MEASUREMENT OF THE ELASTIC- We BOUNDARY

AROUND comwoanD FASTENER HOLES

Dissertation for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
SARAVUT POOLSUK
1977

LIBRARY

Michigan State’-
University "~

 

This is to certify that the
thesis entitled

Measurement of the Elastic—Plastic Boundary Around
Coldworked Fastener Holes

presented by

Saravut Poolsuk

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in Mechanics

 

 

 

Major profes

m, 7% meager

Date July 15, 1977

 

0-7639

ABSTRACT

MEASUREMENT OF THE ELASTIC-PLASTIC BOUNDARY
AROUND COLDWORKED FASTENER HOLES

By

Saravut Poolsuk

A fastener hole which is pre-expanded with an oversize
mandrel is known as a coldworked hole. Coldworking of the hole
creates residual compressive stresses around the hole. These com-
pressive stresses reduce the high tensile stresses procuced by
stress concentration near the hole. This results in the fatigue
life enhancement of the structural component under applied cyclic
loading. The elastic-plastic boundary around a coldworked hole,
caused by the hole expansion with an oversize mandrel, is an impor-
tant measure of the amount of coldworking.

The existing theories developed for the coldworked hole are
discussed, their assumptions and deficiencies pointed out. The
elastic-plastic boundary given by the various theories is not the
same for the same loading conditions. Knowledge of the elastic-
plastic boundary around a coldworked hole is required in both
theoretical analysis and design. The purpose of this thesis is to
measure the elastic-plastic boundary around coldworked fastener
holes and compare with the theoretical predictions.

Two experimental techniques have been developed to evaluate

the existing theories. The two techniques used to locate the

Saravut Poolsuk

elastic-plastic boundary are foil gages and thickness change measure-
ment. Their accuracies have been compared and found to be similar.

To test the validity of the theories, two kinds of aluminum,
7075-T6 and 1100, with two different thicknesses: 6.35 mm. (l/4 in.)
and 3.l8 mm. (l/8 in.), are used as specimens. Holes in the circular
specimens are coldworked with four different sizes of the oversize
mandrel. This industrial coldworking process is found to give non-
uniform deformation through the specimen thickness.

The final results of the measurements around the coldworked
holes obtained by the thickness change method are presented in
dimensionless form for hole size, amount of coldwork, and material.
The comparisOn of the experiments with seven theories shows that the
Nadai theory is the best one to predict the elastic-plastic boundary
around the coldworked 3.18 mm. (l/8 in.) thick plate,while the
Sachs and Rich-Impellizzeri theories are suitable for the coldworked
6.35 mm. (l/4 in.) thick plate for smaller amounts of coldwork.

None of the theories predicts the elastic-plastic boundary well for
the 6.35 mm. (l/4 in.) thick plate for larger amounts of coldwork.
Also, none of them agrees with the experiments of the softer llOO
aluminum. In addition, the results show that the plate thickness is
an important factor in determining the plane condition during cold-
working process. The experimentally determined final diameters of

relaxed holes are also presented and compared with four theories.

MEASUREMENT OF THE ELASTIC-PLASTIC BOUNDARY
AROUND COLDWORKED EASTENER HOLES

By

Saravut Poolsuk

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Metallurgy,
Mechanics and Materials Science

1977

$07048

ACKNOWLEDGMENTS

The author would like to express his heartfelt appreciation
to his major professor and chairman of his guidance committee,
Professor William N. Sharpe, Jr., who suggested the problem and
provided invaluable guidance and encouragement through all phases of
this study.. Thanks are also extended to the other members of his
guidance committee, Dr. Denton D. McGrady, Dr. George E. Mase, and
Dr. Carl L. Foiles. Special thanks are due to Dr. McGrady, who
spent many hours correcting the author's writing.

The author wishes also to thank Mr. Donald L. Childs and
Mr. Robert E. Rose, respectively for their helpful preparation of
specimens and electrical equipment. Financial support was granted
through his major professor's research by the Air Force Office of
Scientific Research, Bolling Air Force Base, Washington, D. C. 20332.
This support is gratefully acknowledged.

Special thanks also are due to his dear wife, Rutchaneeporn,
and daughter, Ketsanee, for their understanding, help, and encourage-

ment throughout the course of this study.

ii

TABLE OF CONTENTS

Chapter Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . v

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

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . l

 

 

 

 

1.1. Coldworked Fastener Holes . . . . . . . . . . . . 1
1.2. Theoretical Solutions . . . . . . . . . . . . . . 6
1.3. The Elastic-Plastic Boundary . . . . . . . . . . . 7
1.4. Coldworking,Procedure . . . . . o . . . . . . . 9
1.5. Overview of Dissertation . . . . . . . . . . . . . 10

 

2, MATERIAL SPECIFICATIONS . . . . . . . . . . . . . . . . . 12
3. HOLE PREPARATION AND COLDHORKING . . . . . . . . . . . . I7

‘36]. H0] P e t. o o q o o o o o o o a o o o o o o 17
302. COIdWQIKEHQ PYQQQdUL§_. o o o o o o o o o o 0.0 o o 19

4. THEORIES OF RESIDUAL STRESSES AROUND A COLDNORKED HOLE . 27

 

 

 

 

 

 

 

 

 

 

 

 

.4.1. Introduction . . . . . . . . . . . . . . . . . . 27
4.20 Nada.i Theory 0 o o a o o o a o o o o o o e o o o o 33
4030 swa1nger Theor¥,- 0 o o o o o o o o o o o o o o o o 35
4.4. Taylor Theonx,. . . . . . . . . . . 36
4.5. Alexander-Ford and Mangas arian Theories . . . . . . 37
4.6. Carter-Hanagud Theory . . . . . . . . . . . . . . . 38
4.7. ’Hsu-Forman Theorr . . . . . . . . . . . . . 39
4.8. Adler-Dupree's F nite Element Method . . . . . . . 42
4.9. Potter-Grandt Theory . . . . . . . . . . . . . . . 43
4.10. Thick-Walled Tube Theory . . . . . . . . . . . . . 46
_4.11. Rich-Impellizzeri Theory . . . . . . . . . . . . . 47
4.12. Discussion . . . . . . . . . . . . . . . . . . . 49

 

 

 

 

5. ELASTIC-PLASTIC BOUNDARY MEASUREMENT TECHNIQUES . . . . . 53
5.1. Nature of Deformation around a Coldworked Hole . . 53
,5.2. Foil Strain Gage Technique . . . . . . . . . . . . 55
5.3. Thickness Change Measurement Technique . . . . . . 74
45.4. Photoelastic Coating Technique . . . . . . . . . . 87

 

Chapter Page
5, ELASTIC-PLASTIC BOUNDARY MEASUREMENT . . . . . . . . . . . 92

6.1. Elastic-Plastic Boundary Measurement with Foil 92

Gages o o a o o o o o o 6 o o o o 6 I o o o o o 0

6.1.1. 6.35 mm. (1/4 in.) Thick, 7075-T6
Specimen Data

6.1.2. 3.18 mm. (1/8 in.) Thick, 7075-T6
Specimen Data.

6.1.3. 3.18 mm. (1/8 in.) Thick, Aluminum Type
1100 Specimen Data

6.2. Elastic-Plastic Boundary Measurement with
ThiCkness Change Measurement . . . . a . . . . . 98

6.2.1. 6.35 mm. (1/4 in.) Thick, 7075-T6
Specimen Data

6.2.2. 3.18 mm. (1/8 in.) Thick, 7075-T6
Specimen Data

6.2.3. 3.28 mm. (1/8 in.) Thick, Aluminum Type
1100 Specimen Data

6.3. Discussion . . . . . . . . . . . . . . . . . . . . . 106-
7. COMPARISON OF THEORIES WITH EXPERIMENTS . . ‘,‘ . . . . . 114
8. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . 127
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . 131

iv

Table

2.1
3.1

3.2

3.3

6.1

6.2

6.3

6.4

6.5

7.1

7.2

LIST OF TABLES

Mechanical properties of materials used . . . . . . . .

Original diameters of holes for a typical set
of 6.35 mm. (1/4 in.) thick, 7075-T6 specimens . . .

Original diameters of holes for a typical set
of 3.18 mm. (1/8 in.) thick, 7075-T6 specimens . . .

Original diameters of holes for a typical set
of 3.18 mm. (1/8 in.) thick, type 1100 specimens. . .

Measured r la and its uncertainty obtained from foil gage
technique when foil gages were applied on one side and on
both sides surfaces of the 6.35 mm. (1/4 in.) thick,
7075-T6 aluminum specimens ...............

Measured r la and its uncertainty obtained from foil
gage tecthque with foil gages applied to one side

and to both sides of the 3.18 mm. (1/8 in.) thick,
7075-T6 aluminum specimen . . . . . . . . . . .....

Measured r la obtained from thickness change
measurement technique for the 6. 35 mm. (1/4 in.)
thick, 7075- T6 aluminum specimens . . . . . . . . . .

Measured rp/a obtained from thickness change
measurement technique for the 3.18 mm. (1/8 in.)
thick, 7075- T6 aluminum specimens . . . . . . . . . .

Measured r p/a obtained from thickness change
measurement technique for the 3.18 mm. (1/8 in.)
thick, type 1100 aluminum specimens with +0. 005 in.
scale of Daytronic amplifier . . . . . . . . . . . .

Percent differences between the measured r p/a
obtained from thickness change measurement for
the 3.18 mm. (1/8 in.) thick, 7075-T6 aluminum
and the Nadai and Hsu-Forman theories . . . . . . .

Percent differences between the measured rp/a
obtained from thickness change measurement for
the 6.35 mm. (1/4 in.) thick, aluminum and
the Thick-Walled Tube and Rich-Impellizzeri
theories 0 O O O O O O O O O I O O I O O O O O O O O

Page

15

20

21

22

93

96

99

103

107

117

118

Table Page

7.3 Average measured r /a obtained from thickness change
measurement and predicted r /a from four theories
for the 3.18 mm. (1/8 in.) ghick, type 1100
a1uminum O O o O O O 0 O O O O O O O O O 0 O O O O O 119

7.4 Measured final diameters of 6.35 mm. (1/4 in.) thick,
7075-T6 specimens after the mandrel was pulled
through the holes . . . . . . . . . . . . . . . . . . 122

7.5 Measured final diameters of 3.18 mm. (1/8 in.) thick,
7075-T6 specimens after the mandrel was pulled
throughtheh01esoa..o..o.......... 123

7.6 Measured final diameters of 3.18 mm. (1/8 in.) thick,
type 1100 aluminum specimens after the mandrel was
pulled through the holes . . . . . . . . . . . . . . 124

7.7 Comparison of measured final diameters with theories
for the 7075-16 aluminum specimens as four different
sizes of the oversize mandrels were pulled through
the holles O O o I O O O O O O O O O 0 O O O O O O O 126

vi

Figure

.1

1

#000000

LIST OF FIGURES

Stress concentration factor at the edge of a hole of
radius "a" in infinite plate due to tensile

stress, 0' .....................

Residual stresses predicted by the Nadai and Hsu-
Forman theories for a 0.305 mm. (0.012 in.)
diametral expansion of a 6.60 mm. (0.26 in.) hole

in 7075-T6 aluminum ................

Elastic-plastic boundary predicted by Hsu-Forman's
theory (8) located where the predicted loading
radial and tangential strains are equal in
magnitude and opposite in sign for a 0.305 mm.
(0.012 in.) diametral expansion of 6.60 mm. (0.26

in.) hole in 7075-T6 aluminum ...........

Stress-strain curves for the two different sheets

of 7075-T6 aluminum ................

Stress-strain curves for the 3.18 mm. (1/8 in.)

thick, type 1100 aluminum .............

Photomicrographs of the two sheets of 7075-T6
aluminum (100x) and one sheet of 1100 aluminum

(500 X) ......................
Specimen configuration ................

Photograph of mandrels and sleeves ..........

Schematic of the J. 0. King coldworking process

Geometry and coordinate system used in coldworking

hole theories ...................

Loading elastic-plastic boundary, B, vs. radial

displacement at the hole, 0 ............

Elastic-plastic boundaries predicted by various

theories ......................

Photograph of the deformed region around the hole

in a 6.35 mm. (1/4 in.) thick specimen (100 X) . . .

vii

Page

13

14

16

18

23
25

28

45

50

54

Figure Page

5.2 Photograph of equipment set up for the foil gage
teChnique O O 0 O I O O I O O O O 0 DO. 0 O O O O 0 57

5.3 Photograph of bonded specimen with radial and
tangential foil gages on the upper side . . . . . . 58

5.4 Radial and tangential strains measured with foil
gages on the upper side of the 6.35 mm. (1/4 in.)
thick, 7075-T6 aluminum (P.18) as the mandrel of
0.076 mm. (0.003 in.) radial expansion was
pulled through the ho1d . . . . . . . . . . . . . 60

5.5 Radial and tangential strains measured with foil
gages on the upper side of the 6.35 mm. (1/4 in.)
thick, 7075-16 aluminum (P.18) as the mandrel of
0.102 mm. (0.004 in.) radial expansion was
pulled through the hole . . . . . . . . . . . . . . 61

5.6 Radial and tangential strains measured with foil
gages on the upper side of the 6.35 mm. (1/4 in.)
thick, 7075-T6 aluminum (P.18) as the mandrel of
0.127 mm. (0.005 in.) radial expansion was
pulled through the hole . . . . . . . . . . . . . 62

5.7 Radial and tangential strains measured with foil
gages on the upper side of the 6.35 mm. (1/4 in.)
thick, 7075-T6 aluminum (P.18) as the mandrel of
0.152 mm. (0.006 in.) radial expansion was
pulled through the hole . . . . . . . . . . . . . 63

5.8 Radial and tangential strains measured with foil
gages on the upper side of the 3.18 mm. (1/8 in.)
thick, 7075-T6 aluminum (P.20) as the mandrel of
0.076 mm. (0.003 in.) radial expansion was
pulled through the hole . . . . . . . . . . . . . 64

5.9 Radial and tangential strains measured with foil
gages on the upper side of the 3.18 mm. (1/8 in.)
thick, 7075-T6 aluminum (P.20) as the mandrel of
0.102 mm. (0.004 in.) radial expansion was
pulled through the hole , . . . . . . . . . . . . 65

5.10 Radial and tangential strains measured with foil
gages on the upper side of the 3.18 mm. (1/8 in.)
thick, 7075-T6 aluminum (P.20) as the mandrel of
.0.127 mm. (0.005 in.) radial expansion was
pulled through the hole . . . . . . . . . . . . . 66

viii

Figure Page

5.11 Radial and tangential strains measured with foil gages
on the upper side of the 3.18 mm. (1/8 in.) thick,
7075-T6 aluminum (P.20) as the mandrel of 0.152 mm.
(0.006 in.) radial expansion was pulled through
the hole ...................... 67

5.12 Typical method used to locate the elastic-plastic
boundary, r , of typical 6.35 mm. (1/4 in.) thick,
7075-T6 aTuRinum (P.18) in which foil gages were
applied on one side ................ 68

5.13 Typical method used to locate the elastic-plastic
boundary, r , of typical 3.18 mm. (1/8 in.) thick,
7075-T6 aluRinum (P.20) in which foil gages were
applied on one side ............. . . . 69

5.14 Typical method used to locate the elastic-plastic
boundary, r , of typical 6.35 mm. (1/4 in.)
thick, 70759T6 aluminum (P.14) ........... 71

5.15 Typical method used to locate the elastic¢p1astic
boundary, r , of typical 3.18 mm. (1/8 in.)

thick, 70759T6 aluminum (P.28) ........... 72
5.16 Calibration curve of the LVDT on $0.005 in. scale

of Daytronic amplifier model 300C . . . .. ..... 77
5.17 Calibration curve of the LVDT on $0.001 in. scale

of Daytronic amplifier model 300C ......... 78
5.18 Calibration curve of linear potentiometer ...... 79

5.19 Photograph of the thickness change measurement
setup ....................... 82

5.20 Photograph of the LVDT and linear potentiometer
mounted in holders ................. 83

5.21 Typical method used to locate the elastic-plastic
boundary, r , from the thickness profile of the
6.35 mm. (T94 in.) thiCk, 7075-T6 aluminum (P.12)
on direction number 2 and $0.001 in. scale after
four different sizes of oversize mandrel were
pulled through the hole .............. 84

5.22 Typical method used to locate the elastic—plastic
boundary, r , from the thickness profile of the
3.18 mm. (178 in.) thick, 7075-16 aluminum (P.28)
on direction number 2 and $0.001 in. scale after
four different sizes of oversize mandrel were
pulled through the hole .............. 85

Figure

5.23

Typical method used to locate the elastic-plastic
boundary, r , from the thickness profile of the
3.18 mm. (198 in.) thick, type 1100 aluminum (P.36)
on direction number 2 and $0.005 in. scale after
four different sizes of oversize mandrel were

pulled through the hole ..............

Isochromatic-fringe patterns and elastic-plastic
boundaries obtained around the coldworked 6.35 mm.
(1/4 in.) thick specimen (P.4) as four different
sizes of oversize mandrel were pulled through the
hole .......................

Comparison of the measured r /a obtained from foil
gages applied on one side and on both sides of the

6.35 mm. (1/4 in.) thick, 7075-T6 aluminum .....

Comparison of the measured r /a obtained from foil
gages applied on one side 3nd on both sides of the

3.18 mm. (1/8 in.) thick, 7075-T6 aluminum .....

Comparison of the measured r /a obtained from thick-
ness change measurement teBhnique with 1 CW. and 4

CM. on one specimen of the 6.35 mm. (1/4 in.) thick,
7075-T6 a1uminum ..................

Comparison of the measured r /a obtained from thick-
ness change measurement teBhnique with 1 CW. and
4 CM. on one specimen of the 3.18 mm. (1/8 in.)

thick, 7075-T6 aluminum ..............

