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This is to certify that the

thesis entitled
A STUDY OF CYCLIC STRESS AND
STRAIN CONCENTRATION FACTORS
AT NOTCH ROOTS THROUTHOUT
FATIGUE LIFE

presented by

Charles H. Bofferding III

has been accepted towards fulfillment
of the requirements for

Masters degeein Mechanics

6;me

Major professor

Date 5,-/l f/? 0

0-7639

 

 

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25¢ per day per item

RETURNING LIBRARY MATERIALS:

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charge from circulation reco

 

 

A STUDY OF CYCLIC STRESS AND STRAIN CONCENTRATION
FACTORS AT NOTCH ROOTS THROUGHOUT FATIGUE LIFE

By

Charles H. Bofferding III

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

1980

 

 

ABSTRACT

A STUDY OF CYCLIC STRESS AND STRAIN CONCENTRATION
FACTORS AT NOTCH ROOTS THROUGHOUT FATIGUE LIFE

By

Charles H. Bofferding III

Notch root stress and strain concentration factors, as related by
Nueber's rule, are examined for various materials and notch geometries
throughout fatigue life. Plates with circular and elliptic notches are
cycled under constant amplitude load control (nominallyelastic,complete—
ly reversed) in a hydraulic closed-loop fatigue system. Long and short
life tests are run for each material and notch geometry. Strain at the
notch root is recorded with the Interferrometric Strain Gage. The notch
root strain is reproduced in a smooth sample, cycled under strain con-
trol, to produce local stress—strain hysterisis loops. These data are

used tocomputer.

 

 

ACKNOWLEDGEMENTS

The work for this thesis was funded by the National Science
Foundation.

I would like to thank Lisa Adrian, Melvin Cartwright, Greg
Hoshal and Tom Mase for their many hours of data reduction and
drawing. Also special thanks goes to Dr. John Martin who gave me

the opportunity, freedom and inspiration to accomplish this work.

ii

 

TABLE OF CONTENTS

Page
LIST OF TABLES iv
LIST OF FIGURES V
LIST OF SYMBOLS Vii
Chapter

1. INTRODUCTION 1

2. THE INTERFEROMETRIC STRAIN GAGE 4

2 1 Fundamentals of the I.S.G. 4

2.2 Calibration of the I.S.G. 15

2.2.1 Strain to Fringe Shift 15

2.2.2 Fringe Shift to Voltage 15

2.2.3 Strain to Voltage 17

2.2.4 Calibration Test 17

2.2.5 Resolution 18

3. SAMPLES AND MATERIALS 20

4. EXPERIMENTAL METHODS 25

5. EXPERIMENTAL RESULTS 29

6. SUMMARY AND CONCLUSIONS 43

LIST OF REFERENCES 46

 

LIST OF TABLES

Table Page
1 Material properties. 24
2 Test data. 30

iv

 

Figure
1

2

10
11
12
13

14

15
16

17

18

19

LIST OF FIGURES

Interferometric strain gage.

Indentations on sample.

Fringe generation principles.

Interference pattern.

I.S.G. schematic.

Electric analog to fringe pattern.

S1 and $2 for current mirror position 185.
Determination of maximum and minimum spacing.
I.S.G. vs. clip gage.

Computer printout of maximum and minimum.
Sample geometry.

Notch geometry.

Photograph of samples.

Indentations for 2024 elliptic sample before and
after testing.

Output of cracked samples.
Test set up.

K5, K0, and Kf versus life for 1018 mild steel
circular notches.

Kg, K0, and Kf versus life for 1018 mild steel
elliptical notches.

K6, K0, and Kf versus life for 2024 aluminum
circular notches.

Page

10
12
13
14
l9
19
21
22
22

23

27
28

31

32

33

 

Figure

20

21

22

23
24
25
26
27

28

K6, K0, and Kf versus life for 2024 aluminum
elliptical notches.

Kg, K0, and Kf versus life for 7475 aluminum
circular notches.

K6, K0, and Kf versus life for 7475 aluminum
elliptical notches.

