EARING IN CUPPING EXPERIMENTS
RELATED TO
ANISOTROPIC PLASTICITY THEORY .

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY *
ROBERT W. BUND
1969

“Thesis I LIB R A R Y

MlCl‘ilgdu State
University

 

This is to certify that the

thesis entitled
Earing In Cupping Experiments
Related To

Anisotropic Plasticity Theory

presented by

Robert W. Bund

has been accepted towards fulfillment
of the requirements for
Ph.D. degree in Mechanics
Department of Metallurgy, Mechanics and

Materials Science

 

/ , \
/ l \ A
j) .1 (I f" O ,1 k.) L C Li ( / v
‘ 5“ \J. 'Li‘t VVJ L“-

Major professor

\

 

 

Date August 13, 1969

 

0-169

J’-

ABSTRACT

EARING IN CUPPING EXPERIMENTS RELATED TO
ANISOTROPIC PLASTICITY THEORY

by
Robert W. Bund

During the operation of drawing a cylindrical cup from a flat,
circular sheet—metal blank, undulations or ears are produced at the
free edge. A study of the earing phenomenon was made from an experi-
mental and a theoretical point-of-view. The general objective of this
investigation was to study the earing phenomenon theoretically using
plasticity theory, and then to compare the results of this theoretical
analysis with the experimental results of cupping tests. The theo-
retical analysis is an extension of the work of Chung and Swift, and
of Hill. The General Electric 265 computer was used during the in-
vestigation to facilitate the computations.

Commercially-produced, aluminum—killed steel sheet was used
to produce blanks (4.800 inch diameter by 0.035 inch thick) for the
cupping eXperiments. A polar-grid pattern was imprinted on the flat
blanks by the electrochemical etching method to eXperimentally de-
termine strain at nine successive stages of partial draws. Polyethy-
lene film was used as a lubricant during the draw operation to preserve
the polar-grid pattern. The double-action draw die.used in the ex-
periments was actuated by a single-action, straight-sided mechanical
press equipped with a pneumatic die cushion.

The theoretical study used Hill's anisotropic yield function

to introduce anisotropy into the plane stress analysis. The direct

Robert W. Bund

method was used to determine the anisotropic parameters by directly
measuring the yield stress at selected orientations. The indirect
method (strain-ratio method) was used as a check on the direct method.
Plastic potential theory was used to derive the stress, strain-
increment equations from the anisotrOpic yield equation. A rigid
work-hardening material was assumed since the elastic strains were
considered negligible compared to the plastic strains.

Strain hardening was introduced into the theoretical study
by means of the three-parameter Ludwik stress-strain relation. It
was assumed that hardening causes the anisotropic yield ellipse to
enlarge without changing its shape (isotrOpic hardening) while pre—
serving its initial anisotropy.

As a result of the investigation, it was found that the theo-
retical analysis did predict strain fields of the type associated with
the 0° and 90° earing which occurred during the experimental study.
However the radial strain field from the theoretical analysis for the
0° and 45° directions indicated strains smaller in magnitude than
occurred during the experimental cupping. It is believed that better
agreement could be obtained by using the indirect method to determine
the anisotropic parameters. Since both theoretical and experimental
results of this investigation indicate prOportional straining, a
total strain theory could be used to replace the incremental theory

used in the investigation.

EARING IN CUPPING EXPERIMENTS
RELATED TO

ANISOTROPIC PLASTICITY THEORY

By

Robert W. Bund

A THESIS

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

1969

ACKNOWLEDGEMENTS

To the many people who have helped and encouraged me, particularly
to Professor L. E. Malvern for his valuable counsel throughout the study,
and to the members of my guidance committee, Professors T. Triffet,

G. E. Mase and R. H. Wasserman.
Fifteen months of advanced graduate studies were made possible by

a National Science Foundation Science Faculty Fellowship.

ii

TABLE OF CONTENTS

LIST OF TABLES O O O O O O O O O O O O O O O I O O O 0

LIST OF FIGURES O O I O O C O O O O I O O O O O O O O

I.