Comparison of the measured r /a obtained from thick-
ness change measurement teBhnique with 1 CW. and
4 CM. on one specimen of the 3.18 mm. (1/8 in.)
thick, type 1100 aluminum with $0.005 in. scale
of Daytronic amplifier ..............

Comparison of the measured r /a obtained from thick-
ness change measurement anB foil ga e (both sides)
techniques of the 6.35 mm. (1/4 in.1 thick,

7075-16 aluminum (P.11,12,l4) ...........

Comparison of the measured r /a obtained from thick-
ness change measurement and foil gage (both sides)
techniques of the 3.18 mm. (1/8 in.) thick,
7075-T6 aluminum (P.25,28,30) ...........

Page

86

90

94

97

101

105

108

111

Figure

7.1 Comparison of the elastic-plastic boundaries
predicted by various theories and experiments
for 7075-T6 aluminum ...............

7.2 Comparison of the measured final relaxed hole

diameters with four theories (Nadai, Hsu-Forman,
Potter-Grandt, and Adler-Dupree) ..........

xi

125

CHAPTER 1
INTRODUCTION

It is known that the stress concentration is very high at the
edge of a hole in a plate which is subjected to a uniform tensile
stress at its ends. Figure 1.1 shows the elastic stress distributions
on one side of an open hole in an infinite plate with a uniform ten-
sile stress at infinity (1,2). The tangential stress, 06, at the hole
wall is about three times the remote stress, 0, and decreases rapidly
away from the hole. These high stresses at the hole edge may locally
exceed the material's yield stress. A crack will then initiate and
lead to an early failure of the plate (2). When the plate is sub-
jected to cyclic loading, its failure will occur at a stress lower
than its yield point, and its service life is much shorter than
usual, viz., in the aircraft structures (3,4,5). One common method
for increasing the service life is coldworking.

The purpose of this thesis is to experimentally measure the
elastic-plastic boundary around coldworked fastener holes and compare

with the theoretical predictions.

l.l Coldworked Fastener Holes

Many investigators (3.4.5.6,7) have indicated that pre-
expansion of the hole before loading reduces the fastener hole stress
concentration effect by creating beneficial compressive residual

1

stresses at the hole edge. These compressive residual stresses are
created by plastic deformation of the material surrounding the hole,
resulting from expansion of the hole and subsequent removal of the

radial load. As a circular hole in a flat infinite plate having a

0W. CV0)

 

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(3

 

 

 

 

(D
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d)
4
1
J»
P
db

 

1.0 ' 2.0 3.0 4.0 5.0 6.0

Stress Concentration F actor
8

Figure l.l.--Stress concentration factor at the edge of a hole of
radius "a" in infinite plate due to tensile stress, 0.

well-defined yield point is expanded either by uniform pressure or
radial displacement, yielding will start along the hole edge. If the
pressure is increased further, the plate will yield in a zone,

a$r<r but will remain elastic beyond it, rarp, (where a and rp

p.
represent the hole radius and elastic-plastic boundary around the
hole, respectively). After the removal of pressure the compressive
residual stresses are created by the "springback" of the elastic
material around the plastic zone.

Fastener holes are by far the most common stress concentra-
tions existing in aircraft structures. These stress concentrations
affect and are usually the controlling factors in fatigue life of
structural components of the aircraft. Gran et a1. (4), in 1971,
indicated that flaws originating at fastener holes are among the
most prevalent sources of fracture in aircraft structures. Allen-
Ellis (3) and Grandt-Gallagher (6) suggested that proper interference
fitting in the structure of aircraft wings not only prevents possible
cracks but also prolongs the fatigue life of the structure. One
common technique for enhancing fatigue life of fastener holes is to
introduce a controlled residual stress field around the fastener
holes by means of an interference-fit fastener (6) or by pre-
expansion of the holes with an oversize mandrel (5), which is known
as the coldworking process. Coldworking differs mainly from inter-
ference-fit applications in allowing the hole to radially unload
before inserting the fastener. Tiffany et a1. (5), in 1973, con-
cluded from their investigation of fatigue and stress-corrosion tests

of fastener holes processed in various ways that coldworking is

advantageous. Fatigue life of the cracked fastener holes has been
shown to be substantially increased through the coldworking operation.
Basic understanding of the residual nonlinear stress around
prestressed holes will come from theoretical and experimental studies
of representative problems. The theories developed have shown that
the compressive residual tangential stress at the edge of a cold-
worked hole is very high and decays rapidly. At a distance approx-
imately equal to the diameter of the hole (depending on the amount of
coldworking used), the residual stress becomes tensile and the maximum
tension stress occurs at the elastic-plastic boundary, rp. The resid-
ual stress distributions predicted by the Nadai (7) and the Hsu-
Forman (8) theories are shown in Figure 1.2. Discussion of the
theories in Chapter 4 and comparison with the measurements in
Chapter 7 will show that this earliest theory by Nadai is still
very useful. The elastic-plastic boundary, rp, predicted by each
theory occurs at the tensile peak of the residual tangential stress.
The value of residual tangential stress at this boundary is equal to
the residual radial stress in magnitude. Figure 1.2 shows a very
large compressive residual stress next to the edge of the hole. This
stress tends to counteract the tensile stress concentration which
would otherwise occur in practice. Analytical, numerical, and finite-
element theories have been used to predict the residual stresses and
strains around a prestressed hole; but the validation of these

theories with experimental results has been extremely limited.

 

 

 

 

 

Ksi MPG
6011400
so.
40:300
Khan
20.
+100
10-
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\ l
3 To \ / Distance in Hole Edgetmm __...—--—-— 0;
\ l I’.‘"".—
g \\\ / a- ”TT
(‘6 ‘ - r" I |
30300 / 1 I
.. / l .
-300 // I l_.rp(Nadai)-=2.10c
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/
Nadai
7° 1500 ll
80 I --—Hsu-Forman
. I
90- 600/
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110‘
120-
'30 Figure l.2.--Residua1 stresses predicted by the
Nadai and Hsu-Forman theories for a
I40 0.305 m. (0.012 in.) diametral
.000 expansion of a 6.60 m. (0.26 in.)

150 hole in 7075-T6 aluminum.

1.2 TheoreticalSolutions‘

 

There are several theoretical solutions to the problem of an
expanding hole in a thin sheet of material, discussed in more detail
in Chapter 4 of this thesis, all of which assume that the hole is
radial loaded (prescribing either pressure or displacement at the
hole) until the material near the hole is loaded above its yield
stress. All of the theories (except the ones by Sachs and Rich-
Impellizzeri) assume that a state of plane stress exists everywhere
in the infinitely large sheet. After the loading is removed at the
hole edge, most theories assume that the material unloads elastically
with no reverse yielding.

The theories may be classified according to their kinds of
solution. Analytical solutions, in which expressions for the stresses
and strains are given in closed form, are desirable because of their
ease of application. Some numerical solutions in which the differ-
ential equations, along with their boundary conditions, are solved
via a computer are also available. As the results of these numerical
solutions are presented in dimensionless form, they are also easy to
use. One finite-element solution which employs an elastic-plastic
(finite-element computer code also exists. Equally as important as
the form of the solution is the assumption about the material behavior
in the plastic region. The constitutive behavior assumed ranges from
perfectly plastic to elastic-perfectly-plastic to a two-parameter
power law. A fair amount of research has been done on this residual

stress problem in the past few years.

1.3 The Elastic-Plastic Boundary

 

Designers encounter considerable difficulty because no plas-
ticity solution is available for the coldworked hole that makes it
possible to evaluate the true stress and strain distributions or the
elastic-plastic boundary. The elastic-plastic boundary not only
determines the residual stress field around a coldworked hole but
also is very important in the design and spacing of hole locations in
practice. Only the theory of Potter-Grandt (9) has provided the
relation between rp and ua in dimensionless form. The plot of this
relationship is useful to the designer in that he can directly pre-
dict the elastic-plastic boundary of his plate geometry and material
associated with amount of coldwork. But the value of the elastic-
plastic boundary predicted by this theory is relatively small when
compared with the others, due to its assumptions as discussed in
Chapter 4. Therefore, there still is need for a theory which will
accurately predict rp for a particular coldworked hole. A simple
and precise technique is required to initially evaluate theories,
which is the objective of this thesis. The elastic-plastic boundary
can be located not only by the residual stress distributions (as
described in section 1.1) but also by the strain distributions and
the change in thickness of the plate.

All theories indicate that the elastic-plastic boundary
around a coldworked hole is exactly at the location where loading or
unloading radial and tangential strains are first equal in magnitUde.
To indicate how to locate the elastic-plastic boundary, loading

strains predicted by the Hsu-Forman theory are plotted in Figure 1.3.

5.00-l

4&X).

 
    
 

 

 

Distance from the Edge of The Hole (mm)

:5 3-°°' /—Radial(Comp.)
.3
E
(7) 2130' /~Tangentia| (Ten)
ILMD .
i
I
A | n J a A
0 2 | 4 6 8 MD
I
I
I

9‘

Figure 1.3.--Elastic-p1astic boundary predicted by Hsu-Forman (8)
located where the predicted loading radial and
tangential strains are equal in ma nitude and opposite
in sign for a 0.305 mm. (0.012 in.§ diametral expansion
of 6.60 mm. (0.26 in.) hole in 7075-T6 aluminum.

This theory may be the most general of all theories discussed in
Chapter IV.
Elasticity theory shows that the radial and tangential

stresses, 0 and 06, are equal in magnitude and opposite in sign.

Y‘

Since 02 = O, the elastic strain equation,

_ l "
E:z _ E'LEZ'U(Or+°QZl’
shows that the transverse strain, 82, vanishes in the elastic region
and at the elastic-plastic boundary. This means that there is no
change in the plate thickness at the elastic-plastic boundary. In
other words, the elastic-plastic boundary can be located precisely

where the plate thickness starts to change.

1.4 Coldworking Procedure
The coldworking procedure studied in this thesis has been

developed by J. 0. King, Inc.

711 Trabert Avenue, N.W.

Atlanta, Georgia 30318
In this procedure a thin-walled sleeve is first inserted into the
hole, after which a tapered mandrel is pulled through the sleeve.
After the mandrel has been removed, the sleeve may or may not be
removed before the fastener is inserted. The specific amounts of
expansion used in this study, 0.076 (0.003 in.), 0.102 (0.004 in.),
0.127 (0.005 in.), and 0.152 (0.006 in.) mm. radial expansions of
a 6.60 mm. (0.26 in.) hole were chosen to study their effects on

the elastic-plastic boundary around the coldworked hole. None of the

modern theories account for the presence of a sleeve. Accordingly,

10

and in order to avoid some errors of measurement for smaller amounts
of coldwork, the sleeve was removed for these experiments.

It may be argued that if one wishes to evaluate theories of
plastic radial expansion, one should match the boundary conditions as
closely as possible. As mentioned earlier, one of the objectives of
this thesis is to initially evaluate the theories. This will be done
by examining (by comparison of experiments with theories) the capa-
bility of the theories to model this practical coldworking process.
An experimental study in which one carefully generated a radial
loading would be quite useful in deciding what simplifying assumptions
are permissible. However, the industrial process would still have to

be studied to determine the theory's applicability.

1.5 Overview of Dissertation

The materials used in this thesis are aluminum 7075-T6 and
1100, whose properties are given in Chapter 2. Care must be taken
in preparing the original hole to exact dimensions and coldworking it
in a manner that is similar to the industrial process and yet con-
trolled enough to permit reproducible tests. The procedures for pre-
paring and coldworking the specimens are given in Chapter 3. The
background of the problem and the existing theories are discussed at
some length in Chapter 4. Various possible techniques for locating
rp, including the instruments, precautions, and procedures are
described in Chapter 5.

The results of a series of coldworking experiments for two

specimen thicknesses and two kinds of aluminum are given in Chapter 6.

11

The comparison of the theories with the experimental results is

discussed in Chapter 7, and closing comments are in Chapter 8.

CHAPTER 2
MATERIAL SPECIFICATIONS

The materials used for these studies were aluminum types
7075-16 and 1100. Two thicknesses--6.35 mm. (1/4 in.) and 3.18 mm.
(1/8 in.) for type 7075-T6 and 3.18 (1/8 in.) for type 1100 were
tested. Aluminum type 7075-T6 is a "7XXX" series (Zn-Al) and high-
strength alloy used for aircraft, space vehicles, and other applica-
tions where great strength is required. It is weldable but is
subjected to stress corrosion and therefore must be protected by
cladding or other surface protection methods. Generally, its precip-
itation hardening heat treatment begins with the solution heat
treatment at 466°C. (870°F.) and then quenching at room temperature
and reheating (artificial aging) at 121°C. (250°F.) for 25 hours.
Typically, this material contains 7% Zn, 2-3% Mg, and up to 1260% Cu
plus a small amount of Cr to reduce stress corrosion susceptibility.
Aluminum type 1100 is a pure aluminum of 99.00% or more purity with
no special treatment.

Tension test specimens were cut from the same sheet as the
specimens and tested according to ASTM E8 procedures. Foil gages
were used to measure the strains. The stress-strain curves were

obtained and are presented in Figures 2.1 and 2.2. Their mechanical

12

13

lWPa. P Kel

6501i

'90
SCI).

 

5504r80

500 a

4501

40C)-

350 -

30C)-

25Cl.

 

Stresses

200 $30 — 6.35 mm.( mm.)

_...-- 3.l8 mm. (I/Bin.)
15C)

H30 -

50'-

 

qt-

1 1 l
I’ I T ' I

o. I 11) i : 233
Strains (96)

Figure 2.l.--Stress-strain curves for the two different sheets of
7075-T6 aluminum.

14

MPO Ksl

 

 

 

A A A A - A I A L n A 4
f v v v v

0 l t 1.00 2.00
Strains 0%)

Figure 2.2.-~Stress-strain curve for the 3.18 mm. (1/8 in.) thick,
type 1100 aluminum.

15

properties, as obtained iron the stress-strain curve and Rockwell

hardness tester, are given in Table 2.1.

TABLE 2.1--Mechanica1 properties oanaterials used.

 

Aluninun
6.35 mn.(l]4 in) 3.18m.(l78 in) 3.18m.(l/8 in)
thick.7075-T6 thick.]075-T6 thick. Type 1100

Ultimate Strength 589.1 527.1 79.2

(MPa.) (85.5 Ksi.) (76.5 Ksi.) (11.5 Ksi.)
Yield Strength

at 0.2% offset 647.8 503.0 32.6

(MPa.) (79.5 Ksi.) (73.0 Ksi.) (4.73 Ksi.)
Modulus of 3 3 3
Elasticity 69.6 x 103 68.2 x 103 67.5 x 10
(MPa.) (10.l)<10 Ksi.) (9.9 x 10 Ksi.) (9.8 x 103Ksi.)
Poisson's

ratio 0.31 0.31 0.28
Hardness 91RB 90RB 28RH

 

Photographs of their microstructures are presented in Figure

2.3 and it is seen that the microstructure is essentially the same

for both thicknesses of 7075-16 aluminum.

6.35 mm. (1/4 in.)
thick, 7075-T6

3.18 mm. (1/8 in.)
thick, 7075-T6

3.18 mm. (1/8 in.)
thick, type 1100

 

Figure 2.3.--Photomicrographs of the two sheets of 7075-T6 aluminum
(100X) and one sheet of 1100 aluminum (SOOX).

CHAPTER 3
HOLE PREPARATION AND COLDWORKING

3.1 Hole Preparation

It is necessary to generate round, nontapered holes to a
specific dimension in order to measure accurately the amount of cold-
working deformation. A certain tolerance on the holes is required if
one is to compare coldworking of various specimens. The variability
in hole size and coldworking process that would be represented in an
industrial atmosphere cannot be permitted here.

The specimens were circular plates of 178 mm. (7.00 in.)
outSide diameter, with a hole of 6.60 mm. (0.260 in.) typical diameter
at the middle, as illustrated in Figure 3.1. The reason for selecting
the 178 mm. circular plate is because of the limitation of the tool-
maker's microscope in measuring the hole diameter of the plate. The
ratio §-= 27 is large enough for treatment as an infinitely large
plate (9 ). Holes were prepared by first drilling them with a
6.35 mm. (1/4 in.) drill and then using a honing machine to bring
the diameter up to 6.60 mm. (0.260 in.). The honing machine pro-
duced straight sides in the hole (no evidence of spiraling) and
square edges of the hole. The size was determined with a plug gauge
with a "go" cylinder of 6.599 mm. (0.2598 in.) and a "no-go" cylinder
of 6.609 mm. (0.2602 in.). Upon receipt from the machine shop, the

holes in the specimens were measured with a toolmaker's microscope

17‘

18

”XS
I, _/_ \‘5.
Q.) I" "‘ "i <2 ".9"
\ \\ ”03
l ~~-=<

@«——Numbered Thickness Change
Measurement Direction

6.6 mm (026m.)lnside Diameter

178 mm. ( 7.0 in.) Outside Diameter

Figure 3.l.--Specimen configuration.

19

equipped with an X-Y stage. The greatest uncertainty in this measure-
ment is in locating the edge of the holes accurately. Measurements
were made along diameters at 450 intervals, and each measurement was
repeated at least three times. The variation in repeated measurements
was usually less than 0.003 mm. (0.0001 in.). Original diameter
measurements were made on both sides of the specimens. The values of
the diameters on both sides were the same, so that only upper side
original diameters of all specimens were recorded and are given in
Tables 3.1, 3.2, and 3.3 for the 6.35 mm. (1/4 in.), 3.18 mm. (1/8 in.)
thick 7075-T6 aluminum and 3.18 mm. (1/8 in.) thick aluminum type
1100, respectively. The measurements show that the holes are of

acceptable roundness and diameter.