Kf versus life
Kf versus life
K versus life
K versus life
K versus life

K versus life

for 1018

for 1018

for 2024

for 2024

for 7475

for 7475

mild steel circular notches.
mild steel elliptical notches.
aluminum circular notches.
aluminum elliptical notches.
aluminum circular notches.

aluminum elliptical notches.

vi

Page

34

35

36

37
38
39
4O
41

42

LIST OF SYMBOLS

3*: Nominal stress equals load divided by gross cross sectional
area.
0*: Local or notch root stress.
e*: Nominal normal strain.
5*: Local or notch root normal strain.
Kt: Theoretical stress concentration factor.
. Ao

KO: Stress concentration factor (AS .

. . Ac
Ks: Strain concentration factor (EE)'

1/2

Kf: [K0 K5]
E: Young's Modulus

*For fatigue these symbols are preceeded by A and are taken as ranges
associated with one cycle.

 

 

l|.u'.'-. -'."-"' - '3' "'

CHAPTER 1

INTRODUCTION

Neuber's original equation (l)* equated the geometric mean of the
notch stress and strain concentration factors to the theoretical stress
concentration factor for a specific geometry and loading scheme, namely
a V notch loaded in shear. Others have applied the relationship to
various notch configurations usually under axial load to find good re—
sults in both elastic testing and, when proper limits are imposed on
notch plasticity, plastic testing (2).

In the attempts to apply Neuber's rule to fatigue the stress and
strain values of the monotonic relationship (S, 0, e, e) are replaced
with the stress and strain ranges of fatigue loading (AS, A0, Ae, Ae).
11/2

In practice [K0 K5 equals a constant Kf which is less than Kt and

Neuber's rule is written as:

TEAM

1
_ <2 _
K _ [K K8] _ (AS Ae

f o

Rearranging and multiplying both sides by E

Kf2 (AS Ae E) = (A0 A8 E) [1]@

 

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

 

This equation shows that a function of changes in nominal stress and
strain need only be multiplied by a constant concentration factor to
obtain the value of the same function of changes in local stress and
strain at the notch root.

Equation [1] reduces to the following form if the nominal stress

and strain are limited to the elastic region (i.e., AeE = AS)

1/
Kf AS = (A0 AeE)2 [2]

The case where AS and Ae are elastic and AG and A8 are inelastic covers
many problems of engineering interest. At even lower values of S, the

notch root remains essentially elastic and Equation 2 reduces to:
Kf AS = A0 [3]

Equations [1] to [3] were derived by Topper et a1. (3). They describe

the relationship between nominal stress—strain behavior and local stress—
strain behavior for the notch root. Equation [1] applies for all values
of nominal stress, Equation [2] for values less than the yield stress,

and Equation [3] for values less than the yield stress dividedIUVKf (i.e.,
elastic local stress). Also it is well known that the difference between
Kt andlqgis greatest for sharp notches (15). It is recognized that these
equations do not account for the complexity of mean stress and its relaxa—
tion nor variations in the stress and strain concentration factors due

to cyclic material properties. However, Wetzel suggests the stress and
strain ranges of smooth samples quickly converge to steady state values
(11) and Blatherwick and Olson (7) and Crews and Hardrath (4) have shown
that the notch root strain range also quickly stabilizes. Crews and

Hardrath (4) assumed that the notch stress could be found by reproducing

 

.
_ n '- .1 ' I . . I‘- 'I

-_ sir-1.12 his

the notch strain in smooth samples. The stabilized local stress was
then used to predict fatigue life of the notched sample with good re—
sults for nominally plastic ranges. Leis et a1 (17) extended this to
nominally plastic ranges and examined K5 and KO at the root to determine

K Kf was found to vary with life and strain ranges but tended to Kt

f'
with life. Using equations [1], [2], and [3] others have equated the
stress—strain state at a notch to that of smooth sample and by applying
cummulative damage methods have predicted notch fatigue life from smooth
sample data (3, 7, ll, 13, 14, 15). (Note, this did not require sub—
jecting smooth samples to notch root strains, only determining the notch
stress—strain state with eq. [1], [2], or [3] and then using smooth sam—
ple fatigue—life data to predict failure).

Wetzel (11) examined these results (3, 4, 7, l3, 14, 15) to find
that equations [1] to [3] accurately relate nominal and local stress—
strain response even in non—linear applications and when used in conjun—
ction with smooth sample data for 0—max loading yield good fatigue life
predictions.