II.

III.

INTRODUCT I ON 0 O O O O O O O O O 0 O O O O O O

1. Preliminary Remarks . . . . . . . . . .
l.
l.

weer-

Of H. W. SWift O O O O O O O O O I O O

1.4 The Contribution of H. W. Swift an Others to

IsotrOpic Cup-Drawing Research . . . .

1.5 More Recent Research Pertaining to Measures of

Anisotropy O I O O O O O O O O O O O O

1.6 Recent Research Pertaining to the Cup-Drawing

Process for AnisotrOpic Metals . . . .
1.7 The Present Investigation . . . . . . .

Some Early Studies Pertaining to AnisotrOpy
Studies of the Cup-Drawing Process up-to-the

EXPERIMENTAL DETERMINATION OF THE TENSILE AND SHEAR YIELD

STRENGTHS O I O O O O O O O O O O O O O O I 0

Preliminary Remarks . . . . . . . . . .
Theory 0 O O O O O 0 O O O O O C O I O O

NNN
WNH

stress 0 O O O I O O O I O O O O I O C

2.4 Experimental Determination of th Tensile Yield

Stress as a Function of Orientation .
THE DERIVED ANISOTROPIC YIELD FUNCTION . . . .
Preliminary Remarks . . . . . . . . . .
Computation of the Shear Parameter N . .

3 1
3 2
3 3 Computation of the Tensile Parameters F,
3.4 An Appraisal of the Computed AnisotrOpic
3 5
3 6

Fourth Quadrant . . . . . . . . . . .

iii

C, an
Parameters
Transformation of the Anisotropic Yield Function .
A Linear Approximation to the Yield Function in the

Experimental Determination of th Shear Yield

H.

Page

vi

10
14
16
18

18
19

.22

23

31

31
32
32
36
43

48

Page

IV. EXPERIMENTAL DETERMINATION OF THE STRAIN-HARDENING

BEHAVIOR I I I I I I I I I I I I I I I I I I I I I I I I 56
4.1 Preliminary Remarks . . . . . . . . . . . . . . . 56
4.2 The Strain-Hardening Assumption . . . . . . . . . 58
4.3 Tensile Test Procedures . . . . . . . . . . . . . 66
4.4 Ludwik's Three-Parameter Stress-Strain Equation . 69
V. THEORETICAL ANALYSIS OF THE CUP-DRAWING PROCESS . . . . 74
5.1 Preliminary Remarks . . . . . . . . . . . . . . . 74
5.2 The Yield Condition . . . . . . . . . . . . . 78
5.3 Stress Analysis Theory for the Flange . . . . . . 82
5.4 Strain Analysis Theory for the Flange . . . . . . 87
5.5 Stress and Strain Analysis for a Rim Element . . 91
5.6 Stress and Strain Analysis for Interior Elements. 94
5 I 7 Results I I I I I I I I I I I I I I I I I I I I I 101
VI 0 CUP-DRAWING EXPERIMENTS o o o o o o o o o o o o o o o o 114
6.1 Preliminary Remarks . . . . . . . . . . . . . . 114
6.2 Producing a Polar Grid Pattern on Sheet-Metal
Blanks I I I I I I I I I I I I I I I I I I I I 115
6.3 Measurement of Grid Spacing . . . . . . . . . . 117
6.4 Experimental Cup Drawing ... . . . . . . . . . . 121
6.5 Procedures Used to Compute Strain from
Experimental Data . . . . . . . . . . . . . . . 123
6 I 6 ReSUItS I I I I I I I I I I I I I I I I I I I I I 131
VII. SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . 141
7.1 Preliminary Remarks . . . . . . . . . . . . . . . 141
7.2 Comparison of the Theoretical Strain Field with
the Experimentally-Determined Strain Field . . 141
7.3 Conclusions and Recommendations . . . . . . . . . 143
MPEND Ix I I I I I I I I I I I I I I I I I I I I I I I I I I I 14 7
BIBLIOGRAPHY I I I I I I I I I I I I I I I I I I I I I I I I 151

iv

Table
203“].