3.2 Coldworking Procedure

 

The J. 0. King, Inc. ACRES coldworking sleeves part numbers:
JK 5535008N04L for the 6.35 mm. (1/4 in.) thick specimens
JK 5535C08N0r for the 3.18 mm. (1/8 in.) thick specimens
were employed in the expansion of the initial hole diameter. Four
associated mandrels were 0. 0. King, Inc. part numbers:
JK 6540-08-251 for the 0.076 mm. (0.003 in.) radial expansion
JK 6540-08-253 for the 0.102 mm. (0.004 in.) radial expansion
JK 6540-08-255 for the 0.127 mm. (0.005 in.) radial expansion
JK 6540-08-257 for the 0.152 mm. (0.006 in.) radial expansion.
The coldworking sleeves were supplied with the mild steel
washer attached (see Figure 3.2), and a dry film lubricant of 0.013 mm.

(0.0005 in.) thick applied to the inside and outside. The function of

20

TABLE 3.l.--Original diameters (in mm.) of holes for a typical set of
6.35 mm. (1/4 in.) thick, 7075-16 specimens.

 

 

Specimen (1) (2) (3) (4)
P.1 6.622 6.622 6.619 6.624
P.2 6.619 6.619 6.619 6.619
P.3 6.614 6.607 6.609 6.614
P.4 6.614 6.614 6.612 6.609
P.5 6.612 6.612 6.607 6.609
P.6 6.622 6.619 6.614 6.617
P.7 6.629 6.629 6.627 6.629
P.8 6.624 6.624 6.624 6.624
P.9 6.619 6.619 6.619 6.619
P.1O 6.614 6.617 6.619 6.617
P.11 6.619 6.622 6.617 6.622
P.12 6.622 6.614 6.614 6.619
P.13 Hole too big

P.14 6.614 6.612 6.612 6.617
P.15. 6.614 6.617 6.617 6.614
P.16 6.612 6.617 6.612 6.619
P.17 6.607 6.609 6.609 6.612
P.18 6.619 6.629 6.622 6.622

 

21

TABLE 3.2.--0riginal diameters (in mm.) of holes far a typical set of
3.18 mm. (1/8 in.) thick, 7075-T6 specimens.

 

 

Specimen (l) (2) (3) (4)
9.19 6.624 6.622 6.619 6.624
9.20 6.622 6.617 6.622 6.629
P.21 Hole too big

9.22 6.622 6.624 6.619 6.619
9.23 6.619 6.617 6.622 6.619
9.24 ,6.619 6.617 6.617 6.619
79.25 6.619 6.614 6.619 6.619
9.26 6.614 6.614 6.614 6.617
9.27 6.612 6.612 6.607 6.614
P.28 6.607 6.609 6.609 6.612
9.29 6.607 6.609 6.604 6.607
9.30 6.614 6.609 6.614 6.614
.9.31 6.617 6.609 6.614 6.612
39.32 6.617 6.612 6.617 6.609
9.33 6.607 6.607 6.607 6.594
9.34 6.609 6.607 6.609 6.604

 

 

22

TABLE 3.3.--0riginal diameters (in mm.) of holes for a typical set of
3.18 mm. (1/8 in.) thick, type 1100 specimens.

 

 

Specimen (l) (2) (3) (4)
.9.35 6.617 6.614 6.614 6.619
9.36 6.604 6.594 6.604 6.604
9.37 6.600 6.604 6.604 6.597
9.38 6.668 6.673 6.634 6.657
9.39 6.607 6.591 6.607 6.607
9.40 6.609 6.609 6.600 6.600
9.41 6.609 6.614 6.614 6.591
9.42 6.612 6.604 6.604 6.602
9.43 6.612 6.609 6.606 6.591
9.44 6.629 6.619 6.619 6.624
P.45 Hole too big

9.46 6.614 6.627 6.629 6.624
9.47 6.622 6.606 6.584 6.612
9.48 6.612 6.591 6.624 6.624
9.49 6.586 6.624 6.627 6.617
9.50 6.627 6.629 6.627 6.602

 

(3)

 

23

 

Figure 3.2.--Photograph of mandrels and sleeves.

24

the mild steel washer is simply to protect the sleeve and specimen as
the mandrel is pulled through; it pops off after coldworking.

The coldworking process involves insertion of a sleeve into
the hole and then drawing a tapered mandrel through the sleeve to
expand the hole, as shown in Figure 3.3. A machine incorporating a
hand-operated hydraulic cylinder was constructed to pull the mandrel
in the laboratory; a photograph of it is included in Figure 5.2. The
tension rod linking the mandrel to the cylindrical piston had been
instrumented with strain gages to permit calibration of the force in
terms of pressure. The peak force was 5357-7145 Newtons (1200-1600
pounds).

The sleeves, part number JK 5535008N04, were provided only
in the length of 6.35 mm. (1/4 in.). In order to prevent the unre-
quired deformation on the upper surface of the 3.18 mm. (1/8 in.)
thick specimens due to the longer sleeve, they were carefully cut
with a lathe so that the length of the sleeve equaled the specimen
thickness. Several sleeves were sectioned, and the average thick-
ness was found to be 0.19 mm. (0.0075 in.), in agreement with
Sharpe (10) and Adler-Dupree (11). This wall thickness is in fair
accord with the sleeve wall thickness of 0.216 $0.013 mm. (0.008
$0.0005 in.) recorded in the manufacturer's literature. The radial
expansion of the hole with the mandrel and sleeve inserted cannot be
measured directly. It must be computed from the dimensions of the

hole, mandrel, and sleeve, as follows:

25

Overs ized' Mand rel

/ Specimen

i
e
é /Fastener Sleeve
1
/é’%

fig /-Lockb01t Head
3

 

 

/Hydraulic Cylinder

 

 

 

 

 

 

 

 

 

Figure 3.3.--Schematic of the J. 0. King coldworking process.

Radial Expansion =

26

Mandrel Diameter + 0.015 in. - 0.260 in.

The relations between hole diameters, mandrel sizes, and radial

expansions are as follows:

Original Hole
Diameter (mm.)

6.604
(0.260 in.)

Mandrel

Diameter (mm.)

6.38
(0.251 in.)

6.43
(0.253 in.)

6.48
(0.255 in.)

6.53
(0.257 in.)

Sleeve

Thickness (mm.)

0.381
(0.015 in.)

Radial

Expansion (mm.)

0.076
(0.003 in.)

0.102
(0.004 in.)

0.127
(0.005 in.)

0.152
(0.006 in.)

It is estimated that this computation expansion is accurate to

$0.013 mm. (0.0005 in.).

Note that the length of the final diameter

of the mandrel is only 3.18 mm. (0.125 in.); it is not possible to

uniformly expand a hole in 6.35 mm. (1/4 in.) thick plate.

CHAPTER 4

THEORIES OF RESIDUAL STRESSES
AROUND A COLDWORKED HOLE

4.1 Introduction

 

The normal geometry that has been considered by experiments
and theories is a flat circular sheet of radius "b" with a circular
hole of radius "a" in the center as shown in Figure 4.1. Many of
the theories that have been developed assume "b" to be infinitely
large, and some of them assume that the thickness of the sheet is
sufficiently small relative to the dimension "b" that a condition of

plane stress exists. Only two theories, those by Sachs (12) and

 

Rich-Impellizzeri (13), assume a state of plane strain when a small

 

ratio of g-and a very large thickness of a plate are considered.
Therefore, the ratio of g-and é-are important in considering a state
of plane condition where t is the plate thickness. Potter-Grandt (9)
considered the effect of g-with a given thickness and found that the
%-> 10 is relatively large enough for the state of plane
stress. They did not consider the effect of %a There is still some

ratio of

question of how big the ratio of the outer radius "b" and the thick-
ness of the sheet should be for plane stress. The solutions for the
residual stresses around a coldworked hole have been obtained in the
references (7,8,9,ll,12,13,l4,16,l7,18,19). These theories differ in
the yield criteria used and in plastic stress-strain relations. Some

27

28

4r-I'

Figure 4.l.--Geometry and coordinate system used in coldworking hole
theories.

29

theories use the Mises-Hencky yield criterion (7,8,9,1l,12,13.14,16).
Others use the Tresca yield criterion (17,19). Some theories, e.g.,
Hsu-Forman (8), Adler-Dupree (11), and Alexander-Ford (14) take into
account strain hardening of the material. In the others no strain
hardening is assumed.

Deformation of the sheet is caused by either a uniform
positive radial displacement, ua, or a uniform negative pressure, -p,
at r=a. The shearing stress and strain vanish by symmetry. The
stress equilibrium equation which must hold for either elastic or

plastic stresses is:

 

r + = o (4.1)

The strain-displacement relation with u as radial displacement

is:

0')
1
II
D.
5:1:

(4.2)

1|:

89
For a radial displacement, ua, small enough that the material

remains elastic everywhere, the stresses, strains, and displacements

are given by (15) for the truly axisynrnetric problems as:

= 123.
or A(l r2)
2

06 = A(1+ 9?)

l‘

30

 

 

 

 

 

 

2
er =1”: (l-u)-(l+u)-E-;
e = A (l-u)+(1+u)9:—
0 El r2
u = % (l-u)r+(l+u)-l3-i
— r .—
Eu
where A a (4.3)

 

= (l-u)a+(l+u)«%f—_

E and u are the modulus of elasticity and Poisson's ratio respec-

tively. The boundary conditions are

u = u =
a at r a

or = 0 at r=b (4.4)

Using the plane stress condition (oz=o), the Mises-Hencky yield
criterion yields:

2 2 .= 2
or +00"0roe . 00 (4-5)

where 00 is the yield stress from the uniaxial stress-strain curve
(which is taken at the 0.2 percent offset yield strength).
The largest radial displacement that can be applied to the

hole without causing plastic deformation is:

- 00a b2
".5 " shag—374 [149419152] (4.61

31

Once uaE (or its equivalent pressure, pE) is exceeded, the
material in the neighborhood of the edge of the hole becomes plastic-

ally deformed within the elastic-plastic boundary, rp, around the

hole. The solution of the problem is nonlinear and no longer easy.

As ua increases, r increases; i.e., the elastic-plastic boundary

P

moves outward. If "b" is assumed to be large, a =-oe outside of r .

r P

Therefore equation (4.5) yields, at r=rp:

o = '09 = (4.7)

-39
/3
The solutions for the stresses and strains in plastic region,
a<r<rp, can be obtained from the theories, depending on which plane
condition and yield criteria are used. The solutions in the plastic

=.-p at r=a) and

region must be found for which u=ua at r=a (or or

the stresses, strains, and displacements match the elastic ones at
r=rp. The various theories fall into three classes: analytical,
numerical, and finite element. The analytical theories produce
closed-form equations based on either an e1astic-perfectly-plastic
stress-strain curve or a two-parameter plastic stress-strain curve.
The numerical ones develop the solutions in terms of incremental
rings between a and b corresponding to the increments on the plastic
stress-strain curve or use a numerical integration to solve the two-
point boundary value problem of finite rings for some range of-E.

The only finite element solution used an elastic-plastic computer

code.

32

The solutions for stresses, strains, and displacements in the

elastic region, rzrp, are:
Wu
r /3 r
2
.3932
09 /§ .r
1+ 2
e =_c10(l)):p_
It 1:73 I
1+ 2
-4 21:21
9 E73 r

C
H

O r
£33 (1+6) 72 (4.8)

Once the hole has been expanded to the prescribed conditions,
the loading is then removed at r=a. This generates, because of the
elastic relaxation of material outside and inside rp, residual stresses
and strains. These quantities can be found by superimposing an
elastic distribution at r=a on the plastic distribution. The elastic

distributions are:

= 9.2
Or °m(r)

a
CO =.Om(?)2 we”
where am is the magnitude of the radial stress generated at r=a by the

loading process. This guarantees that 0 =0 at r=a after unloading.

r

33

4.2 Nadai Theory

Nadai (7), in 1943, developed a theory of plastic expansion
of small tubes fitted into steam condensers and high-pressure boilers.
His paper focussed on a slight enlargement of a small tube, fitted
into the plate of a boiler, by means of a device consisting of three
hardened rollers with a slight taper which were mounted symmetrically
around a long tapered pin. The pressure of the revolving rollers
created a radial plastic distribution of stress in the tube wall and
around the hole in a steel plate. After the removal of the roller,
the residual stresses remain locked in near the edge of the hole, as
well as in the tube, and guarantee a leak-free fit. He considered
both the plastic deformation in the steel plate and copper alloy
tubes. He also solved the plate problem, which is the one of interest
here. His assumptions were:

1. Uniform pressure at the inside edge of the hole in

an infinite plate

2. Mises-Hencky yield criterion

3. Perfectly plastic material response.

In his first investigation of the paper he predicted the

yielding of the plate satisfied "the ellipse of plasticity," namely:

Or2+062'°r°e = 002 (4.10)

and developed the following expression:

20
0' =

r '352 sin (9' £9

34

20
06 = —9$Tn (9416)
/30
2 _ e 2

where 0 was any angle of the points on the ellipse of plasticity. He
found that 0=O° at the onset of the yielding, and 0 = g was the
maximum value. This gave the elastic-plastic boundary rp=1.75a. Not
much was gained in carrying the expansion of the plate beyond this
point.

In his second investigation he used the same assumptions, but
a linear approximation to the Mises-Hencky yield criterion was assumed
instead. He developed the largest radial displacement that could be
applied to the hole without causing plastic deformation as:

"00% ) ‘ ( )
u = 1+0 4.12
aE 73E

In the plastic region, acrsrp, he developed the following

expressions:

0
o r

o = -— (-l+2 1n —)

r ]; t‘p
o

00 = -9 (1+2 1n ~54
V3 9

u = u r/a(l+g-ln E )3
aE 3 r (4.13)

P

35

where the strains are given by:

du/dr

8r

86 u/r (4.14)

After the roller device was removed, the residual stresses
and strains were determined by superposing an elastic distribution of
stresses of the tube joint at r=a on the plastic distribution.

Equation 4.13 is the radial plastic displacement during

loading. One can locate the elastic-plastic boundary, rp, by using

this equation for a given radial displacement, "a’ at r=a.

4.3 Swainger Theony

 

In 1945 Swainger (16) published a theory of plasticity and cal-
culated the stresses and residual stresses around an expanded hole» He
founded his theory on certain observations of plastic behavior that are
well-known today, and treated the material as having a bilinear stress-
strain curve with elastic slope and plastic slope. The material was
assumed to yield according to the Mises-Hencky theory by a uniform pres-
sure applied on the inside of the hole under plane stress conditions.

Two assumptions were made which permitted solution of the
problem:

1. When the yield zone in the bilinear-strain curve is

narrow enough, the plastic slope can be regarded as an
approximate constant through it.

2. In general, when the yield zone is wide, it may be

considered to vary step-by-step over narrow annuluses.

36

He developed the expressions for stresses and strains in his
paper. He also introduced the relation between u and r from which
one can compute rp for his problem. As computed by Sharpe (10), the
rp was about 1.50a for the radial expansion displacement of 0.152 mm.
(0.006 in.) for the 7075-T6 aluminum, which is very much smaller than

the rp predicted by other theories.

4.4 Taylor Theory

 

G. I. Taylor (17) in 1947 published a theory of the expansion
of a circular hole in a thin plastic sheet by radial pressure. His
assumptions were:

1. Perfectly plastic material response

2. Tresca's yield criterior and a flow rule associated with

Mises-Hencky yield criterion

3. Uniform pressure at the edge of the hole.

His solutions started with a pin-hole subjected to a uniform
pressure, and three regions were considered. In the first region, a
s r 6 r2, large plastic strain occurred, and the plastic tangential
stress was negative. The value of r2 was computed to be 0.606rp.
The second region, r2 6 r C rp, had small plastic strain in which
tangential stress was positive. The third region was elastic for
r a rp. Since the initial hole was started with a pin-hole, this
problem is not exactly like the coldworked hole problem.

In the second region, r2 6 r < r , he developed the expres-

p
sions for stresses and strains as follows:

37

0'
_ O Y‘
Or-THt-Z In?)

00 r
06 = T (1+2 1n 73')

 

 

P
30 r T' "
u: 4i? 1 3r "2‘ (4°15)
2.U+§Tn<-,——)2i2 _
P
Er = Bu/Gr
£6 - u/r (4.16)

One cannot compute rp from the third equation of (4.15),
because u at r=r2 is unknown. Therefore, this theory did not give an
explicit relation between rp and ua. He also computed the predicted
variation of thickness with r with experiments in which he slowly
forced a rotating mandrel of very small taper into a hole in a thin

lead sheet. The agreement was very good as the thickness change

decreased to zero at r=r2.

4.5 Alexander-Ford and Mangasarian'Theories

 

Alexander and Ford (14) in 1954 published a theory of expan-
sion by uniform pressure of a hole from zero radius in an infinitely
thin plate. They assumed that the material obeyed the Mises-Hencky
yield criterion and associated Prandtl-Ruess relation for a specific
uniaxial stress-strain curve. The expressions for stresses in the
plastic zone were developed exactly the same as earlier theory of
Nadai (7). For strains, they developed four simultaneous, nonlinear,

ordinary differential equations and solved them by a numerical method.

38

The solutions by this theory are different from the coldworked hole
problem due to zero initial radius of the hole, but one can obtain
the solution to that problem by discarding the parts of the solutions
around the hole that are not required.

Mangasarian (18) in 1960 computed the solutions by both
deformation and incremental theories of plasticity, together with the
Ramberg-Osgood uniaxial stress-strain relation, by a numerical method
and found little difference between the two solutions.

These two theories did not predict rp.

4.6 Carter-Hanagud Theory

A. E. Carter and S. Hanagud (19) in 1974 performed an experi-
mental investigation of the stress corrosion susceptibility of stress-
coined (coldworked) fastener holes in AISI 7075-T651 aluminum. Their
point was that the residual tensile stresses surrounding the hole
could be above the threshold stress to cause stress corrosion crack-
ing. They developed their own theory, which is similar to Taylor (17).
to evaluate loading and unloading stresses. Their assumptions were:

1. Radial displacement at the edge of the hole

2. Tresca yield criterion

3. Elastic-plastic material response.

They also assumed the coldworking mandrel to be rigid and
ignored thickness variations in the plastic region and developed the
following relationship:

_ l+u

uaE - THE-Goa

39

o r:-
u = uaE + 73 %;2(rp2-a2)-2(T—u)aini (4.17)

a

The elastic deformation was included in developing equation (4.17).