Both Crews and Leis et al. reported measurement problems associated
with strain gages at the notch root. The purpose of this study is to
overcome these problems in strain measurement and examine Ke’ Ko’ and Kf
throughout fatigue life. This is done by measuring the notch root strain
with the Interferometric Strain Gage (I.S.G.). The I.S.G. is a noncon—

tacting laser based strain measuring system developed by Dr. William

Sharpe and is fully described in this report.

 

-'.“-‘ = -- ' - - - ' ' --' .5311 HI" ‘--_-I_

CHAPTER 2

THE INTERFEROMETRIC STRAIN GAGE

2.1 Fundamentals of the I.S.G.

 

The Interferometric Strain Gage (20-23) is a noncontacting Laser
based strain measuring system capable of measuring strain over a very
small gage length (50-100 microns). Figure 1 shows the I.S.G.

The I.S.G. measures strain by monitoring the movement of inter-
ference fringes. The fringes are generated by interferring defraction
patterns from two pyramidal indentations placed on a sample with a
Vicker's hardness tester. The indentations generally measure 25 microns
on a side and the center—to—center spacing between them is 100 microns.
A picture of an indentation set is given in Figure 2. Figure 3 shows
the principles involved in fringe generation. Incident monocromatic
coherent laser light is reflected by the indentations. Each indentation
creates a defraction pattern on both sides of the laser. On each side
the defraction patterns interfer to create interferrence fringes. Any
two interferring rays are out of phase by d sina where a is the angle

between the incident laser and the interferring rays. Whenever:
d sina = nA (n=0, i1, :2,...) [4]
where: A = wavelength of the laser light

the rays are in phase interferring constructively causing bright spots,

Figure 4.

 

I-r' -- - . ,... -. ----- -- . - -.- ' -
-'¢'ll:-“ fin . i'...': rut: '1."l."n_'f’l.! FYI—1‘ .'!.-=

 

Figure I Interferometric Strain Goge

 

 

Sample

Indentations on

2

Figure

 

incident Laser Beam

 

 

Figure3 Fringe Generation Principles

 

Wren-'0' “ ‘
\ :

 

Figure 4 Interference Pattern

 

If the interference patterns are viewed from a stationary observa-
tion point defined by do any movement of the indentations is seen as a
fringe shift past this point. For any one fringe in the patter d sina
in Equation [4] is a constant determined by a specific value of n. A
tensile strain (increasing d) implies fringe movement toward the laser
(decreasing a). Likewise a compressive strain will shift the fringes
away from the incident laser. For rigid body motion one pattern will
move toward the laser, the other away from it. If fringe movement to—
ward the laser is measured as positive and movement away negative;
averaging the fringe shift over the two channels cancels out rigid body
motion.

Indentation and fringe movement for a channel are related by:

5 d = —72L—— 6m
Sinao

where: 5m is the number or fraction of fringes passing the

observation point

The two channel average is

 

6 d = A (—J————~(Sm :6”)

SlIIOLO

A schematic of the I.S.G. is shown in Figure 5. It has two channels.

Each monitors the movement of one interference pattern. The movement
from both channels is averaged to cancel out rigid body motion as dis—
cribed.

For a given channel a servo mirror located at the stationary ob—
servation point reflects the interferrence pattern onto a slotted plate
covering the PMT. This allows only a portion of the fringe pattern to

be sensed by the PMT. By rotating the mirror the fringe pattern is

 

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swept over the PMT producing an electric analog to the fringe pat—
tern, Figure 6. The mirror position is controlled by the minicomputer
and each sweep is broken into 256 consecutively numbered steps at which
the corresponding PMT intensity is recorded. For each sweep the mini—
computer determines the mirror position numbers associated with the
maximum and minimum PMT readings implying bright and dark spots of
the fringe pattern. These positions are compared to the positions for
the previous sweep to determine fringe movement. Fringe movement is
output as the number of mirror positions the first two maximum or mini—
mum locations for a channel have moved and is given a sign depending on
whether the shift was toward the laser (+) or away from it (—).