2.4-1

304-1
305-1
306-1

306-2

306-3

4.3-1

501-1

5.5-1

5.6-1

5.7-1

5.7—2

6.5—1
6.5—2

6.6-1

LIST OF TABLES

Single-Shear Test Results . . . . . . . . . . .

Summary Results of the Tensile Yield Stress
Deteminat ion I I I I I I I I I I I I I I I I I

SIGPLT Computer Program Output . . . . . . . .
Computer Output for YIELD Program at a - 0° . .
YIELD3 Computer Program Output for a - 0° . . .

Computer Output from Standard POLFIT Program for

a - 0° I I I I I I I I I I I I I I I I I I I I

Linearized Anisotropic Yield Equations for Five
Orientations . . . . . . . . . . . . . . . . . .

Computer Output from the TENSIL Computer Program for

Three Coupons at a - 0° . . . . . . . . . . . .

Notation for Chapter 5 . . . . . . . . . . . . .

Partial Computer Output for the Stress and Strain

Analysis of a Rim Element at a = 0° . . . . . .
Partial Computer Output for ANI C7 . . . . . . .

The Effect of Increment Size on the Computed Rim
Thickness for a = 0° . . . . . . . . . . . . . .

Computed Strain Ratios for the Rim Element vs.
Rim POSition I I I I I I I I I I I I I I I I I I

Typical Computer Programs for Radial Data . . .
Computer Print-Out for Radial Strains . . . . .

Computer Print—Out for Tangential Strain-Draw
Number 1 I I I I I I I I I I I I I I I I I I I I

Page

26

30
37
45

52

53

54

68

77

97

102

103

108
129

130

134

Figure

101-1

101-2

2.3—1
2.3—2
3.4-1
3.5-1
3.5-2
3.6—1
4.4-1
5.2-1
5.3-1
5.3-2

505-1

5.5-2

5.6—1

5.7—1

5.7—2

507-3

.5.7—4

LIST OF FIGURES

Cross Section of Draw Die Showing Partially-Drawn Cup

Sketch of a Partially-Drawn Cup with Ears in the 0°
and 90° POSit ions I I I I I I I I I I I I I I I I I I

Half-Size Layout of the Single-Shear Test Coupon . . .
Definition of Yield Stress . . . . . . . . . . . . . .
Best—Fit Curve for Tensile Strength vs Orientation . .
Definitions for Transformation of Axes . . . . . . . .
AnisotrOpic Yield Function for a = 0° . . . . . . . . .

Transformation of Axes in or, 0 Stress Space . . . . .

9
Effective Stress-Strain Plot for Three Orientations . .
Yield Conditions for the Isotropic Case . . . . . . . .
Notation Used in Force Equilibrium for a Flange Element
Terminology for a Partially-Drawn Cup . . . . . . . . .

Flow Chart for the Stress and Strain Analysis of a Rim
Element I I I I I I I I I I I I I I I I I I I I I I I

Computer Program for the Stress and Strain Analysis of
a Rim Element at a 3 0° I I I I I I I I I I I I I I I

Flow Chart for the Stress and Strain Analysis of an
Interior Element . . . . . . . . . . . . . . . . . .

Computed Strain History of a Rim Element at a = 0°
During the Cupping Operation . . . . . . . . . . . .