The value of rp can be computed from equation (4.17) for a given ua.

It must be noted that the predicted rp by this theory are larger than

other theories as shown in Figure 4.3 at the end of this chapter.

4.7' Hsu-Forman Theory

 

In 1975, Y. C. Hsu and R. G. Forman (8) developed an exact
elastic-plastic solution for stresses in an infinite sheet having a
circular hole subjected to pressure on the basis of J2 deformation.
Their work is basically the Nadai theory extended to account for work-
hardening. Only small displacements were considered. They assumed:

1. Uniform pressure at the hole

2. The actual stress-strain curve represented by a modified

uniaxial Ramberg-Osgood Law:

0
E-for [0] s o

0')
II

0

n-l
‘%(%) for]o]20
o

("l
I

O

3. Mises-Hencky yield criterion in which R=l, where R was
the ratio of the transverse plastic strain in the plane of
the sheet to the plastic strain through the thickness.
The solutions were developed in terms of a parameter a which
varies between-g-at elastic-plastic boundary and “a at the edge of

the hole corresponding to a radial expansion "a'

40

In plastic zone, a s r s rp, they established the following

equations:

a = 172129)? "

case-(1+2R)'Jésino

 

 

 

-9 . '—
cosa+(1+2R) 251m

0 = 112129)? "

 

 

 

 

 

9. :(h-umzkfi exp Tn-_1)(1+23f2(,_h)'
Co (n-l)(1+2R)155ina+(n+l+2R)cosa_ n2+1+2R I

 

 

 

 

n+1+2R
n2+l+2R

SinoLa
= [Sinaa 1%

(n+1+23Lcosa+(n- 1)(1+2R)zsino 71x

n2 -1)(1+2R)f (a _a)
(n+l+2R)cosda+(n-1)(1+2R)‘Sindaexl

2(n2+1+2k) a

nil-s

 

 

 

v= 4033)— geece (4.18)

 

 

= 06 r + (— - —)(O’6 - jig-R. C) (4019)
5

One can compute the elastic-plastic boundary, rp, from
the last equation of (4.18) by setting a = g-at r = rp.
They also developed the expressions in terms of radial expan-

sion as follows:

41

+
U =""""I|UO’
0 RE 0

a (4.20)
which is exactly the same as Nadai (7) for the largest radial expan-

sion for elastic deformation.

l+u’

J3 o n ,
TT$'(EB) [(1-0 )cosoa+ J3'

 

sinaa] (4.21)

‘1

. =4. he- 44 904-1

where E5 is the secant modulus on the uniaxial stress-strain curve at
the effective stress a:
o = [a 2+0 2- 2R 0 o ]l5 (4.22)
r e TTR' r 0

After the removal of pressure, the residual stresses and
strains could be computed by subtracting unloading stresses and
strains from the loading ones.

In addition, this theory stated that the pressure at the edge
of the hole must be known and “a could be determined from the third
equation of (4.18). This is not possible for a practical coldworking
process; no one can tell exactly what the pressure at the edge is.
But he can solve the problem using a given ua in equation (4.21) to
determine aa. With the determined “a for each ua he can compute rp
from the fourth equation of (4.18) by setting a =4§-at the elastic-

plastic boundary.

42

The 7075-T6 aluminum here are adequately represented by:
n=15, oo=73.0 Ksi (503.0 MPa) for 3.18 mm. (1/8 in.) thick;
n=15, oo=79.5 Ksi (548 MPa) for 6.35 mm. (1/4 in.) thick.

ua (mm.) “a (3.18 mm. thick) ea (6.35 mm. thick)
0.0762 (0.003 in.) 125.00 123.80
0.1016 (0.004 in.) 129.60 128.70
0.1270 (0.005 in.) 132.60 131.80
0.1524 (0.006 in.) 134.60 133.90

The values of the rp predicted by this theory are shown at

the end of this chapter (see Figure 4.3).

 

4.8 Adler-Dupree's Finite Element Method

W. F. Adler and D. M. Dupree (11) in 1974 performed an
elastic-plastic element analysis of the two-dimensional stress field
around a coldworked hole. They assumed the plane stress condition
and accounted for plastic behavior of the material by a Ramberg-
Osgood relation which relates the equivalent plastic strain to the
nth power of the equivalent stress. To insure the geometric symmetry
of the elements, they divided the plate thickness into equal layers.-
The finite element mesh was made up of quadrilateral elements in
nine concentric circles, which were divided into two triangular
elements by a diagonal to different corners on each layer. This
resulted in the elimination of the additional nodes at the diagonal
intersection and four triangles in one quadrilateral. Furthermore,
a data reduction method was used to average the strains of the four

triangles within each quadrilateral area of the model to be the input

43

for the contour plots at middle location of that quadrilateral area.
Therefore, the strains could not be determined right at the edge of

the hole. The closest they could get to the hole edge was 0.64 mm.

(0.025 in.).

They started the finite element computation with the principle
of minimum potential energy for the elastic solution, and then con-
nected the elastic solution to the plastic one with an accumulated
prestrain associated with the Prandtl-Reuss flow rule and the
assumptions used. The specific problem they considered was a
0.152 mm. (0.006 in.) radial expansion of a 6.6 mm. (0.26 in.) hole
in 7075-T6 aluminum and then removal of expansion. This is the cold-
working process of J. 0. King Inc., which is the same as used in this
thesis. Two-dimensional plots of stresses and strains, as a function
of distance from the hole edge for the expanded hole, the relaxed
hole, and an expanded hole subjected to remote uniaxial tensions,

were given.

4.9 Potter-Grandt Theory

R. M. Potter and A. F. Grandt (9) in 1975 applied the
general solution of forced fitting by Potter-Ting (20) to coldworked
holes with the following assumptions:

1. Uniform radial displacement at the hole

2. Mises-Hencky yield criterion

3. Elastic-plastic material response.

They used numerical integration to solve the two-point bound-

b-40, and

ary value problem of finite rings of-g-ranging from-%=3 to-E-

44

presented all results in dimensionless forms. Besides the stresses
around the hole, they also predicted the elastic-plastic boundary
versus the radial displacements used. This is useful to the designer,
as shown by one of their figures on the following page, Figure 4.2.
Their work is similar to the earlier work of Nadai (7), except that
they did not use a linearized yield criterion. The later work of
Hsu-Forman contains their solution for loading as a special case.

There are, however, three important differences between the
Potter-Grandt theory and these other two just mentioned:

,1. Potter-Grandt computed the yielding associated with
unloading, and did not simply superpose an elastic
solution.

2. The geometry they considered was a finite ring of outer
radius "b".

3. Their solution is not presented in closed-form; the
differential equation is solved numerically.

The first version of the Nadai theory shows that there is a

limiting value of r = 1.75a corresponding to the maximum allowable

pressure. This reszriction on the deformation shows up in the
Potter-Grandt theory as a limitation on the allowable radial dis-
placement. For the 7075-T6 aluminum used, the maximum radial
displacement of 0.1346 mm. (0.0053 in.) is allowed by this theory;
therefore the 0.152 mm. (0.006 in.) radial displacement of this
experiment is not permitted. The plot of the rp by this theory was

depicted in Figure 4.3 for 3.18 mm. (1/8 in.), 6.35 mm. (1/4 in.)

thick, 7075-T6 aluminums. Pure aluminum type 1100 is not permitted

45

. ..m 8:238. 22., . m .83; mfi we
gemEmoepqmFu peeve; .m> m aceuczon owumepniuPumepw mcwueogii.~.e acumen

 

 

moa .
Jim 5 201 of 3 2553235 .663".
om. Pg 9 o. m
«I 0 0 «F o.-
m.ND
3
W
._ m...
.m I.
.o... q_.
\- H
92.222264. \ x Q . m.
.m.‘ m>o a \ \ ‘\ o. .O.N 8
z D :5th QQN‘ \\ ‘ m
e x .‘ m u
\\\\ 0.” “av
\\ 0 SN __
. . u . .7.
2.265861 b36060 36.53555 N .m o
:n%m

46

due to the out-of-range of its yield stress and radial displacement

used.

4.10 Thick-Walled Tube Theory

 

In 1975, J. 8. Chang (21) used the elastoplastic solution of
a pressurized thick-walled cylinder (12) to evaluate the residual
stress distribution adjacent to an open hole in a thick plate,
before using it for an analytical prediction of fatigue crack growth
of coldworked fastener holes. The assumptions of this theory were:
1. Perfectly plastic material
. Uniform pressure at the edge of the hole

. Plane strain condition

#WN

. Mises Hencky yield criterion and flow rule associated
with Saint-Venant's theory of plastic flow.
The elastoplastic solutions were developed in terms of
stresses and also in terms of an amount of coldworking and plastic

zone radii (for a s r s r ) as follows:

P

0'0 ( T‘ “Eg2
a = -—- 21n—— - l + )
Y‘ ]? Y‘p

o r 2
66 = —9- (2119;;- + 1 + )

V3 p

47

o r 2
u = E973? (4.23)
a

where p is the pressure applied at the edge of the hole.

Chang used these solutions for 12.7 mm. (1/2 in.) thick,
2024-T851 aluminum with 0.178 mm. (0.007 in.) radial expansion
applied to a 12.7 mm. (0.5 in.) hole. The analytical crack growth
behavior was predicted through the use of a computer. Reasonably
good correlations were obtained when compared to those generated
from 8-1 fracture mechanics properties programs.

The prediction of rp by this theory can be obtained from
either the fourth or fifth equation of (4.23) if the applied pressure
or radial expansion used is known. The plot of rp depicted in
Figure 4.3 was obtained from the fifth equation of (4.23). This

theory gives slightly larger rp for all three materials used.

4.11 Rich-Impellizzerijheory

 

D. L. Rich and L. F. Impellizzeri (13) in 1976 developed an
approximate closed-form solution for residual stresses around cold-
worked holes and interference fit fasteners in their determination of
the improvement in fatigue life of structural components of aircraft.
The solutions for loading stresses were taken from the elastoplastic
analysis of the thick-walled tube with internal pressure (12) with
v=0.5 for the plate. For the residual stress, they did the same as
others by simply superposing an elastic unloading on loading one.

The residual stress distributions were valid as long as compressive

48

yield was not exceeded at the hole edge an elastic unloading. When
the compressive yielding occurred, they developed an approximate

compressive yield zone aszy

 

T
82
rC = -——-——~T—( (21n—E + l --—B ) (4.24)
2(1- 5L- b2
b2
The residual tangential stress became maximum in magnitude at r=rC
and decreased to be:

20
° (-2..- .2 e") (4.25)

(“9)9Es b

a2

whereY=7- Sui-75%;!

at the hole edge. This is the same shape as the residual tangential
stress predicted by Potter-Grandt (9) .

It must be noted that they worked on the tangential stress and
took them from the tube analysis (12). By setting the radial inter-
ference fit equal to the sum of radial displacement of the mandrel

and the plate, they introduced the relation between r and u as:

P
02 r r 2 r 2 E
.% =._2.[o,52(21n7§ + 1 - —E-) + 1.5-—E --21 (4.26)
a a E
J3EE P

where EB and Ep were the modulus of elasticity of mandrel and plate,
respectively.
The predicted rp, for this theory, can be computed from equation

(4. 26) by using the trial and error method for agiven u= -ua .For the tested

49

cases, the rp's were given in Figure 4.3 by assuming 0 = 0.3 and

EB = 30x106 psi for the mandrel.

4.12 Discussion

 

The theories for predicting the stresses and strains around
a coldworked hole, as well as some for predicting rp, fall into two
classes: those that allow work-hardening (Alexander-Ford,
Mangasarian, Hsu-Forman, and the finite element one), and those that
do not (Nadai, Taylor, Carter-Hanagud, Sachs, and Rich-Impellizzeri).
All of the theories considered, except the finite element solution
and Mangasarian, are of the deformation, not incremental, type.

Figure 4.3 shows the elastic-plastic boundaries predicted
by various theories for typical 7075-T6 aluminum.

In the non-workhardening theories (see Figure 4.3), Nadai
assumes plane stress and linearized Mises-Hencky yield criterion
in computing his relation between rp and u which causes his rp to be
smaller than Carter-Hanagud, who assumed the Tresca yield criterion
and included elastic deformation in computing rp. The Potter-Grandt
theory uses the Mises-Hencky yield criterion (not the linearized one)
and gives a very much smaller value of rp. The Rich-Impellizzeri
theory assumes plane strain, linearized Mises-Hencky yield criterion
and accounts for radial compressive displacement of the mandrel. It
allows the assumption of elastic unloading, but computes a compres-
sive yield zone to avoid the violation of yield criterion near the
hole edge. The computed rp by this theory are larger than those of

the plane stress condition but smaller than predicted ones by the

thick-walled tube solution. Within this class, the choice of yield

2.5-»

A.._ A A
' V v

I“
0

Plastic Zone Rodus/ Hole Radius (V0)

[.5 1.

vvvv

'v

 

50

/ Cartenl-lmagud

/ Thick-Walled
Tube Th.(Sachs)

/ 1' Adler-Dupree
. /

Rich-

/ K/ Impellizzeri

/ $ /

 

 

2 4 6 a To 12 14
Dimensionless Radial Expmsion Displacement (fiauofiena'o)

Figure 4.3.--Elastic-plastic boundaries predicted by various theories.

51

'criterion, plane condition, and the decision on whether or not a
rigid mandrel and elastic deformation are accounted for make a lot of
difference. Mendelson (22) summarized much experimental data using
these two yield criterions, and he found the Mises-Hencky yield
criterion to be better. Therefore, the Potter-Grandt theory, in the
sense of yield criterion, plane condition, accounting for elastic
deformation in which the real material behaves, and computing yield-
ing on unloading, is more complete.

The work-hardening theories account for elastic deformation
inside the elastic-plastic boundary, rp. Swainger works on the
incremental rings over the plastic stress-strain curve and produces
all equations in terms of an annulus thickness which is different
from the others (8,11,14), while Mangasarian works on both
Jz-deformation and incremental theories. The finite-element solution
by Adler-Dupree suffers from an inability to produce values near the
edge of the hole due to the quadrilateral area there, and also is not
generally applicable. The Hsu-Forman theory is the most general of
all theories developed; it produces all values needed for a cold-
worked hole in workable closed-form that make for economical
computation.

In all of these theories, small deformations are assumed.
This raises a serious question concerning the large deformation of
material near the edge of the hole. The strains computed by all
theories are typically higher than 0.10 near the hole edge, which is
certainly large enough to cause significant errors in small

deformation assumptions.

52

The assumption of elastic unloading is not correct; the
residual stresses violate the yield criterion near the edge of the
hole. The Rich-Impellizzeri theory avoids this violation by the
restriction of compressive yield zone near the edge, while the
Potter-Grandt theory computes yielding upon unloading. A complete
theory, depending on the real material behavior, would be the Hsu-
Forman theory with computing upon unloading and accounting for radial
compressive displacement of a mandrel instead of uniform pressure.

As to experimental verification of any of these theories, it
would be very difficult to match the boundary conditions at the
expanded hole due to the elastic deformation of the plate and mandrel
used. Taylor did do this by pressing a rotating, slightly tapered
mandrel into a very soft, thin sheet; but that is far removed from a
practical coldworking process, and very soft materials are not used

in structures.

CHAPTER 5

ELASTIC-PLASTIC BOUNDARY
MEASUREMENT TECHNIQUES

Three experimental techniques for locating the elastic-
plastic boundary were examined. These were

a) Foil Strain Gages

b) Thickness Change Measurement,

c) Photoelastic Coating.

The first two were used for the measurements reported in this
thesis.

5.1 Nature of Deformation Around a
Coldworked’Hole

 

Some preliminary experiments were run to verify that the cold-
working process was working properly and to get an idea of the state
of deformation around a coldworked hole. A photograph of the defor-
mation area around the hole in a 6.35 mm. (1/4 in.) thick, 7075-T6
aluminum specimen is given in Figure 5.1. The most notable feature
is the large amount of deformation near the edge of the hole which
becomes smaller within a very short distance away from the edge.
Furthermore, the deformation is so great that individual grains have
rotated. Slip lines are easily visible in Figure 5.1. The material
in the neighborhood of the edge of the hole cannot be considered as

a homogeneous, isotropic continuum on a local scale.

53

54

 

Figure 5.l.--Photograph of the deformed region around the hole in a
6.35 mm. (1/4 in.) thick specimen (100X).

55

'5.2 Foil Strain Gage Techniques
The resistance strain gage is an excellent tool for measure-
ment of strain in most structural material, as it closely approaches
the characteristic requirement of strain measurement. A few
researchers (3,23,24) used foil gages to measure the strains near
interference-fit fasteners and were satisfied with the measurements.
However, one report (24) questions the accuracy of data obtained near
a fastener with short gage-length strain gages. Recently, Sharpe (10)
used foil gages with a gage length of 0.38 mm. (0.015 in.) to measure
the strains around a hole during coldworking and due to a static
load. He found that foil gages gave an acceptable measure of the
average strain within the gage length even in the presence of large
strains and large gradients.
For this technique, the following gages and cements were
purchased from Micro-Measurement Co., Romulus, Michigan:
Foil gages: Gage type EP-08-031ME-120 for radial strain
EP-08-031MF-120 for tangential strain
Resistance 120 $0.50% ohms
Gage length 0.79 mm. (0.031 in.)
Gage factor 2.14 $1.0%
Cements: M-bond 200 Adhesive and 200 Catalyst
There are ten gages in one strip, with 2.13 mm. (0.08 in.)
between gage centerlines. Due to the limited number of channels of
the amplifier, three gages were carefully cut out of each strip with

small scissors and used.

56

Electrical circuit used:

6-Channel Amplifier with a Wheatstone bridge circuit in each

channel, by B. & F. Instrument Inc., Philadelphia,
Pennsylvania

Recorder:

Minicomputer - Computer Automation model L51 2

X-Y Recorder - Hewlett Packard, model 7046 A
The instruments are shown in Figure 5.2 for the setup.

