 

 

 

   

STRESS vs.
TIME A51
A82
TIME
STRAIN vs.
TIME AcI
be
2 TIME
____.,.
NEUBER vs.
TIME
TIME
_____..
C ' KZAS be
1 1 1

 

l) Simulate values of Ag and Ae
by using a smooth specimen.
2) Superimpose the product (Ao)(Ae) onto the Neuber vs. Time plot.

FIGURE 25 CONSTRUCTION OF NEUBER VERSUS
TIME GRAPH

47

placed on a recorder so that the product of stress and
strain from a smooth specimen could be replayed upon it.
This process will now be explained in more detail.

The tests were run under strain control using a
smooth axial specimen at room temperature and the hour-
glass type at 1,200°F. The stress and strain values from
a smooth specimen represented A0 and A: respectively.
These quantities werenmltiplied.together on-line with an
analog computer so that the Neuber relation, Aer==(Kt'F ASAe
could be satisfied for each reversal. The changes in
stress and strain (A0 and AG) are defined with respect to
the starting point of each reversal. This idea is illust-
rated in Figure 26. A method was needed for setting up a
new origin from which these changes could be measured during
a smooth specimen test. In this study, the circuit shown
in Figure 27 was used to determine the product of A0 and As
on a reversal by reversal basis. The load cell and exten-
someter voltages from a smooth specimen were plugged into
the analog circuit. All of the operational amplifiers
were unity gain types and were used for addition or subtra-
ction of two voltage levels. The four switches marked 82
could be thrown simultaneously to either the right (R) or
left (L) as could the two switches marked 81. With switches
81 thrown to the left, the circuit was in the calibration
mode. Reference voltages were set by turning pots #01 and

#02 to calibrate the multiplier. After the calibration was

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FIGURE 26 THE NEUBER RELATION IS DEFINED
FOE EACH SEPERATE REVERSAL

49

completed, switches 51 were thrown back to the right so
that the load cell and extensometer voltages were input to
the circuit.

For the first reversal of loading, the changes in
stress and strain from the smooth specimen were multiplied
together and the product was used to follow the Neuber
versus time plot. This was done by throwing switches 82
to the right and setting pots #03 and #05 equal to zero
volts. The summing amplifiers #01 and #02 received only
the stress and strain voltages which were initially zero
at the start of the test. Simply multiplying the actual
stress and strain voltages together would not separate
their changes for each particular reversal. At the end of
the first reversal it was necessary to establish a new vol—
tage origin for calculating the changes in stress and strain
during the second reversal. The positive stress and strain
voltages had to be reset to zero so that changes could be
measured from the new origin. The two digital voltmeters
labeled DVM in Figure 27 were used to monitor the sum of
voltages from amps #03 and #04. Negative voltages from
pots #04 and #06 were used to offset the positive stress
and strain voltages at the end of reversal #1 so that the
DVM's both read zero volts.

To execute the second reversal, switches 82 were
thrown to the left. The negative voltages which had been

set by pots #04 and #06 were now being subtracted from the

50

stress and strain voltages in amps #01 and #02. Therefore,
both inputs to the multiplier were zero volts. The changes
in stress and strain for the second reversal would be defin-
ed in terms of this starting point. Upon completion of the
second reversal another new origin was required. Positive
voltages from pots #03 and #05 were set to exactly cancel
the negative load and strain values as monitored on the DVM's.
Switches 82 were then thrown back to the right and the vol-
tage outputs from amps #01 and #02 were again zero. The
third reversal was then executed. This procedure was re-
peated until all of the reversals on the Neuber versus time
plot had been retraced.

This technique proved to be a very basic solution to
the problem of electronically simulating the Neuber plots.
The method of superimposing current test data onto previous-
ly obtained plots was the key element throughout this experi-
ment. The interesting point was that the researcher became
'part' of the closed loop by having to manually control the
system. The fact that this method allowed information to
be replayed with respect to time was very important. It
meant that time dependent parameters such as creep and re-

laxation could be exactly reproduced in smooth specimens.

51.