Comparison of Computed Rim Thickness at a = 0° and

a = 45° I I I I I I I I I I I I I I I I I I I I I I I

Comparison of Computed Rim Radial Strain at a = 0°
and a = 45° I I I I I I I I I I I I I I I I I I I I I

Computed Thickness Strain for Flange Elements at

a = 0° I I I I I I I I I I I I I I I I I I I I I I

vi

Page

24
25
38
46
47
49
73
79
83

86

95

96

100

104

105

106

109

Figure Page

5.7-5 Computed Thickness Strain for Flange Elements at
a = 45° ' o o o e e o o o o o o o o o o o o o o o e 110

5.7-6 Computed Radial Strain for Flange Elements at
a 3 0° I I I I I I I I I I I I I I I I I I I I I I 111

5.7—7 Computed Radial Strain for Flange Elements at

a = 45° . . . . . . . . . . . . . . . . . . . . . 112
5.7-8 Computed Circumferential Strain for Flange Elements . 113
6.5—1 Flow Chart for Chord Program . . . . . . . . . . . .. 125
6.5-2 Chord Program . . . . . . . . . . . . . . . . . . . . 126
6.5-3 Flow Chart for Radial Program . . . . . . . . . . . . 127
6.5-4 Radial Program . . . . . . . . . . . . . . . . . . . 128

6.6-1 Experimental Radial Strain for Flange Elements at
a . 0° I I I I I I I I I I I I I I I I ’ I I I I I 135

6.6-2 Experimental Circumferential Strain for Flange
Elements at a = 0° I I I I I I I I I I I I I I I I 136

6.6-3 Experimental Radial Strain for Flange Elements at

a a 45° I I I I I I I I I I I I I I I I I I I I I 137
6.6-4 Experimental Circumferential Strain for Flange

Elements at a 8 45° . . . . . . . . . . . . . . . . 138
6.6—5 Depth of Draw vs. Rim Position . . . . . . . . . . . 139

6.6—6 Experimental Strain Ratios vs. Draw Number for the
Element r0 = 2 I 35 I I I I I I I I I I I I I I I I 140

7.2-1 Comparison of Theoretical and Experimentally-
Determined Circumferential Rim Strains . . . . . . 145

7.2-2 Comparison of Theoretical to Experimentally-
Determined Radial Rim Strains . . . . . . . . . . . 146

vii

I. INTRODUCTION

1.1 Preliminary Remarks

 

The drawing process, by which a flat circular blank is trans-
formed into a cup, has been studied by many investigators since the
beginning of the twentieth century. Several of these studies will be
reviewed in Sections 1.3 to 1.6. Some of these investigations were
metallurgically oriented, while others were based on solid mechanics
and continuum theory; some were experimental while others were ana-
lytical, and many were combined experimental and analytical studies.
The cup-drawing process is illustrated in Figure 1.1-1.

Earing is the name given to the development of waviness or
undulations at the free edge of a cylindrical cup which has been drawn
from a flat circular blank; this is illustrated in Figure 1.1-2. Be-
cause of the greater trim allowance required, a larger blank is needed
to produce a certain size cup from sheet stock which develops ears than
from sheet stock which does not ear during the draw operation. The
expense of this trimming operation has encouraged research on the ear-
ing phenomenon, resulting in hundreds of publications during the past
fifty years. One indication of the importance of this problem is that
an earing test has recently been proposed to the industry [1, 49]. A
good current review of the earing phenomenon and the associated liter—
ature was published by Wright [53] in 1965.

An introduction to the present investigation is given in Sec-

tion 1.7.

moo camualuaamwuumm mcfizonm ofio 3mua mo coauomm mmouo HIH.H wuswwm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. 1 J 1
women mmnqomxz<qm maH>omm oe .
$9235 a? 8. amaumzzoo sz II/I
I» _
mmamqom
_ w .I a
r}, 525 \IT—
. a
II .u n mDHg mozam
I W
m
«masomxz<gm 1
as u maHa<m queommann
osz mHa

 

 

 

 

 

 

LLTI So 6639 $338

an n mDHS OZHM “.39 III

 

 

Z

THICKNESS DIRECTION—D TRANSVERSE DIRECTION

 

 

 

- /x
/ -\4//’
4 ‘v ROLLING DIRECTION

 

 

I /4-TRANSVERSE DIRECTION

‘ZTDIRECTION 0F ROLLING

  

 

 

Figure 1.1-2 Sketch of a Partially-Drawn Cup
with Ears in the 0° and 90° Positions

1.2 Some Early Studies Pertaining to Anisotropy

Many of the early metallurgical studies, pertaining to aniso-
tropy, involved research with single crystals, but some investigators
reported results of polycrystalline research. Two early investigations
involving polycrystalline research are reported here.