The specimens were polished with silicon carbide grinding
papers, grit numbers 320 and 400, followed by 600 grit to insure a
smooth surface. Surfaces were finally cleaned with acetone. Three-
gage sets were bonded to specimens following the bonding procedure as
suggested by the manufacturer. The measured strains were in the neigh-
borhood of the elastic-plastic boundary which was far away from the
edge of the hole. The nearest location to the edge of the hole of the
centerline of the first gage was at least 1 mm. from the edge, and the
third one was 5.30 mm. The prepared specimen is shown in Figure 5.3.

Two attempts were made to measure the strains. First, both
radial and tangential strains were measured by bonding both kinds of-
gages at 90° apart to the upper side of the specimen as shown in
Figure 5.3. Secondly, only radial strains were measured by bonding
the radial foil gages to the upper and lower sides of specimens in
the same radial line of the hold. In both attempts to gage locations
were measured with a microscope, and gage resistance was checked.
Then the following procedure was used after the gages were con-

nected to the signal conditioner:

57

 

Figure 5.2.--Photograph of equipment set up for the foil gage
technique. Minicomputer (l), teletype 2),
X-Y recorder (3), 6-channel amplifier (4),
bonded specimen (5), coldworking apparatus (6),
hydraulic pump (7), Bridge Amplifier Meter (8).

58

 

Figure 5.3.--Photograph of bonded specimen with radial and tangential
foil gages on the upper side.

59

a) Balanced all Wheatstone bridges to get zero output;
-R
b) Calibrated all gages with e = g ;
F(Rg‘LRcal)

 

c) Set the gain of the 6-Channel amplifier to 5 volts per
1.00% strain.

The voltage change as the mandrel was pulled through the
hole was temporarily recorded by the minicomputer and was later
plotted on the X-Y recorder. Typical plots, as given in Figures 5.4
to 5.11, of the loading strains versus applied load were obtained
from the foil gages applied on the upper side of the specimens
(P. 18,20). The forces required to pull the mandrel through the hole
were about 5357-7143 N (1200-1600 lb.) and 4464-5357 N (1000-1200 lb.)
for the 6.35 mm. and 3.18 mm. thick, 7075-T6 aluminum respectively.

Strains at the peak of the average strain (typical plots are
in Figures 5.4 to 5.11) were plotted (as in Figures 5.12 and 5.13)
versus gage location to establish the elastic-plastic boundary.
The elastic-plastic boundary, rp, would be where the radial and
tangential strains were equal in magnitude to the maximum elastic
strain before yielding when radial and tangential strains were

measured. The elastic strain is given as

_ 1+0

8 — - ———.—

r 6 RE 0
which were 0.60, 0.56, 0.04% for the 6.35 mm. (1/4 in.), 3.18 mm.

(1/8 in.) thick, 7075-T6 aluminum, and 3.18 mm. (1/8 in.) thick,

llOO aluminum, respectively.

 

 

 

 

 

 

 

”'9095
Radkn(0anp) "gi‘, ; lhngenflalllenJ
at 1.47am from edge : "6° 9' at 1.40mm. from edge
8'2"” g
m
'¥ -4o:z
8 °‘
.1 s.'30
4 "'20
2140 (a)
20 13 Ho 06 o 03 E0 lb 20
Strains 1%)
5 no
14 so
at 360mm. from edge at 153m. from edge
(b)
20 13 15 03 o 05 lo 15 £0
16 ,0
at 6.73 mm from edge at 6.66 mm. from edge
(0)
20 13 1b 03 o 65 Co 13 2b

Figure 5.4.--Radia1 and tangential strains measured with foil gages
on the upper side of the 6.35 mm. (1/4 in.) thick,
7075-T6 aluminum (P.18) as the mandrel of 0.076 mm.
(0.003 in.) radial expansion was pulled through the
hole.
(a) . . . the first (nearest) gage
(b) . . . the second gage
(c) . . . the third gage.

 

 

 

 

 

 

101,70
N N
9"reo g
I
12. a
9 2° 5
Radial (Coma) E 10140 g ngentiallTen.)
a. 2
at 1.47m from edge 3 ,30 at 1.40mm. from edge
6.
4:20
2 ,1-10 (0 )
axT’ 16 15 05 o 03 1b i '
$trains(%) 6 2°
16
170
14‘ o
12-r6
'50
at 3.60 mm. from edge - oteesmm ham edge
1.
I ‘P
(b)
25 13 to as o 06 ID is 20
18 7°
14 60
at 6.73am from edge at 666mm.from edge
(0)
£0 L3 L5 03 o 05 1b L5 i0

Figure 5.5.-—Radial and tangential strains measured with foil gages
on the upper side of the 6.35 mm. (1/4 in.) thick,
7075-T6 aluminum (P.18) as the mandrel of 0.102 mm.
(0.004 in.) radial expansion was pulled through the hole.
a . the first (nearest) gage
(b) . . . the second gage
(c) . . . the third gage.

 

 

 

 

 

 

Radial (Coma) 161.70 Tangential 1 Ten.)
N
N .
9 l4 ‘0 g
atl.47mm " 12- at 1.40m.
3 ~66 5
flOflIdMOO is KL heatedge
.40 Z
1 °‘
3 "3°
4 "20
2,10 (0)
20 L6 do 03 o 66 60 £6 £0
Stratum) '
e-qo
"ieo
12-
at 360mm from edge 0 ~60 at 3.63mm. from edge
I .1
.40
.. (b)
26 16 IO 06 o as 10 16 20
16 “1.70
atersmm from edge J at e.eemmtrom edge
1
,. (c)
225 i3 it oh. <9 6. R: 13 20

Figure 5.6.--Radial and tangential strains measured with foil gages
on the upper side of the 6.35 mm. (1/4 in.) thick,
7075-T6 aluminum (P.18) as the mandrel of 0.127 mm.
(0.005 in.) radial expansion was pulled through the
hole.
(a) . . . the first (nearest) gage
(b) . . . the second gage
(c) . . . the third gage.

63

Radial ( Comp.)

,‘q ‘fllnmnnmllinj
at 1.47am. from edge

17 at 1.40am. from edge
H-wg
X

“5‘

.40:

i 2

£0

-20

‘40 (0 7

 

 

 

24) Te L0 03 c1 06 to 13 ‘TEb
Strains (‘16)
"‘1.70
"160
121
~60

at 360m. from edge 4° a13.53mtrom edge

0 '30
“920
24.10 (b)

 

230 16 00 03 o 06 1b (6 2:0

at 5.73 mm. from edge 4 at 6.66 m. fran edge

11 (c)

I—.

 

 

 

L A
v

2.0 13 1.0 on o as 1.0 136 2.0

Figure 5.7.--Radial and tangential strains measured with foil gages
on the upper side of the 6.35 mm. (1/4 in.) thick,
7075-T6 aluminum (P.18) as the mandrel of 0.152 mm.
(0.006 in.) radial expansion was pulled through the hole.
(a) . . . the first (nearest) gage
(b) . . . the second gage
(c) . . . the third gage.

 

 

 

 

14~
Radial (Coma) 690° Tangential(Ten.)
N 2 _ _
at 1.32m. from edge 8' £0 x at 1.23mm. from edge
an.
E *‘°
' 8' Z
'0 '30
86-
.1
4_-2o
1:. 1.8 i
0.5 $170105 (0,.) 05 1.0 l5
' 260
12‘
'50
IO~
at 3.45m.from edge -40 at 3.36m. from edge
8-
”30
6c:
:2
- (b)
15 1b as o 05 15 Tie
14 6°
12
50
10
at asenm from edge at 5.49am from edge
‘ (c 1
13 f0 03 o 03 10 ’T5

 

Figure 5.8.--Radial and tangential strains measured with foil gages
on the upper side of the 3.18 mm. (1/8 in.) thick,
7075-T6 aluminum (P.20) as the mandrel of 0.076 mm.
(0.003 in.) radial expansion was pulled through the hole.
(a) . . . the first (nearest) gage
(b) . . . the second gage
(c) . . . the third gage.

RadioMComp.)
atl.32 mm. from edge

01
U1

3
1|
8

‘T’ '1’
S 8
02

O
I
l
on
0

Load- Pounds 11102
O
Nevnons xI

&
1

Tangential (Ten. )
at 1.23am from edge

(a)

 

at 3.45mm from edge

 

at 336 mm. from edge

 

at 5.58 m. from edge

 

at 5.49mm. from edge

(c)

 

15 To 65

Figure 5.9.—-Radial and tangential

 

06 Th b

strains measured with foil gages

on the upper side of the 3.18 mm. (1/8 in.) thick,
7075-T6 aluminum (P.20) as the mandrel of 0.102 nm.
(0.004 in.) radial expansion was pulled through the hole.

(a) . .
(b) . .
(c) . .

. the first (nearest) gage
. the second gage
. the third gage.

RadiallComp.) '4 1150 Tangential (Ten)

at I32mm. from edge

(2 . 019 at tzsmm. from edge

Lood-Fbmds 11102
a
2'3
Newtons

 

c
Strains 1%)
'4‘L60

2'50

O-
-40

at 345m. from edge 3 ‘ at 3.36 mm from edge

'30

6-

4:20

2.910 I b)

—g_ - - +2

[5

:."50
at 558mm. from edge .0: at 549 mm. from edge
I
(c)

1.6

Th 0 as To 13

 

 

LOO as 1b 15

Figure 5.lO.--Radial and tangential strains measured with foil gages

on the upper side of the 3.18 mm. (1/8 in.) thick,

7075- T6 aluminum (P. 20) as the mandrel of 0.127 mm.

(0. 005 in.) radial expansion was pulled through the hole.
(a) . . . the first (nearest) gage

(b) . . . the second gage

(c) . . . the third gage.

Radial (Comp) +60 Tangentiall'l’enl
a? team flan: edge '2‘ at L23 mm. from edge

 

 

 

 

:3 no as .a do lb lb
Strains (96)
use
l2-
~50

L40

at 3.45mm.fmn edge a . at 3.36 mm. from edge
'30

2-40 (b)

 

A
T

[5 L0 03 0 do LO is

at 6.58 mm from edge " at 5.49tnmfrom edge

-’ (c)

 

is (a 63 0 de Co 13

Figure 5.ll.--Radial and tangential strains measured with foil gages
on the upper side of the 3.18 mm. (l/8 in.) thick,
7075-T6 aluminum (P.20) as the mandrel of 0.152 mm.
(0.006 in.) radial expansion was pulled through the
hole.
(a) . . . the first (nearest) gage
(b) . . . the second gage
(c) . . . the third gage.

 

68

u- O.|52mn.(0.006ln.)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 r
1 9.2240
l (I: 2.2lfl3.0'/e
[_____.. qs217a
I - |
0.60%
O : ‘ ? t y 5 ; .L 4' 1
o é 4 6 8 i0
Distance from Hole Edge (mm.)
2 .. u-o.i27mm.(o.oosin.)
‘-—————-ié=2.090
m=2.080*i.0%
F———.r|=2.060 '
l + i
,‘o 1 a .1 .e - a
2\: .
2
g 2 V u=0.|02m.(0.004in)
”’ .=rflsea
ibvflflfio ‘5.0°/o
= I150 '
l4 I E
o - - . ; Less
2 M0076mn.(0.003in.)
V
:.' 3 [.670 ‘
[ ' r -I.64o-I.0%
IT :tz-LGOa av.
l ‘ N GTang.Strain
\ i ARad. Strain
0 ¢ e c 4‘

 

 

A A A A
f v v v 1

Figure 5.12.--Typical method used to locate the elastic- lastic
boundary, r , of typical 6.35 mm. (l/4 in. thick,
7075-T6 aiuBinum (P.l8) in which foil gages were
applied on the upper side only.
r1 . . r located by tangential strain
r2 . . . r: located by radial strain.

Strains (°/e)

 

69

 

 

 

 

 

 

 

2r uncieznnxoooema
'i '2‘“ w anew-10%
l . I'_"'2' 2220 '
I 0.56%
o 3 t t 3 t 3 ‘ g a
o 2 4 6 8 ib
2 1; u-O.l27m.(0.006ln.)
‘ r.- r2!I 9;".- 2.03a‘0.0%
I . I
|
o e e - e,
2 1, u-O.l02m(0.004in.)

 

 

F——. r. - 2.000
I’ ' |.9| 0 ‘ 5.0.5

———.r ' LBZO

2

 
    

 

 

 

 

=4] =L600
l wimatso'x.
r =H.800
I 2 (D Tang. Strain
K I A Red Strain

 

A A A A

A A A I .J
f v fiv

Distance From Hole Edge (mm.)

Figure 5.13.--Typical method used to locate the elastic- lastic

boundary, r , of typical 3.l8 mm. (l/8 in.

thick,

7075-T6 aluminum (P.20) in which foil gages were
applied on the upper side only.

r . .
1
r2 . .

. r located by tangential strain
. rp located by radial strain.

P

70

As found by Sharpe (10), the lower side strains are signifi-
cantly larger than the upper side strains. This leads to an error of
the rp location by measuring the upper side strains only. Another
important factor is that the radial and tangential gages were bonded
to the specimen on different radial lines from the hole where the rp
are not exactly equal, as later found by the thickness change mea-
surement technique; therefore the equal value of Sr and 86 on the
plot cannot be found at the same location. For these reasons, a
second procedure was examined. Two three-gage sets of radial foil
gage were bonded to the specimen in the same radial line of the hole
on both sides. Following the same procedure, strains at the peak of
the average radial strains on both sides of the specimen were plotted
versus gage location. The elastic-plastic boundary, rp, would be
where the measured strains were equal in magnitude to the maximum
elastic strain before yielding (see Figure 5.14 and 5.15). A sig-
nificant difference of the rp/a obtained from the first and second
attempts occurred on the smaller amounts of coldwork as listed in
Tables 6.1 and 6.2.

The advantages of applying radial foil gages on both sides
of a specimen relative to radial and tangential on one side may
be stated as follows:

a) Foil gages can be applied in the same radial line of the
hole, with the result that the strains in the same
direction are obtained.

b) It is more accurate to locate the elastic-plastic bound-

ary around a coldworked hole which is slightly different

71

   
       

 

 

 
  

 

 

  
 

 

 

   
 

 
   

 

 

2 4
u=0.l52 mioooen.)
l < \
...... \ - ——---O.BO%
:T“-
o . A - [I A A L .
r f ' 'l V ' ' j
:1
2 ‘ L 82.360
L :8: 252° rp (du)=2.34otl.00%
A l 1 uso.l27mm.(0.006in.l
32 ______
.g o 4 l.l . . . . J
E w Iv : v V ' ‘ fl
a : l 32.3”
_ 2 . L__ =2.l60 rp(av)=2.240*3.00%
o
B
a \
tr \
o I .
.5 u=0.l02 nun. (0.004ln.)
m ----------
o
a. i l
g 0 t 4 I - : 4 f 3 fi‘
0 : L-®=zoso
'5 2 ‘ L__-®“.9|° rpimk2000t450‘5
W
\ 0 Upper Side
I \\\ “30,073 am. A LOW" Sid.
, (0.003h)
.._.. Hsu-Forman
i i
O . a! .' . a . . - a .
o 2: ' 4 e To
i lL-®=i.92a
: r iav.)= Leaatsoox
L---®=i]3a 9

Distance from Hole Edge (mm.)

Figure 5.14.--Typical method used to locate the elastic- lastic
boundary, r , of typical 6.35 mm. (l/4 in.§ thick,
7075-T6 aluminum (P.14).
(l) . . . r located by the upper strain
(2) . . . rg located by the lower strain.

72

  
 
    

 

 

 

 

 

     

 

 

 

     

 

2 .
\\
‘ uxo.Iaz mm.(OOOOIIL)
| 4
| I
o r A :| 3 ‘ ; ¢ ¢ %
l '- 2.l7a
2 |_ I, ZIOO rp(av.)- 2.l4aazoox
l \
\\
\ u-o.I27m.(o.ooeIn.i
S5 '4
g ..... _
.E “
. H - a . ¥ L . .
en 0 e :7 V'L' . - v . . .fi
— I 'QF 2.06a
° r .)=zoeato.eox
g 2 ‘l \ L..®; 20“ p(OV
\
(r \
g u =0.i02 m.(0.004ln.)
.. li
n
‘8
a. I.
E
o o - . '.| . i J J v. J .
U ' a '1 i-Qa LSOa fl
' -
I5 2 1 ‘ L_-®3La6. 'p (W--)-|92°*3~00%
W l O Um SIC.
\
\
\\ “30.0” m. A LNG! SI“
l - \ (0.003in.)
---Heu-Feman
I I‘: -------
o H: ‘ t c . . 4 _
o 2 i'L_®_ a“ 6 8 I0
' - . =l.830e
‘- '(D- “On '9 (av ) 200%

Distance from Hole Edge (mm.)

Figure 5.l5.--Typical method used to locate the elastic-plastic
boundary, r , of typical 3.18 mm. (l/8 in.) thick,
7075-T6 aluminum (P.28).
l . . . r located by the upper strain
(2) . . . r: located by the lower strain.

73

from uniform pressure or radial expansion displacement
assumed theories.

c) The elastic-plastic boundary can be quickly obtained by
averaging the upper and lower rp, giving an actual
average rp of the coldworked hole.

Figures 5.4 to 5.7, the typical radial and tangential strains

obtained from the gages applied on the upper side of a 6.35 mm.
(l/4 in.) thick, 7075-T6 aluminum (P.18) as the mandrel (0.076,
0.102, 0.127, 0.l52 mm.) was pulled through its hole are shown as
a function of applied load.

Figures 5.8 to 5.ll are also the typical radial and tangen-
tial strains on the upper side. These are obtained from the 3.18 mm.
(l/8 in.) thick, 7075-T6 aluminum (P.20).