To find the maximum and minimum locations during a sweep the mini—
computer compares the sum of the PMT intensities associated with the
current mirror position and the last four positions (81) to the sum of
the intensities of the last five mirror positions (82), Figure 7. The
sign of (81—82) indicates the slope of the intensity curve. The sign of

(S —SZ) for the current mirror position is compared to (51—52) for the

1
previous position. The slope only changes sign at a maximum or minimum
point. If the sign changes from + or 0 to — the mirror position is re-
corded as a maximum location. If a — or 0 to + shift occurs the mirror
position is recorded as a minimum location. The average number of posi-
tions between maximum points (called average maximum spacing) is computed
by dividing the difference between the first and last maximum location

by the total number of maximum locations minus one. The average minimum

spacing is computed the same way, Figure 8.

 

12

g

l
I
_J
_l

“L

l

 

 

P M T
Reading
(volts)

mem

 

5'0 «'30 I530 260 250 360

Mirror Position

Figure 6 Electric Analog to Fringe Pattern

 

l3

 

5.4
PMT 5.3
Reading 5 2
(volts) '
5.l
5, O f I r f r fir
I80 l8l I82 l83 l84 I85
Mirror Position
8' = (5.425 + 5.350 + 5.275+5.250 + 5.l25) = 26.425
32: (5.350 + 5.275 + 5.250 + 5.l25 + 5.025) = 26.025
3, '32 = Positive Slope = 0.400
Figure 7 SI and 82 for Current Mirror Position [85

 

l4

 

 

8..
PMT 6‘
Reading
(volts) 4-
2-
1
O r I r 1 T 7 1 v r r ,
II5 I45 I75 205 235 265
Mirror Position
M S , (243 ' I19) 62
ax. acin = -————— =
p g (3: I)
, S , _ (208‘l52) _ 56
Min. pacmg - (2_ l) -
Figure 8 Determination of Maximum and

Minimum Spacing

 

15

2.2 Calibration of the I.S.G.

 

The I.S.G. was calibrated to obtain reliable data without the use
of supplementary measuring devices.

The I.S.G. outputs a voltage proportional to the fringe shift.
Out put voltage and strain are related by first relating strain tofringe

shift then fringe shift to voltage out put.

2.2.1 Strain to Fringe Shift

 

The equation fundamental to the I.S.G. is:

6d=—>‘—6m [5]

Sinai0

Dividing[5]157thedistance between indentations gives:

8 _ 6d _ A 6m
d sina d
o
rearranging this gives:
5m=wfiold [6]
A
where: d = distance between indentations

6d = change in distance between indentations

e = normal strain between indentations

A = laser wave length

6 = angle between the incident laser and observation point

6 = fringe movement in fractions of fringe

2.2.2 Fringe Shift to Voltage
The fringe patterns are swept over photo multiplier tubes (PMT) with
mirrors controlled by a minicomputer. Each sweep is broken into 256

consecutively numbered positions at which the PMT voltage is read. The

 

l6

mimicomputer locates the mirror positions corresponding to maximum and
minimum PMT voltage for each sweep and compares them to those for the
previous sweep. Also the number of mirror positions between maximum
and minimum positions, called fringe spacing (S) is calculated. For
each channel the number of positions the first two maximum or two mini—
mum points shift is added (called channel shift total (CST) and equals
twice the fringe shift). For rigid body motion the channel patterns
shift opposite to each other but shift in the same direction for re-
lative displacement. Therefore, the CST summed over both channels equals
four times the relative displacement component of the patterns in terms
of mirror positions moved. The CST sum is output as an integer (N).
The D/A has 12 bit (i.e. 212 = 4096 increments) converters and a 0—10
volt range. Therefore, one bit of output equals:

10 Volts

4095 = 2.44 Millivolts

If MV equals the total output in millivolts then

MV = N(2.44) [7]

Since N represents four times the shift of the fringe pattern in terms
of mirror positions moved dividing N/4 by S gives the shift in terms of

fractions of fringes:

6 m =

N_/4 [8]
S

Rearranging [7] yields:

2.44 [9]

l7

rearranging [8] gives:

N = 6 111 (3)4 [10]

equating [9] and [10] leads to:

 