In 1926, Kbster [4] presented the results of experimental
studies of heavily cold-rolled copper sheet which was then annealed at
selected temperatures. He found observable differences in metallurgical
characteristics and mechanical properties as a function of the orienta-
tion with respect to the direction of rolling. This was one of the early
studies of mechanical anisotrOpy of sheet metal resulting from its proc-
essing history.

Sachs, in collaboration with Goler, published the results of a
study [5] of the rolling and recrystallization texture of regular, face-
centered metals with emphasis on aluminum and copper. This presentation
had five parts; parts 1 and 2 were published in 1927, while the remain-
ing three parts appeared in 1929. In part 5, Gfiler and Sachs reported
on the formation of ears in cups drawn from sheet aluminum exhibiting

directional prOperties.
1.3 Studies of the Cup-Drawing Process up-to-the Time of H. W. Swift

One of the earliest scholarly attempts to study the cupping
Operation was reported by Sommer [10] in 1925. This article presented
some of the results of Sommer's doctoral dissertation at the Technische
Hochschule in Berlin and included both experimental and theoretical

developments. His work was an attempt to so analyze the process as to

be able to predict the greatest depth of draw consistent with a given
cup diameter. Sommer notes earlier technological studies by P. Ludwik

in 1903 concerning sheet bending, as well as Ludwik's book Elemente der

 

Technologischen Mechanik published in 1909. Sommer resolves the draw
force into components, including the force to overcome the induced tan—
gential stresses in the flange, the force to bend the sheet metal at the
die profile radius, and the force to overcome friction.

In 1926, Eksergian [11] published an article which included an
experimental and a theoretical analysis of the condition at the die—
profile radius during the cup-drawing process. He indicated reasonable
correlation between his theoretical predictions and the experimental
results. Eksergian's experimental apparatus permitted an investigation
of bending under tension with or without including friction. The ex—
perimental set-up appeared to be eminently suited to the task, but the
theoretical analysis was only a first approximation to this problem.

In 1928, Geckeler [12] considered the problem of plastic buck-
ling of the walls of hollow cylinders and the associated wrinkling
problem of the flange metal in a cup during the draw operation. This
analysis is still considered the authority for the wrinkling problem
during the cup-drawing process.

In 1932, Linicus and Sachs [13] presented their analysis of the
influence of the blankholder on the deep-drawing operation. Sachs was
the Director of Research of the Metals Laboratory at the Metallgesell—
schaft A.-G.; Linicus and Herrmann were two scientists working with him
to study the cupping Operation. Their objective was to determine the

significance of the many variables on the force and energy requirements

and on the maximum-permissible depth of draw, consistent with a quality
cup. A cup was considered defective if (1) the bottom was torn out, or
(2) undesirable wrinkling had occurred. In this 1932 publication, they
considered three types of blankholders: spring—actuated or air-actuated
blankholders and rigid blankholders (at a fixed distance from the die
face.) An approximate theoretical analysis of metal thickening during
the draw operation was presented for an element at the flange rim, assum-
ing uniaxial compression in the tangential direction. This analysis was
the basis for a discussion of the required clearance between the die face
and a rigid blankholder and also between the punch diameter and the die
diameter for a "pure" draw condition. If the clearance was less than the
current metal thickness at any point, then "ironing" (metal thinning)
would be superimposed on the "pure" drawing Operation.