These plots are similar. The strains start to increase as
the mandrel is first pulled through the hole and decrease to residual
strains as the mandrel is completely pulled through the hole. As
the second, third, and fourth mandrels are used, the loading strains
will start to increase from the residual strains of the last mandrel
used. Figures 5.l2 and 5.13 show the typical method used to locate
rp/a. The radial and tangential strains are taken from Figures 5.4
to 5.7 and Figures 5.8 to 5.ll, respectively. The uncertainty (plus
or minus sign in the figures) is computed from the offset of the rp/a
located by the radial and tangential strains from the averaged rp/a.

Figures 5.l4 and 5.15 illustrate the typical method used to
locate rp/a when strains were measured on both sides. The strains

plotted here are taken from the specimens (P.14, 28) with the radial

74

foil gages applied on both sides as all mandrels were pulled through
their holes. In these plots, the strains predicted by the most
general theory by Hsu-Forman (as stated in Chapter 4) are also shown.
The experimental strains on the upper surface show very good agree-
ment with the theory on the smaller amounts of coldwork. The rp/a
located by the lower side strain are somewhat larger than the one by '
the upper side strain for the 6.35 mm. thick specimen and very close
to each other on the larger amounts of coldwork. For the thinner
specimen, the rp/a located by both side strains are not very much
different. The final rp/a is computed by averaging the rp/a located
by the measured strains on both sides. The uncertainty (plus or
minus sign in the figures) is computed from the offset of the rp/a
located by the strains on both sides from the actual rp/a and also

liSted later in Tables 6.l and 6.2.

5.3 Thickness Change Measurement Technique

When a circular hole in a plate is expanded by a uniform
pressure or radial displacement, the metal near the edge of the
hole is thickened and extended out along both sides of the surface,
due to the condition of volume constancy (€r+€e+€z=0)'

Based on elasticity theory, during coldworking process the

well-known transverse strain is given as

_ l
62 - E'[az'u(°r+09)]
which is valid everywhere in the elastic region around the hole. At

a given radial loading, all theories described in Chapter 4 agree

that the values of radial and tangential stresses are equal in

magnitude but opposite in sign in the elastic region. Since 02 = 0,
the above equation leads to £2 = 0. This means that there is no
change in thickness of the plate in the elastic region. The plate
thickness will start to change right at the elastic-plastic boundary,
due to the condition of volume constancy and inequality of the
stresses in the plastic region. Therefore, a thickness change of

the plate exists in the plastic region and vanishes in the elastic

region. The elastic-plastic boundary of a coldworked hole

can be found at the location where the thickness first starts to
change. Sharpe (l0) measured the thickness change of a coldworked
hole by J. 0. King coldworking process, which is the same as this
experiment, by focusing the lOOX-lens on a microscope for some
indentations marked along a radial line of the hole. This technique
suffers from inaccuracy in focusing the lens and dees not measure
total thickness change.

Among displacement-measuring devices, the linear variable
differential transformer (LVDT) is an appropriate one. Durelli
et al (25), in 1966 used a LVDT as one of their methods to determine
the sum of principal stresses in two-dimensional problems. They used
an LVDT of 10'6 in. sensitivity to measure a change in thickness at a
point of a loaded plate, and then the proportionality relationship
was used to determine the sum of principal stresses. In l968
Coleman and Ward (26) also used two LVDT for the change in specimen
cross section in inhomogeneous deformation measurement. The diameter
change could be measured over 0.2 in. with a maximum resolution of

better than l0 x lO'6 in.

76

The LVDT used in this technique was purchased from the R.I.S°
Incorporated Sales and Service, 20245 Van Dyke, Detroit, Michigan,
and was model R.I.S. DT-100-2-A with 0.04 in. (1.02 mm.) carbide
ball contact point. The operating range is 10.05 in. (1.27 mm.) with
0.05% at $0.02 in. (0.51 mm.) linearity. A Daytronic amplifier model
3000 was modified to accept the DT-100-2-A transducer to have output
of 10 volts 0.0. A linear potentiometer by Bourns Inc., Riverside,
California, with maximum range of 1.00 in. (25.4 mm.) was used to
locate the radial distance from the edge of the hole.

Four thickness change measurements were run in each specimen
along the numbered directions shown in Figure 3.1. The LVDT and
linear potentiometer were checked and the linearity found satisfac-
tory. The calibration curves of the LVDT on the $0.005 and 10.001
in° scales of Daytronic amplifier are given in Figures 5.16 and
5.17. The best resolution obtained was 50 x 10"6 in. and 10 x 10'6
in. on the $0,005 and £0,001 in. scales, respectively. The finer
scales of $0.0005 and 10.0001 in. were checked, but the linearity
was found to be unsatisfactory. The calibration curve of the linear
potentiometer is shown in Figure 5.18 with a calibration factor of
0.183 in. per volt.

The setup is shown in Figure 5.19. The specimen was held by
a sturdy C-clamp attached to a stationary X-Y translation stage,
which could be moved in and out between two identidal carbide balls
that activated the LVDT. The lowest ball was attached to the steel
frame and could not move. This ball was designed to be in the same

vertical line as the LVDT's contact point. The second ball was

77

   

Output ( Volts)

     

25: see 75.2
Distance x I63 mm.

43.2 -5o.e ~2s.4

 

 

“'0 (b

Figure 5.16.--Calibration curve of the LVDT on 10.005 in. scale of
Daytronic amplifier model 300C.

78
IO T

   
   

0:

Output ( Volts)
«b

L

508 046 152.4
Distance XIO mm

L

452U4 dCfiGi

 

 

-lO .1.

Figure 5.17.--Ca1ibration curve of the LVDT on :0.001 in. scale of
Daytronic amplifier model 300C.

Output ( Volts)

79

 

 

A A A A ‘ L A A
v r v v v V v v v

0 5.l l0.2 I55 20.4 25.4
Distance (mm.)

Figure 5.18.--Calibration curve of linear potentiometer.

80

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.Anv mewEOVHcmuoa emmcrp .fimv mumpm.cowumpmcmeu >nx
.Amv cmugoomg >ux .Aev coewqum .Amv gunpoc ho>4 we»
.5 3:295 38.5.3. .3 5: e: co 25:28
.azumm acmEmgammwe wmcazu mmwcxuwzu on» mo saugmouosmuu.m_.m mgzmwd

 

81

attached to a horizontal small strip sheet. This sheet was tightened
on the steel spacer block and used to control the LVDT spring pres-
sure on the specimen by lifting its free end up and down with a small
threaded rod connecting it with the steel block. This middle ball
must be set in the same vertical line as the lowest and the LVDT's
balls (see Figure 5.20) in order to have a measurement of the thick-
ness change at the same point. The horizontal level for specimen
thickness was adjusted by two steel spacer blocks--one for each
thickness-—that control the position of the middle contact ball to
fit the specimen. The most important requirement of the setup is
that the X-Y translation stage and the LVDT holder must be tightened
to insure no relative motion during testing. The LVDT output was
connected to the X-Y recorder from Daytronic amplifier output. The
output of the linear potentiometer, mounted on the specimen holder,
was directly connected to the same recorder so that it could provide
a permanent profile of the change in thickness of a specimen as a
fdnction of the distance from the edge of the hole.

Since only one LVDT was used, the ideal original thickness of
a specimen would be the zero volt output (null) line when the null was
set at a starting point; but it was very difficult to have a uniform
thickness in the real specimen. Two precautionary steps must be made
before the testing. First, one must set the contact point of the
balls as close to the centerline of the hole as possible in order to
get the real rp around the hole. Second, the null of the LVDT should
be set at the nearest location to the original starting point to
obtain the same path as the original one when the coldworking process

was completely done.

82

 

Figure 5.20,-_Photograph of the LVDT and linear potentiometer mounted
in holders.

83

The following procedure was used for the test:

a) The original thickness of the non-coldworked specimen was
recorded, using the methods described above. A small weight was used
to balance the specimen weight on the lowest ball when the specimen
was horizontally moved. The starting point was made at about 20 mm.
away from the edge to insure that it would not be in the plastic
region.

b) After each mandrel was pulled through the hole, the
sleeve was removed for final diameter measurement and to avoid the
possibility of contact of the lowest ball and sleeve lip edge near
the edge of the hole. The same procedure was then followed to
record the thickness change by each mandrel.

c) The thickness profile recorded in b) was superimposed
on the original one. The point in which the second curve deviated
from the first one was the elastic-plastic boundary around the cold-
worked hole by each mandrel.

This technique is straightforward. Only one suggestion may
be pointed out: the first deviation of the thickness profile of the
6.35 mm. (1/4 in.) thick coldworked specimen from the original one
came from change in thickness due to non-uniform deformation. The
locations where the thickness started to change on each surface were
checked by using a very thin piece of glass attached to the other
surface in both original thickness and the thickness after each
mandrel was pulled through the hole. The small marks on the left and

the right sides of the arrow index (see Figure 5.21) are the actual

 

 

 

 

Figure 5.21.--Typical method used to locate the elastic-plastic
boundary, r , from the thickness profile of the 6.35 mm.
(1/4 in.) tRick, 7075-T6 aluminum (P.12) on direction
number 2 and $0.001 in. scale after four different
sizes of oversize mandrels were pulled through the hole.
(0) . . . original thickness of non-coldworked hole.
(1) . . . after 0.076 mm. (0.003 in.) mandrel
(2) . . . after 0.102 mm. (0.004 in.) mandrel
(3) . . . after 0.127 mm. (0.005 in.) mandrel
(4) . . . after 0.152 mm. (0.006 in.) mandrel.

 

 

 

 

 

 

 

 

 

 

 

Figure 5.22.--Typical method used to locate the elastic-plastic
boundary, r , from the thickness profile of the 3.18 mm.
(1/8 in.) tRick, 7075-T6 aluminum (P.28) on direction
number 2 and 10.001 in. scale after four different sizes
of oversize mandrels were pulled through the hole.

(0) . . . original thickness of non-coldworked hole
(1) . . . after 0.076 mm. (0.003 in.) mandrel

(2) . . . after 0.102 mm. (0.004 in.) mandrel

(3) . . . after 0.127 mm. (0.005 in. mandrel

(4) . . . after 0.152 mm. (0.006 in.) mandrel.

86

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5.23.--Typical method used to locate the elastic-plastic
boundary, r , from the thickness profile of the 3.18 mm.
(l/8 in.) tRick, type 1100 aluminum (P.36) on direction
number 2 and 10.005 in. scale after four different sizes
of oversize mandrels were pulled through the hole.
(0) . . . original thickness of non-coldworked hole

(1) . . . after 0.076 mm. (0.003 in.) mandrel
(2) . . . after 0.102 mm. (0.004 in.) mandrel
(3) . . . after 0.127 mm. (0.005 in.) mandrel
(4) . after 0.152 mm. (0.006 in.) mandrel.

87

locations where the thickness started to change. The actual rp around
the hole of the thicker specimen could be located by averaging the
upper and lower rp. This was not done on the 3.18 mm. (1/8 in.) thick
specimen because the starting changes in thickness on both sides are
very close to each other.

Figures 5.21 to 5.23 illustrate the typical methods used to
locate the rp in each kind of specimen. Only one of four directions
is shown. The spikes (as seen in the plots) occur at the location of
the scale marks scribed on the specimen. This enables one to compare
the various graphs. The arrow index is the final location of the

elastic-plastic boundary measured with this technique.

5.4 Photoelastic Coating Technique

Photoelastic coating is a simple technique. A thin layer of
birefringent coating is bonded to the surface of a metallic specimen.
When the specimen is loaded and strained, the birefringent coating
responds; and the resulting fringe pattern observed in a reflected-
light polariscope can be interpreted in terms of the surface strains
of the metallic specimen. An important advantage of the birefringent
coating over other means of strain measurement is that the strains

are obtained over the entire coated area in a single picture. But

the thickness effect of the coating and the high strain gradients
constitute limitations (27,28,29,30).

Dixon and Visser (31) used this method to measure the stress
distribution and plastic deformation ahead of a crack in aluminum

alloy and mild steel; but, admittedly, their fringe orders at the

88

tip of the crack, where the strain gradients were very high, suffered
from the inaccuracy of the birefringent coating method. Calcote and
Bowman (32) used this method of isochromatics to predict the elastic—
plastic boundary of a rectangular beam having a single concentrated
load acting at its free end, but they found good agreement only on
the tension side. Recently the same method was used to visualize
the deformation around holes (33) and on interference-fit fastener
(23) but was found to be unsatisfactory. This method would be useful
only when the elastic-plastic boundary is far away from the hole.
The plastic coating used was purchased from Photoelastic Inc.
Plastic sheet type PS-lD with 0.022 10.002 in. thick as
suggested by references 29,31, and 0.15 K-factor
Cements: resin type PC-8
hardener type PCH-8
A polariscope model LF/MU by Instruments Budd Division and a Nikon
F-lOO camera with Kodak film Panatonic-X were used.
Specimens were polished with silicon carbide grinding papers,
grit number 320, and then followed by 400 grit. The fringe order
at the elastic-plastic boUndary is giVen as
N =.Z(li£l.o
rm 0
which were 3.37, 3.24, and 0.2 for the 6.35 mm. (1/4 in.), 3.18 mm.
(1/8 in.) thick, 7075-T6 aluminum, and the 3.18 mm. (1/8 in.) thick,
aluminum 1100, respectively.
It was quickly discovered that the photoelastic coating

would not stick in the very high gradient area near the edge of a

89

cold-worked hole. Some preliminary experiments were tried, by cutting
a hole on plastic the same size as the original hole of the specimen.
to get an idea of fringe patterns at the elastic-plastic boundary.
The bonding procedure as suggested by the manufacturer was used.
The fringe orders on the specimen, when observed through the analyzer
on the polariscope, showed up and then disappeared during the cold-
working process. Peeling of the plastic from the adhesive was found
at the edge of the hole. The maximum elongation of the plastic itself
is about 10 percent so that it cannot stand for the very high strain
gradient there.

A 6.35 mm. thick, 7075-T6 specimen was tried. The hole in
the plastic was cut about 2 mm. bigger than the original hole diameter
of the specimen. Some photographs were taken through the analyzer
with camera attached to the telemicroscope on the polariscope during
the coldworking process with ”mandrel in" condition as shown in
Figure 5.24. The rp's located by this technique (see Figure 5.24)
were located at the location of the calculated isochromatic fringe
order, which was the same as all mandrels were pulled through the
hole. The rp measured by this technique is smaller than the foil
gage technique with foil gage on the upper side surface. This means
that it is certainly smaller than the actual rp found by two
previous techniques.

Therefore, photoelastic coating is not applicable for meas-
uring the elastic-plastic boundary around the coldworked hole, due
to the following limitations:

6) The plastic bent along the specimen surface near the

90

 

(a) 0.0762 mm. (0.003 in.) (b) 0.1016 mm. (0.004 in.)
Rad. Exp. rp = 1.61 a Rad. Exp. rp = 1.91 a

   

(c) 0.127 mm. (0.005 in.) (d) 0.1524 mm. (0.006 in.)
Rad. Exp. rp = 2.06 a Rad. Exp. rp = 2.21 a

Figure 5.24.--Isochromatic-fringe patterns and elastic-plastic
boundaries obtained around the coldworked 6.35 mm°
(1/4 in.) thick specimen (P.4) as four different sizes
of oversize mandrels were pulled through the hole.

b)

91

edge of the hole leading to errors in reading the fringe
order;

It is difficult to apply the plastic on both sides of
the specimen surface;

The outer diameter of sleeve lip on the lower side of
the specimen is about 11 mm., which is large enough to
cause some trouble in reading the fringe order in case

the plastic is applied on both sides.

CHAPTER 6

ELASTIC-PLASTIC BOUNDARY MEASUREMENTS

In this chapter the elastic-plastic boundary, as measured by
foil gages and thickness change, are compared. The photo-elastic
coating was no longer employed after the debonding was found around
the coldworked hole.

6.l Elastic-Plastic Boundary
Measurement with Foil Gages

 

 

6.l.l 6.35 mm. (1/4 in.) Thick,
7075-T6 Specimen Data

The elastic-plastic boundary measurements by this technique
were made on seven coldworked 6.35 mm. (1/4 in.) thick specimens,
four specimens for gages applied on one side and three specimens for
gages applied on both sides. Table 6.1 lists the measured rp/a for
different expansions and specimens. The original hole diameter in
these specimens was very slightly oversized, as shown in Table 3.1.
The rp was located after each mandrel was pulled through the hole.
Since the gage application was made on different radial lines of the
hole for gages on one side, the uncertainty of the measurements of
rp/a located by the one side strains is larger than of the one located
by both sides strains (see Table 6.1). This uncertainty is computed

from the offset of the rp/a located by the radial and tangential

strains (or by the radial strains on both sides) from the actual rp/a.

92

93

Figure 6.1 illustrates the difference in the measured rp/a by
both procedures. The average of the measured rp/a was plotted (four
measurements by upper side foil gages and three by both sides foil
gages) as a function of radial expansion displacement used. This
plot shows that the actual rp/a obtained by averaging the measured
rp/a on both sides was significantly larger than the rp/a obtained

using the foil gages on the upper side only, and this was to be

expected by Sharpe AFOSR report (10).

TABLE 6.l--Measured r /a and its uncertainty obtained from foil gage
technique Rhen foil gages were applied on one side and on
both sides surfaces of the 6.35 mm. (1/4 in.) thick,
7075-T6 aluminum specimens.

 

Upper side foil gage Both sides foil gage

 

1‘ a r a

(fie/J ‘eL’
Specimen P.7 P.8 P.9 P.18 P.11 P.12 P.14
Radial
Displacement
(mm.)
0.076 1.62 1.65 1.72 1.64 1.91 1.69 1.83
(0.003 in.) 15.0% 11.0% 12.0% 11.0% 14.0% 10.0% 15.0%
0.102 1.84 1.95 1.96 1.85 2.16 2.05 2.00
(0.004 in.) 17.0% 12.0% 13.0% 15.0% 15.0% 12.5% 14.5%
0.127 2.13 2.12 2.23 2.08 2.21 2.27 2.24
(0.005 in.) 19.0% 17.0% 16.0% 11.0% 13.0% 12.0% 13.0%
0.152 2.30 2.30 2.29 2.21 2.45 2.39 2.34
(0.006 in.) 112.0% 19.0% 17.0% 13.0% 11.0% 12.0% 11.0%

 

94

 

 

 

2.5
i
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8 i
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Figure 6.l.--Comparison of the measured r /a obtained from foil gages
applied on one side and both Sides of the 6.35 mm.
(1/4 in.) thick, 7075-T6 aluminum.