MV
6 = ____._____
m 2.44 (5)4 [11]
2.2.3 Strain to Voltage
Equating [6] and [11] leads to:
s (sin 0L0)d = MV [12]
A 2.44 (8)4
rearranging gives:
MV = —————o——5 (51313” 2.44 (8)4 [13]

where: MV = Millivolts of output
6 = normal strain between indentations
a = angle between incident laser and observation point
d = distance (in microns) between indentations

S = average spacing between maximum mirror positions

2.2.4 Calibration Test

A specimen was loaded under strain control. Strain was measured
with the I.S.G. and a MTS extensometer (model #632. 13B 20). The strain
range was i .0048 e. For this range the extensometer was calibrated at

0.01% error. The I.S.G. was calibrated using Equation [13] with:

A = 0.6328u
a = 41.170

d = lOOu

 

18

e = .004

S = 84

Note, S must be determined before you can calibrate. This is easily
done by the computer.

Figure 9 shows a plot of I.S.G. strain vs. clip gage strain.
Figure 10 shows a sample computer printout listing maximum and minimum

locations and spacing.

2.2.5 Resolution
For these tests the average spacing S was 65, a0 = 430 and A =
0.6328u. Therefore, the smallest displacement measurable is:

= 1/65 0.6328

G d 2 sin 430

= 0.0071374u
For the 100 micron gage length used the smallest measurable strain

theoretically is: 949%%%;Z£ = 0.007%. However, the actual resolution
varies between 0.01% and 0.023% for the test run. This is due to in—
accuracy in measuring the exact mirror position for the maximum and

minimum locations.

 

l9

 

 

 

 

l

_.Cllp gags;

 

Figure 9 ISG v3. Clip Gage

INTERFEROMETRIC STRAIN GAGE COMMANDS:

** PUSH INT T0 JUMP OUT.

** D-DISPLAY TO OSCILLOSCOPE.
** P-PRINT MAXIMUM & MINIMUM.
** R-RUN WITH STRAIN CONTROL.
** U-USER CONTROL.

** H-HELP: PRINT COMMANDS.

D,P,R,U,H?P

CH1 3 MAXS AT: +001¢6 +00172 +00237
2 MINS AT: +09138 +¢¢2¢4

CH2 3 MAXS AT: +¢¢117 +aa182 +¢0247
3 MINS AT: +0¢081 +0¢149 +0fl215

AVG SEPERATION WINDOW
CH1 +00065 +fl0021
CH2 +¢0¢65 +¢0021

Figure IO Computer Print Out of Max and Min

 

CHAPTER 3

SAMPLES AND MATERIALS

Dimensions for the notched plates and smooth samples are given in
Figure 11. Both circular and elliptic notches were examined for each
material. Figure 12 shows the specific dimensions for these notches.
Figure 13 is a photograph of a smooth sample flanked by notched samples.
Figure 14 shows before and after photographs of the indentations for
the long life 2024 elliptic sample and the fatigued sample. From
Peterson's book (18) Kt was found to be 3.10 for the circular notch
and 5.10 for the elliptic notch. However, since the indentations were
placed in a finite distance from the notch root (40 microns) the variance
in KC had to be examined. Using the equations developed in Coker's book
(19) the variation of Kt was found to be 1.695 percent for the circular
notch and 3.202 percent for the elliptical notch. Therefore, Kt equals
3.05 for the circular notch and 4.94 for the elliptic notch. It was
thought that longer samples would have to be used to overcome end effects
but tests showed uniform stress flow for the sample used. Threematerials
were tested, 2024 T351 aluminum alloy, 7475 T7351 aluminum alloy and 1018
mild steel. All materials were used as supplied. Table 1 gives material

properties.

20

 

21

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I_—_gv—f—I_i_‘_.__‘l

 

F—_——¥___/

 

f\
k.)