In 1934, Sachs and Herrmann [14] presented further developments
in this continuing study. They first clearly proved from experimental
studies that the stress state in the side-walls of the partially—drawn
cup was not simply uniaxial tension. This condition was compared with
an element in a hollow tube stressed under combined tension and internal
pressure as discussed by Lode [7]. A second finding was that the force
to tear the bottom out of a cup was maximum for a ratio of punch profile
radius to punch diameter of 0.33, all other parameters being of negli-
gible consequence. Sachs carefully distinguished between the "force to
draw" and the "force to tear the bottom from the cup." If the bottom
did not tear out, then the punch force was the force to draw. The force
to draw was practically independent of the punch-profile radius, while
the force to tear the bottom out was highly dependent on the punch-

profile radius; within reasonable limits, larger radii required lower

drawing forces. However, for large die-profile radii, the sheet metal
wrinkled as it moved over the die radius and was no longer influenced
(squeezed) by the blankholder. The investigations of Herrmann and
Sachs showed that to prevent wrinkling during these final stages of the
draw, the die-profile radius must be less than twenty times the metal
thickness. The authors attempted to generalize their results by using
dimensionless ratios and similarity laws.

In March, 1935, Sachs [15] reported further on these cupping
investigations. In this paper, the author set up the differential
equation for an element in the flange and integrated it for certain
elementary cases assuming that Tresca's yield condition was applicable.
Sachs determined the strain distribution numerically (using simplifying
assumptions of isotrOpy, zero work-hardening, zero friction, constant
wall thickness, and plane stress) for elements in the flange in 1930;

this is reported on page 265 of Hoffman and Sachs [60].

1.4 The Contribution of H. W. Swift and Others to Isotropic CuprDrawing

Research

H. W. Swift and his colleagues from Sheffield University, Eng-
land began publishing the results of their investigations of sheet metal
working in 1940. Swift's 1940 article [16] reviewed previous papers
pertaining to sheet metal drawability and proposed that the cylindrical
cup-drawing test be more completely investigated for this use. The
author then gave the results of a comprehensive eXperimental investi-
gation of cylindrical cup-drawing for a two-inch diameter cup using a
sub—press fitted either with a rigid blankholder or an hydraulically-

actuated blankholder (as desired). This sub-press was set into a

commercial press or a testing machine for its energy source. Swift
reported general agreement with the detailed systematic eXperimental
investigations of cylindrical cup drawing both by S. Fukui [17] of
Japan in 1938 and by Sachs [14] of Germany in 1934. In the brief
discussion of his earing studies, Swift concluded that the phenomenon
of ear develOpment was obscure.

Chung and Swift [21] reported in 1951 on an extensive study of
cup-drawing both experimentally and theoretically. Their experimental
work utilized a specially designed single-action, straight-sided, me-
chanical press [20] rated at 50 tons with an air cushion in the bed to
actuate the blankholder. The press stroke could be varied from 3 to
10 inches, and the press speed could be varied from 5 to 60 strokes per
minute. The cups drawn were 4 inches in diameter (twice the diameter
of the cups produced on their sub-press experiments as reported in
1940). The experimental tests provided a systematic evaluation of the
draw process by examining the effect of important process variables on
the process. The process variables considered included the blankholder
type, blankholder force, blank diameter, blank thickness, punch and die
profile radii, and punch—die clearance. The objective was to determine
the effect of these process variables on quantities such as the punch
load, the process work, and the principal strains. General agreement
with earlier work was noted. Experimental results from partial draws
were not reported.

The analytical part of this 1951 publication by Chung and Swift
presented a real improvement over any previous treatment. The analy-

tical treatment investigated the stress and strain history of an element,

first while undergoing radial drawing in the flange, and then when
subjected to bending and frictional forces at the draw profile radius.
No attempt was made to predict the strains which occurred in the

stretch forming region around the punch. The authors used plasticity
theory as described in [18] including the root mean square measure for
representative stress and strain. The analysis for plastic bending
under tension had been published earlier [19] by Swift, and had also
been considered by Lubahn and Sachs [22] in 1950. Swift utilizes Hill's
finding [27] that the representative strain in any flange element can
be approximated by the circumferential strain at that instant.