95

6.1.2 3.18 mm. (1/8 in.) Thick,

7075-T6 Specimen Data

Elastic-plastic boundary measurements were made on four
specimens for gages applied on one side and three specimens for gages
applied on both sides as listed in Table 6.2. The plot of rp/a
versus radial expansion displacement in Figure 6.2 was taken from
Table 6.2. The differences of the measured rp/a by both procedures
are not as large as the ones of the 6.35 mm. thick specimens. This
is true because the thickening of the metal near the hole of thinner
specimens, when the mandrel is pulled through, is less than for the
thicker specimens. In addition, the force required to pull the
mandrel through the hole is reduced. The result is less sleeve
washer effect, which explains the smaller difference in strains on
both sides. This plot agrees with the thicker specimens in that the
rp/a measured on both sides was larger than the rp/a measured on the
upper side only. The scattering of the measurements is not as wide
as in the previous specimens. Even allowing for the biggest dif-
ference in the 0.076 Inn. (0.003 in.) radial expansion, this tech-
nique gives a close agreement in both procedures. This means that
the coldworking process by J. 0. King does give more uniform
deformation around a coldworked hole in the thinner specimen.
6.1.3 3.18 mm. (1/8 inch) Thick,

Aluminum type 1100 Specimen

The foil strain gage technique was used on some specimens of
type 1100 aluminum, but the coldworking process was not successful.
Radial strain of these specimens became smaller on the free upper

surface and larger on the lower surface as bigger mandrels were used.

TABLE 6.2--Measured r /a and its uncertaint

96

y obtained from foil gage

 

 

technique Bith foil ga es applied to one side and to both

Sides of the 3.18 mm. (1/8 inch) thick, 7075-T6 aluminum

spec1men.

Upper side foil gage Both sides foil gage
(rp/a) (r la)
P

Specimen P.19 P.20 P.22 P.24 P.25 P.28 P.30
Radial
Displacement
(mm.)
0.076 1.71 1.70 1.68 1.72 1.86 1.83 1.76
(0.003 in.) 13.0% 15.0% 14.5% 16.0% 12.0% 12.0% 12.0%
0.102 1.87 1.91 1.89 1.90 1.95 1.92 1.96
(0.004 in.) 12.0% 15.0% 11.0% 15.0% 13.0% 13.0% 12.5%
0.127 2.02 2.03 2.12 2.00 2.01 2.05 2.02
(0.005 in.) 11.0% 10.0% 11.5% 10.0% 11.5% 10.5% 10.0%
0.152 2.09 2.19 2.12 2.16 .2.05 2.14 2.12
(0.006 in.) 11.0% 13.0% 11.5% 13.0% 11.5% 3:2.0% i3.0%

 

The bending of the metal itself occurred and caused thickening on

the lower surface much more than on the upper surface.

In addition,

the sleeve was tightly wedged into the hole and the metal at the edge

surface was also pulled from the upper to the lower surface by fric-

tion between the sleeve and the edge surface.

Thus the coldworking

process by J. 0. King did not give uniform deformation in the very

soft metal and caused a lot of shearing effect during processing.

Usable data were not obtained for type 1100 aluminum with the foil

gage technique.

97

 

 

 

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Figure 6.2.--Comparison 0f the measured r /a obtained from foil gages
applied on one side and both sides of the 3.18 mm.
(1/8 in.) thick, 7075-T6 aluminum.

98

6.2 Elastic-Plastic Boundary
Measurement by Thickness Change
Measurement

 

 

 

6.2.1 6.35 mm. (1/4 in.) Thick,
7075-T6 Specimen Data

Thickness change measurements were made on the same specimens
that were used for the foil gage technique. In each case the thick-
ness change measurement was taken after the foil gage technique for
each mandrel used. For seven (P.7,8,9,11,12,l4,18) specimens, four
successively larger mandrels were used, measurements being taken
after each coldworking operation. For eight (P.1,2,3,5,10,15,16,l7)
specimens, only one mandrel was used; i.e. the coldworking was not
done in an incremental manner. The specimen numbers associated with
the measured rp/a are given in Table 6.3. Three specimens (P.7,8,9)
were tested using both scales (10.005 and 10.001 in.) on the
Daytronic amplifier. The overall rp/a obtained from both scales
were similar. The average of rp/a by both scales is the measured
rp/a by this technique. The rp/a measurements were made for four
radial directions around the coldworked hole. One value of the rp/a
represents one measurement. The average of the scattering of measure-
ments was computed by using a standard deviation method (34) based on
25 measurements for each amount of coldwork.

Figure 6.3 compares the average rp/a obtained from seven
specimens in which four mandrels were pulled through the hole of
each specimen, with that obtained from two sets of specimens in which

only one mandrel was pulled through the hole. The plot shows a very

99

 

 

 

 

 

 

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Radial Expansion Displacement (mm.)

Figure 6.3.--Comparison of the measured rp/a obtained from thickness
' change measurement technique with 1 CW. and 4 CW. on
one specimen of the 6.35 mm. (1/4 in.) thick, 7075-16
aluminum.

102

similar elastic-plastic boundary around the coldworked hole for the
two coldworking procedures. In other words, it makes little dif-
ference whether a given radial expansion is achieved in one step or
several steps. However, the rp/a for each mandrel in four directions
was not equal, indicating that the elastic-plastic boundary around
these coldworked holes was not truly round because the original hole
was not perfectly round, as shown in Table 3.1.
6.2.2 3.18 mm. (1/8 in.) Thick,

7075-T6 Specimen Data

Elastic-plastic boundary measurements were made on nine cold-
worked 3.18 mm. (1/8 in.) thick specimens after four mandrels were
pulled through the hole of each specimen. Three of them were tested
On both scales as for the thicker specimens. Two specimens (P.19
and 22) were tested by employing all three examining techniques.

The photoelastic coating failed and only one direction rp could be
measured, as given in Table 6.4. Furthermore, only one set of speci-
mens was tested with one coldworking operation per specimen (four
specimens for four coldworking levels).

Figure 6.4 shows no difference in the elastic-plastic
boundary around the coldworked hole for incremental coldworking
versus single coldworking. This plot was made using the average
rp/a of the nine specimens (30 measurements) which were coldworked
incrementally, and the average rp/a of four specimens (four measure-

ments per one specimen) which were coldworked in one operation.

103

 

 

 

 

 

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Radial Expansion Displacement (mm)

Figure 6.4.--Comparison of the measured r /a obtained from thickness
change measurement technique with 1 CW. and 4 CH. on
one specimen of the 3.18 mm. (1/8 in.) thick, 7075-T6
aiuminum.

106

6.2.3 3.l8 mm. (l/8 in.) Thick,
Aluminum Type llOO Specimen
Data

Elastic-plastic boundary measurements were made on four 3.18
mm. (l/8 inch) thick, aluminum llOO specimens after four mandrels
were pulled through the hole of each specimen. The scattering of
the measurements was very large due to the softness of the metal.

The surfaces on both sides were scratched by the carbide balls during
the movement of the specimen° This made too many ripples on the
final plot from the X-Y recorder. Careful judgment must be made to
decide where the thickness change starts. All specimens were run
with :0.005 in. scale to have a better curve from the X-Y recorder.

A final plat from the X-Y recorder is shown in Figure 5.21.

Table 6.5 lists the numbers of the specimens and the cold-
working used for each. The table shows that the measured rp/a around
the coldworked holes are relatively larger than the rp/a obtained for
the 7075-T6 samples. This is because the yield stress of this
specimen is much lower.

Figure 6.5 illustrates the comparison of the elastic-plastic
boundary around the coldworked holes caused by incremental and single
coldworking Operations. This plot also indicates the consistency of

no difference in the rp/a obtained from the incremental and single

cold-working operations.

6.3 Discussion

 

As for the techniques used to measure the elastic-plastic

boundary around the coldworked hole by the J. 0. King coldworking

107

 

 

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Radial Expansion Displacement (mm.)

Figure 6.5.--Comparison of the measured r /a obtained from thickness
change measurement technique with T CW. and 4 CW. on one
specimen of the 3.18 mm. (l/8 in.) thick, type 1100
aluminum with :0.005 in. scale of Daytronic amplifier.

109

process, only two techniques--the foil strain gage and the thickness
change measurement--are workable. The photoelastic coating is not
usable due to its limitation as explained in section 5.4. Besides
the limitation of using it with a very soft metal, the foil strain
gage technique has another limitation: only three gages can be used
within 4.26 mm. This method would give more accurate rp if more and
smaller gages could be used within the same distance. However, the
strains obtained from the 7075-T6 aluminum increase monotonically
from the elastic region to the first gage location. None of them
deviates much from the smooth curve connecting those three points
(see Figures 5.l2 to 5.l5). This means that the strains obtained
from this method and the curve connecting them can be a good repre-
sentation of strain distribution around the coldworked hole. As
discussed in Chapter 4, the Hsu-Forman is the mOSt general of all'
present theories developed. The strains on the upper free surface of
the specimen agree well with the predicted strains by this theory
(see Figure 5.14 and 5.15) for smaller amounts of coldwork. The
nearest gage location to the edge of the hole was between T and 2 mm.
whereas the predicted strains by most theories are less than 2%.
The obtained strains were also never more than 2%, yet the foil
strain gages had a maximum strain range of :lO percent. With its
excellent characteristic of measuring infinitesimal strain, the foil
strain gage is an accurate technique for the measurement of elastic-
plastic boundary around the coldworked hole.

The lack of uniform deformation through the specimen thick-

ness in the experimentally measured surface strains makes it impossible

110

r
The actual average rp of the coldworked hole was finally determined by

to locate rp by determiningsurface strains (8 and 86) on only one side.

averaging the rp of both side surfaces of the specimen. The thickness
change measurement, though tedious on the very soft metal, is workable
in all metals. However, both techniques give useful data on the
harder metal 7075-T6 aluminum. The measured rp/a obtained from both

techniques have been shown here to be consistent with each other.

In order to clarify their accuracies, two extra plots are
plotted in Figures 6.6 and 6.7. Figure 6.6 is the plot of average
rp/a obtained from the 6.35 mm. thick, 7075-T6 specimens, taken from
Tables 6.l and 6.3. The average rp/a obtained from the thickness
change measurement were made from twelve measurements (four measure-
ments per one specimen) while three measurements were taken from the
foil gage (one measurement per one specimen). Percentage differences
are based on the distance between rp and the edge of the hole and
are ll.00, 7.50, 1.60, and 2.00 for the 0.076(0.003 in.), 0.l02
(0.004 in.), 0.l27 (0.005 in.) and 0.152(0.006 in.) mm. radial
expansion displacements, respectively.

Figure 6.7, the average rp/a obtained from'both techniques
of the 3.l8 mm. thick, 7075-T6 specimens are taken from Tables 6.2
and 6.4 and compared. The percentage differences are 8.50, l.00,
2.00 and 2.73 for the incremental mandrel sizes, respectively.

Results from both techniques are in excellent agreement for
larger deformation, but show somewhat greater variations for smaller
deformation. The overall percentage differences of both techniques

are 5.52 and 3.56 for the 6.35 and 3.l8 mm. thick specimens, which

111

2.50«

l

l"

o

o
l

P

 

—Thickness Change
l.50 4 Measurement.
-—---- Foil Gage on Both Sides

( R ll, l2, I4)
max.
t
mln.

1.00 4 t : 0- A0 ‘0 . .
O 0.05 DJ

Radial Expansion Displacement (mm)

Plastic Zone Radius/ Hole Radius ( r / a)

 

0.076 0.102 01127 0.152
, L, . .1 , , ,1 0
OJ!)

 

L
1

Figure 6.6.--Comparison of the measured r /a obtained from thickness
change measurement and foil gage (both sides) techniques
of the 6.35 mm. (l/4 in.) thick, 7075-T6 aluminum
(P.ll,l2,l4).

112

 

 

 

2.501

1

1
c?
\
‘3
“200 «
a
.2
'3
tr
.2
O
I
\
:3
G’ i
. __ Thickness Change
3 15° ‘ Measurement
§ . _..._ Foil Gage on Both Sides
0
E , (9.25, 23. 30)

max.
‘ § av.
min.
0076 0.102 0.127 0.152
1.00 + .¥: + : 4. 10L: :4105 ;l: ; :1:
0 0.05 0.1 0.15

Radial Expmsion Displacement (mm)

Figure 6.7.--Comparison of the measured r /a obtained from thickness
change measurement and foil gage (both sides) techniques
of the 3.18 mm. (1/8 in.) thick, 7075-T6 aluminum
(P.25,28,30).

113

are well within the acceptable range of experimental accuracy. There-
fore, the thickness change measurement is also an accurate technique

for the elastic-plastic boundary measurement. With its advantages of
being reusable and efficient, the thickness change measurement may be

recommended for elastic-plastic boundary measurement in the future.

CHAPTER 7
COMPARISON OF THEORIES WITH EXPERIMENTS

The seven theories used for comparison in the experiments are
the Potter-Grandt, Hsu-Forman, Nadai, Rich-Impellizzeri, Adler-Dupree,
Sachs, and Carter-Hanagud. The predicted rp/a were computed on a
typical specimen of each material for the particular expansion con-
sidered and were separately plotted on a dimensionless form in
Figure 4.3, and replotted in Figure 7.1 for comparison with the
experiments. This is a general form in which the predicted rp/a
associated with different amounts of coldwork for each theory can be
concisely compared. Only one value of rp/a predicted by Adler-Dupree
is given, due to the fact that only one specimen thickness and radial
expansion were computed via the finite-element method in their report.
Regardless of yield criteria used, the seven theories fall into two
classes: plane stress and plane strain conditions. As shown in the
plot in Figure 7.l, the predicted rp/a by plane strain theories
(Rich-Impellizzeri and Sachs) are larger than the ones by plane
stress theories assumed (except the ones by Carter-Hanagud and
Adler-Dupree). Rich-Impellizzeri modified the thick-walled tube
theory (l2) by including the compressive radial displacement of the

mandrel under the same assumptions which cause smaller rp/a than the

114

115

 

 

 

/Carter-Hmagud
. /
.0 .. /
. . Thick-Walled
denotes standard de at'on /
J I V' ' , Tube nixSocns)
Adler-Dupree
‘l Impellizzeri
“~215-
2
he.
.3
‘6 .
m i
5
1:
\
.3 20
8
c:
“s“
N
is:
O
1:
L5 .
'0 0 ‘r t 4 4 t c :
O 2 4 6 8 10* 12 14

Dimensionless Radial Expansion Displacement10= LL fie Io 05)

Figure 7.l.--Comparison of the elastic-plastic boundaries predicted
by various theories and experiments for 7075-T6
aluminum.

116

(original Sachs solution. Nadai considers the perfectly plastic
material response and produces a higher rp/a than the predicted value
by the work-hardening theory of Hsu-Forman. The Potter-Grandt theory
predicts the smallest rp/a under the perfectly plastic material while
Carter-Hanagud predict the largest value of all theories under Tresca
yield criterion assumption.

Figure 7.1 compares the theories with experiments for the
7075-T6 aluminum. The measured rp/a, with the thickness change meas-
urement, are taken from Table 6.3 and 6.4 for the 6.35 mm. (l/4 in.)
and 3.l8 mm. (l/8 in.) thick, 7075-T6 specimens respectively. The
comparison of 3.l8 mm. thick specimen data with the theories is excel-
lent. The experimental data lie between the two plane stress theories
(Nadai and Hsu-Forman). The percentage differences between the
average experimental and the predicted rp/a by these two theories are
given in Table 7.1. The percentage difference is based on the dis-
tance from the edge of the hole to the elastic-plastic boundary, and
a positive sign means that the experimental value is larger than the
value predicted by theory. By this comparison, only two plane stress
theories agree quite well with the experiments. The Nadai theory, in
which perfectly plastic material is assumed, is in nearly complete
agreement with the experiments. As found by the foil gage technique,
the experimental strains on the upper side of the specimen, where the
supposed uniform deformation should be, agreed with the predicted
strains by the Hsu-Forman theory better than the ones by Nadai. This
means that the Hsu-Forman theory has the best chance of agreeing with

experiments if uniform deformation exists through the thickness of

117

TABLE 7.l--Percentage differences between the measured r la obtained
from thickness change measurement for the 3.1 mm. (l/8 in.)
thick, 7075-T6 aluminum and the Nadai and Hsu-Forman

 

 

theories.
Radial Displacement x Difference of Experiments with
(mm.) Nadai Hsu-Forman
0.076 (0.003 in.) +l.32 +6.95
0.l02 (0.004 in.) 0.0 +8.24
0.127 (0.005 in.) -2.89 +5.2l
0.l52 (0.006 in.) -5.26 +3.85

 

the specimen. According to the measured strains obtained from the
foil gage on both sides of the specimen, the difference between the
strains on each surface became smaller as a bigger mandrel was pulled
through the hole. One may suggest that for larger amounts of cold-
work a nearly uniform deformation exists through the thickness of
3.18 mm. (l/8 in.) thick, 7075-T6 specimen, and a condition of plane
stress is attained. In consideration of conservative design, two
optimum theories (Nadai and Hsu-Forman) may be suggested to a designer
for spacing the hole locations in practice. He may predict the
plastic zone size around a coldworked hole by the Nadai theory when
requiring more precision on 3.l8 mm. (l/8 in.) thick plate.

The comparison of the experiments with the theories shows the
non-existence of plane stress condition on the 6.35 mm. (l/4 in.)
thick specimens. It tends to be either plane strain or generalized
plane strain condition. The experimental rp/a, in the plot, are very

much larger than the predicted rp/a of the plane stress theories.