K)
\/

 

 

l
KP

ta

9.0

 

 

 

 

 

 

Notched Sample

O.lO 0.15
F
:2

'- 0135

L

0.48

 

 

 

 

. Smooth Sample

Figurell Sample Geometry

 

 

 

 

 

22

0.375

I.—
_l

 

‘f—__ A

*— Ol87

b. Elliptical

0. Circular

Figure l2 Notch Geometry

 

Figure l3 Photograph of Samples

 

 

 

Figure l4 Indentations for 2024 Elliptic Sample Before
and After Testing

 

 

24

Table 1. Material Properties

MATERIAL j; §y* Cyclic Strain
Hardening e§ponent n'

1018 30000 KSI 57.8 KSI 0.211

2024 T351 10000 KSI 27.3 KSI 0.075

7475 T7351 10000 RSI 41.1 RSI 0.124

*Actually this is the proportional limit and is given to show that
all nominal ranges are indeed elastic.

The 2024 aluminum cyclically hardens at all inelastic strain ampli—
tudes, the 7475 aluminum cyclically softens for all strain ranges tested
and the 1018 mild steel softens for strain amplitudes lower than 0.30
percent, hardens for ranges greater than 0.42 percent and is approxi-

mately stable at ranges between these values.

 

CHAPTER 4

EXPERIMENTAL METHODS

All notched tests were run in nominally elastic ranges. The

nominal stress (AS) is taken as the load divided by the cross—sectional

area of the plate if no notch were present. The nominal strain (Ae) is

simply:

Where E is Young's Modulus

Local strain (A9) is measured during testing with the I.S.G. at

periodic intervals and plotted as A6 vs. AS. Reproducing this strain

in the smooth specimens gives local stresses (A0). Inherent in this
method is the assumption that shear stresses at the notch root are ne-

gligible. Also mean strain is not accounted for as only the ranges were

duplicated.

All tests were run to failure or a minimum of 1,460,000 cycles.

For notched samples failure was determined by the I.S.G. output. If a

crack started between the indentations small tensile stresses created

large measured strains. Little or no strain is recorded in tension if

25

 

26

the stress between the indentations is relieved by a crack outside the
gage length. No output is possible if a crack goes through an indenr
tation. Figure 15 shows output associated with various crack modes.
However, some cracks were detected sooner than others. For the smooth
samples failure was easily determined by conventional methods. A
closed loop servo controlled hydraulic testing machine was used for all
tests. Figure 16 shows the test setup. Mounting stresses were reduced
with a "woods" metal pot. All tests started in tension and were com—

pletely reversed.

 

 

          
  

 

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Test Set Up

 

 

CHAPTER 5

EXPERIMENTAL RESULTS

The samples are divided into three groups according to material
type. Circular and elliptic notches were tested for each material and
a long and a short life test was made for each notch geometry. Table 2
gives the nominal stress range for each notched sample, the number of
cycles to failure for both the notched and smooth samples and the ranges
through which K0’ K8 and Kf vary for each test. These ranges do not
reflect trends during life (other than scatter) but are helpful when
comparing trends due to materials, life and notch geometry. Figures
17 through 22 are plots of Ko’ Ke’ and Kf versus life. Each plot shows
a long and short life test for a given material and notch geometry.
Scales for these plots were chosen to best show the distribution for
each set of tests. Figures 23 through 28 are duplicates of Figures 17
through 22 showing only K and are all drawn to the same scale to facili—

f

tate comparisons between the test sets.

29

 

30

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CHAPTER 6

SUMMARY AND CONCULSIONS

For all tests K was less than Kt and for all materials the

f
variance was greater for the elliptic notches than for the circular
notches. For 2024 aluminum the greatest difference between Kt and Kf
in terms of percent of Kt for elliptic versus circular notches was 8
percent versus 5 percent; for 7475 aluminum, 23 percent versus 11 per-
cent and 23 percent versus 7 percent for 1018 mild steel. Note the Kf
values for the 2024 samples were closest to Kt for both notch geometries
and for 7475 the farthest from Kt for all materials tested.

Table 2 shows excellent correlation between fatigue life of the
smooth and notched samples. In all cases but one Nf for the smooth
sample was less than Nf for the corresponding notched sample. This is
just as expected because notched samples were run until the I.S.G. out—
put was effected by a crack and that happens only after the crack extends
through the indentation site. This strongly suggests that notch root
strain can be modeled with smooth samples in which only the normal notch
root strain is reproduced.