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CHAPTER 5

EXPERIMENTAL RESULTS AND DISCUSSION

5.1 Room Temperature I.S.G. Measurements

 

The results are presented in a chronological sequence
leading up to a comparison of notch root behavior obtained
by two methods: 1) direct strain measurement and stress
simulation and 2) Neuber prediction. This chapter contains
a large amount of actual data since the experiments were
qualitative in many respects.

Figure 28 shows I.S.G. measurements of strain vs.
applied load for a notched specimen during the first three
cycles of loading. The strain and load data were also
plotted on a time scale in Figure 29. The initial loading
pattern consisted of constant amplitude completely reversed
loading between i 14 KN.

The most noticeable effects in notch root behavior
were caused by cyclic hardening. The tensile peaks showed
a large decrease in strain for each successive cycle due
to strain hardening. The compressive strains experienced
much less variation during the three cycle period. As a
result, an initially tensile mean strain was shifted in the

compressive direction. Creep effects were also present in

52

53

theseinxxntemperature data. This is shown more clearly in
the time plot of local strain (Figure 29). The largest
amount of creep took place during the first 100 second hold
time and then diminished with each successive reversal. It
should be noted that the time scale was manually switched
from 5 sec/cm to 50 sec/cm for the hold periods.

For remote behavior, cyclic hardening again caused the
total strain to decrease for each plotted loop. The effects
of creep were minimal for the remote location. The amount
of creep at both locations in the specimen decreased as the
material stabilized.

When a sufficient number of cycles had been applied to
stabilize the material, the I.S.G. was used to record data

at four different cyclic load levels which are listed:

Level # Load (Kn)
l i 14.0
2 i 14.5
3 i 15.5
4 i 16.0

Strain measurements were obtained at each of five locations
across the notched specimen (locations were defined in
Chapter 3). Figures 30 through 33 each show results for a
different load level. (These plots should be viewed without
reference to a strain origin since the I.S.G. had to be
repositioned for each location.) These figures illustrate
the effects of cyclic loading at various distances from the

notch. The amount of plastic strain diminished significantly

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.....‘.
...
...
.1.

 

60

as the distance from the notch increased. Also, when the
load was raised from Level 1 to Level 4, the strain at the
remote location (#5) increased by 21% while the local
strain (#1) experienced a 50% increase. This gives an

indication of the strain concentration near the notch.

5.2 Room Temperature Stress Simulation

 

These results were obtained by replaying the strain
histories shown in Section 5.1 onto smooth specimens. Time
plots of strain which had been measured with the I.S.G. were
placed on a dual-pen recorder. As the pen moved across the
recorder, the strain signal from a smooth specimen was con-
trolled so that the original I.S.G.-measured strain plot
would be retraced. The resulting stresses were plotted on
the same recorder. This procedure was used to determine the
stresses at the local and remote locations for both initial
and stable behavior. The hysteresis 100ps for the first
three cycles are shown in Figure 34.

Since these tests were run under strain control, stress
was the independent variable and would increase if strain
hardening occured. This effect is shown clearly for initial
notch root data. There is an interesting relationship be-
tween the decrease in strain range and the increase in
stress, both caused by cyclic hardening. The decrease in
strain was experienced while cycling between constant load
limits when I.S.G. measurements were recorded. The increase

in stress peaks during the simulation was due to material

61

.l-_.___.-._.-r_v_ ._..‘———

........ . . _
+-—.—.—...—...__

+ Remote Response Jr

- -.- -.'—. .._-_._.. *-¢'-.’—. —._V._.. -

1

~ 4" Local Response

 

_I;-,;.-; Cycle 3 ”/1-

 

 

 

 

FIGURE 34 SMOOTH SPECIMEN SIMULATION OF INITIAL STRESSES

62

properties. The fact that the stress and strain have this
upward and downward trend is not surprising. It is difficult,
however, to separate the various factors which determine
whether the stress or the strain is the more dominant
characteristic. This study was more concerned with trying
to duplicate this behavior than in trying to explain these
interactions.

The remote location exhibited a small amount of strain
hardening during the first three cycles. Figure 35 shows
the remote strain and simulated stress on a time scale.