118

The experimental results do not vary as greatly with the amount of
coldwork as do the predicted rp/a of the plane strain theories. Two
plane strain theories (Sachs and Rich-Impellizzeri) agree fairly well
with the experiments as shown in Table 7.2 and Figure 7.l. For the
same reason as given for the thinner specimens, these two theories
may be the only ones which have a better chance to agree with the
experiments on the smaller amounts of coldwork if the deformation is
uniform. The experiments on this thickness do not agree with any
theory for the larger amount of coldwork used, 0.l52 mm. (0.006 in.),
which is the largest amount typically employed on the 6.35 mm. (l/4
in.) thick plate. Two theories--Sachs and Rich-Impellizzeri--may be
used to predict rp/a around a coldworked hole of the 6.35 mm. (l/4
in.) thick plate for smaller amounts of coldwork. But one must make
measurements to get the actual rp/a if a larger coldwork is required,
in order to avoid an overlap of the rp with an adjacent coldworked

hole.

TABLE 7.2--Percentage differences between the measured r /a obtained
from thickness change measurement for the 6. 35m
(l/4 in.) thick, 7075- T6 aluminum and the Thick- Walled
Tube and Rich- Impellizzeri theories.

 

 

Radial Displacement % Difference of Experiments with
(mm.) Thick-Walled Tube Rich-Impellizzeri
0.076. (0.003 in.) +9.52 +l7.95
0.l02 (0.004 in.) +l.77 +8.49
00127 (0.005 ine) ‘8e00 '2031

0.152 (0.006 in.) -15.00 -11.11

 

119

Table 7.3 shows the predicted rp/a by some theories (Nadai,
Carter-Hanagud, Sachs, and Rich-Impellizzeri) and the average rp/a by
thickness change measurement of the 3.18 mm. (l/8 in.) thick, alum-
inum type 1100 specimens. The predicted rp/a computed from three of
these theories, as shown in this table, are incredibly large. Only
the Nadai theory gives results that compare closely to the experi-
mental work for the soft 1100 aluminum specimens. The variations in
the rp/a values are about 20%, this difference perhaps caused either
by an error in locating rp/a from thickness change curves or by the
theory itself. As explained in the chapter on techniques, many extra
ripples are caused by the scratched surface; this leads to difficulty
in locating the change in thickness. Pure aluminum, however, is

rarely used for industrial applications.

TABLE 7.3--Average measured r /a obtained from thickness change
measurement and prBdicted r /a from four theories for the
3.18 mm. (1/8 in.) thick, type 1100 aluminum.

 

 

Average rp/a Nadai Carter-Hanagud Sachs Rich-Impellizzeri
3.47 3.08 8.52 7.43 7.39
3.66 3.19 9.80 8.57 8.54
3.76 3.27 10.93 9.59 9.55
3.84 3.33 11.96 10.50 10.47

 

Another experimental condition which can be compared with the
theories is the final diameter of the relaxed hole after unloading.

Such data also help the designer to determine the amount of coldwork

120

needed for his final hole size before fabricating in practice. Two
researchers (9,11) have determined these data. Potter-Grandt comb
puted the residual radial displacement of a coldworked hole after
unloading and plotted this information in dimensionless form as a
function of the amount of coldwork. The final diameters of the
relaxed holes of the 7075-T6 specimens were computed by this theory
and are shown in Figure 7.2. Adler-Dupree compared experimental
relaxed hole diameters with the computed ones by the finite-element
method on the 6.35 mm. (1/4 in.) thick, 7075-T6 aluminum. The exper-
imentally permanent radial deformation of 0.112 mm. (0.0044 in.) was
found after a 5.53 mm. (0.257 in.) mandrel was pulled through a

6.60 nun (0.26 in.) hole, while a final deformation of 0.127 mm.
(0.005 in.) was found in this experiment.

The final relaxed hole diameters are given in Tables 7.4,
7.5 and 7.6 for all kinds of the specimens. Those values are the
average of four measurements around the hole (the same as the origi-
nal hole measurement). Four theories (Nadai, Hsu-Forman, Potter-
Grandt, and Adler-Dupree) are compared with the experiments. The
predicted relaxed hole diameters, by the Nadai and Hsu-Forman
theories, were computed from the residual tangential strains right
at the edge of the hole.

Figure 7.2 compares the experiments and the theories. Only
one curve of the 3.18 (l/8 in.) thick, 7075-T6 aluminum predicted by
each theory is plotted for clarity. The computed values by the Hsu-
Forman and Nadai theories vary with the amount of coldwork as a

straight line connecting those values, and so do the experiments.

121

Only one value is given by the Adler-Dupree theory, due to the reason
stated earlier.

Table 7.7 and Figure 7.2 show larger values in the experiments,
which is the same as the result of Adler-Dupree (11). This may stem
from the difficulty of matching boundary conditions at the experiment
hole with theories, and from lack of a precise theory for computing
the finite strains near the edge of the coldworked hole. The Nadai
theory, as computed on the aluminum type 1100, shows unusable data.
For computation on very low yield stress material, the Nadai theory
gives an exaggerated residual strain at the edge of the hole, and
results in a too large relaxed hole as compared with the experiments.
Therefore, there is no comparison on the aluminum type 1100 in

Figure 7.2.

122

TABLE 7.4.--Measured final diameters of 6.35 mm. (1/4 in.) thick, 7075-T6 specimens
after the mandrel was pulled through the holes.

 

Final diameters (mm.) by radial expansion of

 

 

Specme” or‘emat‘m 0.076 0.102 —o.127 —o.152 m.
P.1 (1) 6.721
2 6.721
3 6.718
4 6.726
P.2 1 6.729
2 6.723
3 6.723
4 6.716
P.3 1 6.782
2 6.772
3 6.787
4 6.785
P.5 1 6.795
2 6.807
3 6.782
4 6.797
P.7 1 6.734 6.795 6.856 6.883
2 6.739 6.795 6.840 6.871
3 6.736 6.792 6.835 6.792
4 6.739 6.797 6.848 6.863
P.8 1 6.731 6.784 6.835 6.871
2 6.736 6.782 6.835 6.868
3 6.739 6.784 6,838 6.873
4 6.741 6.787 6.833 6.876
P.10 1 6.873
2 6.864
3 6.873
4 6.876
P.11 1 6.729 6.805 6.838 6.886
2 6.718 6.800 6.828 6.886
3 6.726 6.789 6.825 6.861
4 6.711 6.782 6.822 6.868
P.12 1 6.736 6.777 6,833 6.858
2 6.736 6.795 6.835 6.881
3 6.731 6.787 6.835 6.878
4 6.739 6.784 6.825 6.863
P.15 1 6.751
2 6.746
3 6.762
4 6.759
P.16 1 6.830
2 6.802
3 6.825
4 6.833
P.17 1 6.838
2 6.838
3 6.838
(4) 6.843
Average 6.729 6.782 6.825 6.868
(0.2649 in.) (0.2670 in.) (0.2687 in.) (0.2704 in.)
St. Deviation 20.008 $0.016 $0.018 $0.0127

(:0.0003 in.) (:0.0006 in.) (10.0007 in.) (10.0005 in.)

 

TABLE 7.5.--Measured final diameters of 3.18 mm. (1/8 in.) thick,
7075-T6 specimens after the mandrel was pulled through

the holes.

Specimen Orientation

Final diameters (mm.) by radial expansion of

 

 

0.076 0.102 0.127 0.152 mm.
P.25 (1) 6.736 6.782 6.807 6.873
2 6.764 6.795 6.820 6.873
3 6.739 6.787 6.815 6.878
4 6.772 6.782 6.820 6.871
P.27 1 6.731 6.792 6.840 6.864
2 6.723 6.787 6.845 6.864
3 6.731 6.769 6.822 6.873
4 6.746 6.792 6.830 6.881
P.28 1 6.729 6.784 6.840 6.863
2 6.721 6.774 6.835 6.868
3 6.726 6.792 6.843 6.863
4 6.718 6.784 6.840 6.863
P.31 1 6.713
2 6.716
3 6.726
- 4 6.716
P.32 1 6.789
2 6.795
3 6.782
4 6.784
P.33 1 6.863
2 6.843
3 6.822
4 6.848
P.34 1 6.864
2 6.871
3 6.871
(4) 6.866
Average 6.731 6.787 6.833 6.871
(0.2650 in.) (0.2672 in.) (0.2690 in.) (0.2705 in.)
St. Deviation 10.018 10.008 $0.016 $0.005

(10.0007 in.)(i0.0003 in.)(:0.0006 in.)(t0.0002 in.)

124

TABLE 7.6--Measured final diameters of 3.18 mm. (l/8 in.) thick, type
1100 aluminum specimens after the mandrel was pulled through
the holes.

 

Final diameters (mm.) by radial.expansion of

 

Specimen Orientation

 

0.076 0.102 0.127 0.152 mm.
P.38 (1) 6.802
2 6.804
3 6.777
4 6.792
P.39 1 6.862
2 6.894
3 6.855
4 6.873
P.4O 1 6.789
2 6.820
3 6.762
4 6.784
P.4l 1 6.825
2 6.840
3 6.805
4 6.820
P.42 1 6.863
2 6.853
3 6.853
4 6.853
P.43 1 6.894
2 6.904
3 6.909
4 6.922
P.44 1 6.952
2 6.929
3 6.911
4 6.911
P.46 1 6.927
2 ,6.888
3 6.891
4 6.927
P.47 1 6.779 6.845 6.876 6.906
2 6.772 6.837 6.864 6.919
3 6.817 6.858 6.878 6.909
4 6.807 6.850 6.888 6.906
P.50 1 6.789 6.840 6.919 6.967
2 6.815 6.840 6.914 6.942
3 6.784 6.833 6.909 6.960
(4) 6.822 6.858 6.914 6.980
Average 6.795 6.845 6.891 6.922
(0.2675 in.) (0.2695 in.) (0.2713 in.) (0.2725 in.)
St. Deviation $0.018 i0.020 $0.023 $0.025

(i0.0007 in.)(i0.0008 in.)(i0.0009 in.)(10.001 in.)

 

125

 
    

 

mm. in.
701-27150
' I Standard Deviation
A 3.18mmthick, iypelloo
3.9532740 a 3.18m.i'hlck, 7075-1'6
0 636mmihick, 7075-1‘6
_...._ Nadai
590 ,-272°_.._.Heu-F0rman
—-—Fbiier-Grandi
O Adler-Dupree
(iasquRlD
.2
E am.-2680
'5
3
3
267532660
5 y ,
E! ’///fi .
19 - ."’//’ .,/’
36-70-1526“ ¢u ./
O /
E .
/
665352620
’33
0 w ( . . _ ; : 4' : : - 5
0 ‘ 0.05 010 0.15

Rad. Expansion Displacement (mm.)

Figure 7.2.--Comparison of the measured final relaxed hole diameters

with

four theories (Nadai, Hsu-Forman, Potter-Grandt,

and Adler-Dupree).

126

TABLE 7.7.--Comparison of measured final diameters with theories for
the 7075-T6 aluminum specimens as four different sizes
of the oversize mandrels were pulled through the holes.

 

 

Radial - Experiments larger than (mm.)
. Thickness
Displacement Nadai Hsu-Forman Potter-Grandt
(mm.) (mm.)
0.076 6.35 (1/4 in.) 0.044 0.036 0.065
(0.003 in.) 3.18 (1/8 in.) 0.043 0.036 0.063
0.102 6.35 0.051 0.043 0.076
(0.004 in.) 3.18 0.054 0.046 0.076
0.127 6.35 0.048 0.041 0.075
(0.005 in.) 3.18 0.054 0.046 0.077
0.152 6.35 0.047 0.039 0.072
(0.006 in.) 3.18 0.044 0.038 0.067

 

CHAPTER 8

CONCLUSIONS

In this experimental study, three simple techniques were
examined for usefulness. Only two techniques (foil strain gage and
thickness change measurement) are workable; they gave very close
results. With its reasonable accuracy, simplicity, and economy, the
thickness change measurement technique may be used for the elastic-
plastic boundary measurement around the coldworked hole in the future.

5 and 4 x 10"3 in. of the LVDT and the

The resolution of 10 x 10'
linear potentiometer are high enough to generate an accurate value

of the location where the original thickness of the specimen starts
to change.

As to comparison with the theories, the results obtained from
the thickness change measurement on the coldworked 3.18 mm. (1/8 in.)
thick, 7075-T6 aluminum, made by the J. 0. King coldworking process,
agree quite well with two plane stress theories (Nadai and Hsu-
Forman). According to the plane stress assumption of these two
theories, it is further indicated that a state of plane stress exists
in the plastic region during the coldworking process. In other words,
the plane stress assumption appears to be correct for the 3.18 mm.
(1/8 in.) thick plate. However, the results of the 6.35 mm. (1/4 in.)

thick, 7075-T6 aluminum show a larger rp/a than the values predicted

by the plane stress assumed theories (except the one by Carter-

127

128

Hanagud). Two plane strain assumed theories (the thick-walled tube
and Rich-Impellizzeri) agree fairly well with the experiments for
the smaller amounts of coldwork. None of the theories agrees with
the experiments on the larger amounts of coldwork. The condition of
plane stress does not exist in this thicker plate; it tends to be
either plane strain or generalized plane strain. It is interesting
to point out that as the thickness of the same material is increased
(the ratio of a hole-diameter-to-thickness is decreased), the results
of the rp/a given by the plane stress assumed theories (except the
one by Carter-Hanagud) converge to that of the plane strain assumed
theories. For the large coldworking levels, the experimental results
do not vary greatly with amount of coldwork. Therefore, for a given
hole diameter the specimen thickness makes a considerable difference
in the location of the elastic-plastic boundary around the hole.

Aluminum type 1100 shows unexpected results due to its soft-
ness and very low yield strength, and none of the theories agrees
well with the data obtained from the thickness change measurement
technique.

The results from the foil strain gage technique show that the
industrial coldworking process does not produce the uniform radial
expansion of the hole edge that all theories assume. The shearing
deformation associated with the pulling of the mandrel is important.

The result for the final diameter of relaxed holes after
unloading shows some differences with the calculated values of three
theories (Nadai, Hsu-Forman, and Potter-Grandt). The measured

diameters are larger than the calculated values, perhaps due to the

129

difficulty in matching boundary conditions at the experiment hole with
theory and to the question of computation of the finite strains near
the edge of the hole.

From the conservative design viewpoint, the thickness change
measurement technique as developed in this work is quite useful in
selecting a theory for a particular use. The results obtained from
this technique indicate that some theories can be directly used to
evaluate the elastic-plastic boundary around the coldworked hole on
two different thicknesses. In the absence of direct measurement, the
Hsu-Forman and Nadai theories are the two best adapted for calculat-
ing the rp of the 3.18 mm. (1/8 in.) thick plate with the required
precision. The thick-walled tube and Rich-Impellizzeri theories are
optimum for the 6.35 mm. (1/4 in.) thick plate associated with small
amounts of coldwork. For larger amounts of coldwork and more preci-
sion, it is necessary to make the thickness change measurement.

Chang (2]) and Rich et al (13) were conservative when they assumed
plane strain on the residual stresses computation of the 12.7 mm.
(0.5 in.) and 19.05 mm. (0.75 in.) thick plates, respectively. But
their radial expansions of 0.18 mm. (0.007 in.) and 0.1521mn.

(0.006 in.) were too large to directly evaluate the elastic-plastic
boundaries around their coldworked holes from the theories used.

The experimentally measured strains on the upper free surface
of the specimen agree quite well with the values predicted by the
Hsu-Forman theory. This theory should be the best of all the theories
for predicting the elastic-plastic boundary around a coldworked hole

of the 3.18 mm. thick plate 1: uniform deformation is performed,

130

whereas the Sachs and Rich-Impellizzeri theories should be the best
for use with the 6.35 mm. thick plate associated with small_amounts
of coldwork.

The gap between each level of coldwork considered (0.025 mm.
(0.001 in.)) is small enough for one to examine the variation of the
elastic-plastic boundary with confidence. However, only one material
(7075-T6 aluminum) was studied. In the future, the thickness change
measurement technique should be applied to several thicknesses of a
variety of other metals for determining the elastic-plastic boundary

around coldworked holes.

REFERENCES

10.

11.

REFERENCES

Timoshenko, S., and J. N. Goodier, Theory of Elasticity (McGraw-
Hill, New York, 1951), pp. 80-81.

 

Knott, J. F., Fundamentals of Fracture Mechanics (John Wiley &
Sons, New York-Toronto, 1973), pp. 25-45.

 

Allen, M., and J. A. Ellis, "Stress and Strain Distribution in
the Vicinity of Interference Fit Fasteners" (Technical
Report AFFDL-TR-72-153, Wright-Patterson AFB, Ohio,

Jan. 1973).

Gran, R. J., et al., "Investigation and Analysis Development of
Early Life Aircraft Structural Failures" (Technical Report
AFFDL-TR-70-149, Wright-Patterson AFB, Ohio, March 1971).

Tiffany, C. F., et al., "Fatigue and Stress-Corrosion Test of
Selected Fastener/Hole Process" (Technical Report ASD-TR-lll,
Wright-Patterson AFB, Ohio, Jan. 1973).

Grandt, A. F., and J. P. Gallagher, "DevelOping an Infinite Life
Design Procedure for Fastener Holes Utilizing Fracture
Mechanics" (Technical Report AFML/LLP-72-3, Wright-Patterson
AFB, Ohio, Sept. 1972).

Nadai, A., "Theory of the Expanding of Boiler and Condenser Tube
Joints through Rolling" (Trans. ASME 65, Nov. 1943),
pp. 865-880.

Hsu, Y. C., and R. G. Forman, "Elastic-Plastic Analysis of an
Infinite Sheet Having a Circular Hole under Pressure"
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Potter, R. M., and A. F. Grandt, Jr., "An Analysis of Residual
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