The variance of K0’ K6’ and K depends on the material and the 1018

f
mild steel and 2024 aluminum show variances with strain range. For 1018

mild steel long life circular and elliptical tests K5, K0, and Kf are

stable throughout life. During short life cycling KO decreases while

43

 

 

 

 

44

K6 increases and Kf tends to decrease with life. For both notch types
Kf is lower for short life.

Kf for low cycle 2024 aluminum samples approaches stable long life
values. This is best seen in the elliptic samples (Figure 18). A large
jump in Kf, Ko’ and KS occurs at cycle 200 for the short life circular
sample. This transition is suspect because the local hystersis loops
radically change shape for the smooth sample but do not for the notched

sample between cycle 100 and cycle 200 and N for the smooth sample is

f
much less than Nf for the notched sample. It is possible that while

reproducing the notch strains in the smooth sample a system malfunction

caused an overload. Due to lack of samples another smooth sample was

 

not run. It is interesting to note that for the entire long life cir—
cular test K0 was greater than Ke° All values do not stabilize for the
2024 circular samples until after twenty cycles whereas the elliptic
long life values are stable throughout the entire life. The elliptic
short life values never stabilize but shift smoothly toward the stable
long life values with life.

K K0, and KE for 7475 aluminum stabilize after the first ten

f,
cycles for the circular samples and after the first twenty cycles for
the elliptic. K8 is less than KO throughout the entire long life

elliptic test and for portions of the short life test. For both sets of

tests stable values of Kf for short life are equal to or near stable

 

long life values of Kf.
In general the elliptical samples exhibit stable (or for 2024 low
life, consistent) notch concentration factors throughout life while the

circular samples (with the exception of 1018 mild steel) required ten

to twenty cycles to stabilize or develop smooth trends. It must be

 

45

noted that the 2024 elliptic low life sample never stabilized even
though Nf = 254. For bottnaluminumsstrain range had little or no

effect on stable Kf values for circular samples, however, for elliptical

samples 7475 showed two close but distinct values for long and short

life while short life 2024 tended toward stable long life values of Kf.

The 1018 mild steel reacted consistently for both notch types with a

lower diminishing K for short life and smooth stable trends throughout

f

life for all strain ranges.

To summarize: The problems associated with mounted strain gages
have been overcome with the I.S.G. and the techniques of reproducing
notch strain in smooth samples is a good one for obtaining notch stresses.

The characteristics of cyclic K seem to be material dependent. For 1018

f

mild steel Kf is stable throughout life with no initial stabilization

period and decreases with increasing strain range. Also for larger

strain ranges Kf decreases with life. For 2024 and 7475 aluminum cir—

cular notches Kf stabilizes after twenty cycles to a given value for

each material reguardless of strain range. For elliptic samples 2024

short life K values tend toward stable long life values which 7475 ex—

f
hibits decreasing values for Kf with increasing life. In general, for
any material elliptic samples stabilized quicker but showed greater var-
iance from Kt than circular samples.

There is still much to learn in this area. These tests should be
extended to shorter life regions and comprehensive life—Kf trends de-

veloped and modeled for various materials. Presently work is being done

in these areas.

 

 

    

. "a “a seals." 9
A

 

LIST OF REFERENCES

 

 

10.

LIST OF REFERENCES

Neuber, H., "Theory of Stress Concentration for Shear Strained
Prismatic Bodies with Arbitrary Non-linear Stress and Strain

Law," Journal of Applied Mechanics, Vol. 28, No. 4, Dec. 1961,
pp. 544-550.

Gowda, C. U. B., and T. H. Topper, ”On the Relation Between
Stress and Strain Concentration Factors in Notched Members in

Plane Stress," Journal of Applied Mechanics, Vol. 37, No. 1,
Mar. 1970, pp. 77-84.

Topper, T. H., R. M. Wetsel, and JoDean Morrow, "Neuber's Rule
Applied to Fatigue of Notched Specimens," Journal of Materials
JMLSA, Vol. 4, No. 1, Mar. 1969, pp. 200—209.

Crews, J. H., Jr., and H. F. Hardrath, ”A Study of Cyclic Plastic
Stresses at a Notch Root," Experimental Mechanics, Vol. 6, No. 6,
June 1966, pp. 313—320.