The strain values in this Figure were replayed onto a smooth
specimen to determine the stresses. The remote stress and
strain information was used to conduct the Neuber prediction-
of notch root behavior as explained in Section 4.4.

The stable data are plotted in Figure 36 at each of four
load levels. (Load levels refer to those used during I.S.G.
measurements from Section 5.1.) The data for Level 1 show
that the amount of plastic strain had decreased from the
initial values in Figure 34. {Huahysteresis loops for the
other three levels simply show increased stress and plastic
strain at each location. Time plots of stress and strain
for stable local response are shown in Figure 37 for Levels
3 and 4. Note the difference in the rate of increase bet-

ween the stress and strain values.

63

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64

Response 1"::‘ '

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FIGURE 36 SMOOTH SPECIMEN SIMULATION OF STABILIZED STRESSES

 

65

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66

5.3 Room Temperature Neuber Prediction

 

The remote stress and strain versus time plots were
multiplied by CKt'F so that.(Kt'F ASAe could be plotted
(Section 4.4). Such a plot is shown in Figure 38 for
initial behavior at Load Level 1. The stress-strain in-
formation from Figure 35 was used to construct this Neuber
prediction curve. The same type of plots were drawn to
predict stabilized local behavior in a notched specimen at
four load levels.

A smooth specimen was manually controlled in the MTS
testing system so that the product of stress and strain
would follow the Neuber prediction curves. The resulting
stress and strain values constituted the predicted notch
root behavior. The Neuber prediction was then compared
with data from the stress simulation of local response
(Section 5.2).

The first three cycles were plotted in Figure 39.
During the first cycle, the Neuber simulation was slightly
high on stress which caused lower strain peaks to occur
due to the multiplication. Actually, the tensile and
compressive strains were only 9% low for the first cycle.
The predicted tensile strain on the second cycle was low by
8% while the compressive strain was 13% lower than the stress
simulation.

The stable stress-strain curves in Figure 40 were
superimposed to show the excellent results obtained with

the Neuber predictions. For Load Levels 1 and 2, the

6'7

 

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NEUBER PREDICTION

  

 

68

 

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NEUBER PREDICTION AND STREss SIMULATION OF

INITIAL LOCAL BEHAVIOR

FIGURE 39

69

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70

Neuber method was approximately 6% high in predicting ten-
sile and compressive strains. Load Levels 3 and 4 show

nearly a perfect correlation between the two sets of curves.

5.4 High Temperature I.S.G. Strain Measurement
Notch root and remote strains were also measured at
1,200°F in a Hastelloy X specimen which had been cyclically

stabilized. Four load levels were again used as follows:

Level # Load (KN)
l i 10.5
2 i 11.3
3 i 12.3
4 i 13.3

Hysteresis loops showing applied load vs. local strain at
four different load levels are shown in Figures 41 and 42.
Strain versus time plots for each of the corresponding
load levels are shown in Figures 43 through 46.

’Small increases in load produced large strains at this
temperature; especially strain due to creep. During the
100 second hold time the amount of creep strain at each

load level was as follows:

 

Level # Creep Strain
1 0.05%
2 0.10%
3 0.13%
4 0.18%

These values were approximately equal for tension and com-

pression. Note that the total measured strains were nearly

 

 

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FIGURE 41 LOAD VERSUS LOCAL STRAIN AT 1,200°F

72

 

 

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LOAD VERSUS LOCAL STRAIN AT 1,200°F

FIGURE 42

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78

1.0% in both tension and compression during Load Level 4.
The interferometric strain measurements were consistant
even for these high strain ranges.

The remote data for the stabilized material exhibited
much less creep and plastic strain. The load versus strain
plots from each of the four load levels are shown in Figure
47. (Note: the four load levels used during the elevated
temperature tests were different than the levels used at
room temperature.) The amount of creep during the hold

times was very small.