It should be noted that other similar studies of the stress and
strain history of elements in the flange of the partially-drawn cup
have been made for isotropic materials. In 1949, Jackson [28] pub—
lished a simplified theoretical analysis of radial drawing. In chapter
eleven of his text [27], Hill presents the necessary mathematics for
this radial drawing analysis. In 1958, Fukui, Yuri and Yoshida [62]
published the results of their studies of radial drawing using the total
strain theory. In 1964, W00 [63] extended the work of Swift and Chung
to permit the blankholding force to be distributed over an area near
the rim of the flange; he also suggested a method of analysis for
stretch-forming over the punch.

In 1960 Alexander [64] published an excellent appraisal of
the theory of deep drawing up to that time, and suggested areas for
further study.

In 1954, Swift [23] reported some experimental results relative

to stretch-forming of sheet metal over a metal punch in an ordinary die

10

or by means of an hydraulic bulge fixture. In some tests the punch
was flat-faced; in other tests a spherical punch was used. The effect
of lubricant was investigated in the extreme cases of an excellent
lubricant vs. no lubricant. During hydraulic bulging, where friction
can be assumed zero, the fracture point was very close to the pole
point. However, the point of fracture moved out part way on the spher—
ical punch with good lubrication, and moved out much further if no
lubrication was used. These results demonstrate the importance of
lubrication on the instability condition for the essentially biaxial-
tension stress state of the sheet metal over the punch head where
stretch-forming occurs. The author noted his earlier publication [24]
on plastic instability under plane stress, which in turn acknowledged

work by Brown and Sachs [25] in 1948 and by Hill [26] in 1950.
1.5 More Recent Research Pertaining to Measures of Anisotropy

There have been numerous publications [29-57] relating the
processing history of rolled sheet metal to its resultant anisotrOpy.
AnisotrOpy may manifest itself in respect to physical properties and
metallurgical properties.

Klingler and Sachs [36] listed three distinct types of ani-
sotropy: anelastic, mechanical, and crystallographic. Anelastic
anisotropy [44] results from residual stresses caused by previous cold-
work and is related to the Bauschinger effect [6, 59]. Since the
Bauschinger effect is usually minimized commercially by stress-relief
annealing, anelastic anisotropy is not considered responsible for ear-
ing. Mechanical anisotropy, as described by Phillips and Dunkle [30],

results from the directional extension of segregated constituents such

11

as dendrites, slag, gas holes, carbides and sulphides. Since these
inclusions are normally minimized in current mill practices, earing

in commercially-produced sheet metal is associated only with crystal-
lographic anisotropy [45]. This crystallographic orientation, called
texture, of the sheet metal results in anisotropy as measured by such
physical constants as yield strengths, tensile strengths, uniform elon-
gations or strain ratios of the test coupons oriented in different dir-
ections in the metal [31, 34].

The earing phenomenon is influenced by many things. Blade [48]
has reported on research which investigated the effect of composition
and constitution on the earing of aluminum. The effect of the method
of producing the ingot metal on the earing of aluminum sheet has also
been studied [39, 41—43]. The introduction of the less costly, con-
tinuous casting process in place of casting in ingot molds (about 1950)
aggravated the earing problem for aluminum; certain variations in the
process have been successful in mitigating this problem of earing.

As part of a study to develop an earing test for sheet metal,
Blade and Pearson [49] reported in 1962 on the effect of certain die
parameters on earing of aluminum. In 1963, Siebel and Mack [51] reported
on additional experimental research studying the effect of die conditions
on the amount of earing. In 1965, Wright [52] reported the results of
his systematic experimental investigation. Wright found that such fac—
tors as depth of draw, punch-profile radius, blankholder force, and the
amount of ironing significantly affected the degree of earing; however,
Wright found that the die-profile radius had no effect, and the type of

lubrication had only a slight effect.