Box, W. A., "The Effect of Plastic Strains on Stress Concentration,"
Proceedings, Society for Experimental Stress Analysis, Vol. 8, No.

 

21, 1951, pp. 99—110.

Griffith, G. E., "Experimental Investigation of the Effect of Plastic
Flow in a Tension Paper with a Circular Mole," NACA Technical Note
1705, 1949, National Advisory Committee for Aeronautics.

Blatherwick, A. A. and B. K. Olsen, "Stress Redistribution in Notched
Specimens Under Cyclic Stress," ASD Technical Report 6l~451, 1961,
Aeronautical Systems Division, Wright-Patterson Air Force Base,
Dayton, Ohio.

Gowda, C. V. D. and T. H. Topper, "Cyclic Deformation Behavior of
Notched Mild Steel Plates in Plane Stress," Experimental Mechanics,
Vol. 12, No. 8, Aug. 1972, pp. 359—367.

Leis, B. H., "Cyclic Inelastic Deformation Behavior of Thin Notched
Plates," M.A.Sc. Thesis, University of Waterloo, Canada, 1972.

Evans, R. J. and K. S. Pister, "Constitutive Equations for a Class
of Nonlinear Solids," International Journal of Solids and Structures,
Vol. 2, 1966, pp. 427-445.

 

46

 

   
 
  

EEQK‘MTMX ’-"‘. Tit-1'1

...|Ir.1"._. ;_.!1 . _ . .. 1'. .'.. .lf '_!-’dus“

11.

12.

13.

14.

15.

16.

17.

18.

19.

23.

Wetzel, R. M., "Smooth Specimen Simulation of the Fatigue Behavior
of Notches," Journal of Materials, JMLSA, Vol. 3, No. 3, Sept. 1968,
pp. 646—657.

Morrow, JoDean, "Cyclic Plastic Strain Energy and FatiguetfifMetals,"
Internal Friction, Damping and Cyclic Plasticity, ASTM STP 378,
American Society for Testing and Materials, Philadelphia, 1965, pp.
45—87.

Manson, S. S. and M. H. Hirschberg, "Crack Initiation and Propaga—
tion in Notched Fatigue Specimens," Proceedings of the First Inter-
national Conference on Fracture, Vol. 1, Japanese Society for
Strength and Fracture of Materials, 1966, pp. 479-498.

Peterson, R. E., "Fatigue of Metals in Engineering and Design,"
Thirth—Sixth Edgar Marburg Lecture, American Society for Testing
and Material, 1962; Materials Research and Standards, Vol. 3, No.
2, Feb. 1963, pp. 122—139.

Dolan, T. J., "Non—linear Response Under Cyclic Loading Conditions,"
Proceedings of the 9th Midwest Mechanics Conference, Aug. 1965,
pp. 3—21.

Peterson, R. E., "Notch—Sensitivity," Metal Fatigue, Sines and
Waisman, eds., McGraw—Hill, New York, 1959, pp. 293—306.

Leis, B. N., C. V. B. Gowda, and T. H. Topper, "Some Studies of

the Influence of Localized and Gross Plasticity on the Monotonic

and Cyclic Concentration Factors," Journal of Testing and Evaluation,
JTEVA, Vol. 1, No. 4, July 1973, pp. 341—348.

Peterson, R. E., "Stress Concentration Factors," John Wiley and
Sons, Inc., 1974, pp. 150—196.

Coker, E. G., and L. N. G. Filon, "A Treatise on Photoelasticity,”
2nd Ed., Cambridge University Press, 1957.

Sharpe, W. N., Jr., "The Interferometric Strain Gage," Experimental

Mechanics 8, April 1968, pp. 164—170.

Sharpe, W. N., Jr., "Interferometric Surface Strain Measurement,"
International Journal of Non-Destructive Testing 3, 1971, pp. 51—76.

 

Sharpe, W. N., Jr., "A Short Gage—Length Optical Gage for Small
Strain,” Experimental Mechanics, 14, 1974, pp. 373—377.

Sharpe, W. N., Jr., "Development and Application of an Interfero-

metric System for MEasuring Crack Displacements," Final Report on
Grant NSG 1148, June 1976.

47

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