5.5 High Temperature Stress Simulation

 

The measured remote and local strains shown in Section.
5.4 were imposed on an hourglass specimen at 1,200°F. The
hourglass specimen was cyclically stabilized before perform-
ing the stress simulation. The resulting stress versus
strain curves for stable notch root response at four dif-
ferent load levels are plotted in Figures 48 and 49. The
relationship between stress and strain in time is illustrat-
ed in Figure 50 for Load Level 3. The increase in strain
which was imposed during the hold period helped to balance
the stress relaxation of the material. At Levels 1 and 2
there is almost no stress relaxation while Levels 3 and 4
show just a slight amount.

The remote data from the stress simulation ii; plotted
in Figure 51. The time plot for Level 3 stresses and strains

is shown in Figure 52. Very little creep had been measured

79

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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FIGURE 48 LOCAL SIMULATED STRESS VERSUS STRAIN AT

 

80

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Load Level I. 777 _;.l;_._;l_.___-._~__

 

 

 

 

 

FIGURE 49 LOCAL SIMULATED STRESS VERSUS STRAIN AT 1,200°F

81

  

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84

with the I.S.G. at the remote location. Therefore, when
these measured strains were replayed onto the smooth speci-
men,aalarge amount of stress relaxation occured during the
hold times. This caused the stress vs. strain plots to take
on an almost rectangular appearance.

The stress simulation data shows that as the distance
from the notch increases, the amount of stress relaxation
also increases. This is related to the fact that there is
less creep strain to balance the relaxation of stress at
the remote location. This information is valuable when
trying to construct an overall picture of high temperature

behavior in a notched plate.

5.6 High Temperature Neuber Prediction

 

The Neuber prediction of local behavior was entirely
dependent upon remote stress and strain data. These remote
data had been determined by the stress simulation described
in Section 5.5. Therefore, it is not surprising that when
the remote stress and strain values are multiplied by 0&3)2
to construct the Neuber prediction curves, that these curves
have decreasing slopes during the hold times. An example of
this is shown in Figure 53 for Level 3. This figure shows
the actual curves which were retraced during the smooth speci-
men Neuber prediction.

A cyclically stabilized hourglass specimen heated to

1,200°F was used to establish notch root behavior at four

load levels. A comparison was made between the measured

85

strain/stress simulation and the Neuber prediction in
Figures 54 and 55. The most noticeable trend at all four
levels was the amount of stress relaxation predicted by the
Neuber relation. For Load Levels 1 and 2, the stresses

at the end of the 100 second hold times were low by 23%
and 27% respectively. The stresses were predicted more
accurately at the higher Load Levels. At load level 3 the
stresses were 22% low and at Level 4 the stresses were 15%
lower than the stress simulation. In terms of strain
range, the error in predicting Level 1 strains was 20% low
while the Level 4 strains were predicted within 10%.

The behavior during the hold times was quite interest-~
ing. The stress relaxation caused the product of stress and
strain to decrease. This meant that the Neuber prediction
curve's downward slope was automatically satisfied. However,
continued stress relaxation forced the operator to increase
the level of strain in the specimen so that the Neuber rela-
tion would still be satisfied. This accounted for the slanted

corners of the Neuber hysterisis loops.

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FIGURE 54 NEUBER PREDICTION AND STRESS SIMULATION AT 1,200°F

88

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LOAD LEVEL 3

 

NEUBER PREDICTION

371----

-~_-..._.-- -.

' ff' 7' STRESS SIMULATION

 

.. _ 1‘ . I

 

FIGURE 55 NEUBER PREDICTION AND STRESS SIMULATION AT 1,200°F

CHAPTER 6

CONCLUSIONS

Neuber control of a smooth specimen predicted the
notch root stress-strain behavior of a circular center
notched plate that was made of Hastelloy X with excellent
agreement to direct experimentally measured notch root
strains and simulated stresses at room temperature. The
agreement was good for initial behavior during cyclic
hardening and for the stable condition at four different
load levels. At 650°C and for the stable conditions,
agreement with experimental data were acceptable with the
maximum error at 20%. At this higher temperature, the
direct experimental data showed primarily creep strain
during hold times. The Neuber prediction showed both creep
and stress relaxation. This difference in the general be-
havior resulted in significantly larger errors at this

elevated temperature than those for room temperature.

89

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10.

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