12

The number of ears on a drawn cup and the orientation of the
ears with respect to the direction of rolling have also been investi-
gated. In 1958, Thorley and Tucker [46] reported that ears on cups
drawn from aluminum alloy sheet stock may vary in position, number,
and size. The two most common arrangements are: (i) Two ears in the
rolling direction plus two more at right angles to this direction;
(ii) Four ears at 45° to the rolling direction. Other less common
arrangements are: (iii) Eight ears consisting of a combination of the
two previously-mentioned types: (iv) Eight ears at 22%0 and 67%0 to
the rolling direction; (v) Eight ears at 30° and 60° to the rolling
direction. In 1950, Bourne and Hill [56] reported that copper gave
four ears at 45°; brass gave four ears at 50°; and brass gave six ears
at 0° and 60°.

Hill's anisotrOpic theory [27, 55] was proposed for metals,
such as aluminum-killed steel, producing four ears. Aust and Morral
[40] reported in 1953 that strain—ratio measurements for 28 aluminum
were in good agreement with Hill's theory for plastic anisotropy.

Anisotropy implies a variation in material prOperties at a
point as a function of direction. In the study of the cup-drawing
operation, any anisotrOpy of the sheet material is often associated
with the formation of ears on the drawn cup [29, 32, 33, 35]; however,
a more careful analysis indicated that it is anisotropy in the plane
of the sheet metal (particularly as regards the yield strength or the
strain ratio), which is related to the formation of ears [50]. This

is called planar anisotropy.

13

In a 1948 publication, Jackson, Smith and Lankford [37] intro-
duced the idea of normal anisotropy. During the plastic flow of a
tensile test coupon made from sheet metal, the ratios, "R," of the
logarithmic width strain to the thickness strain were computed. If
the material were isotropic, this strain ratio "R" would be unity for
any orientation of the test coupon. Any variation of this strain ratio
from unity is a measure of the normal anisotropy, while any variation
of this strain ratio with orientation of the test coupon is a measure
of planar anisotropy [50,70].

In an important follow-up paper to [37], Lankford, Snyder and
Bauscher [38] in 1950 presented the results of further work at the re—
search laboratories of Carnegie-Illinois Steel Corporation. This work
clearly indicated that good drawability of sheet steel was associated
with a high value of normal anisotropy as measured by the strain ratio
"R." While planar anisotrOpy is associated with the earing phenomenon,
normal anisotropy is associated with the degree of drawability in a
cupping operation. The material used in their study included forty-
six lots of aluminum-killed, low carbon, deep drawing steel sheets.
For this material they found that the effective stress, effective
strain curves for plastic deformation could accurately be represented
by the formula 5 = KEn, where "n" can be shown to be equal to the
uniform strain corresponding to the tensile strength or the maximum
load in the tensile test. Whereas normal anisotropy correlates with
drawability, "n" correlates with stretchability. Their work indicated
that normal anisotropy (ratio of width to thickness strain) remains

essentially constant during work-hardening (elongation).

14

In 1961, Tucker [47] drew cups from aluminum single crystals
and used a simple theoretical approach to predict all of the important
features of the earing during cupping. Extension of this theory to
polycrystalline metal was discussed briefly.

In 1966, Wilson [58] reviewed the current understanding of the
relationships of preferred orientation and plastic anisotrOpy (as meas-
ured by the strain ratio R), to practical performance of the sheet

metal in the draw die.

1.6 Recent Research Pertaining to the Cup-Drawing Process for Ani-

 

sotropic Metals

 

In 1962 Warwick and Alexander [65] presented an approximate
method of determining the limiting drawing ratio during cup—drawing by
a plane strain instability condition and compared their results with
experimental results (as well as with Whitely's "Strain Ratio" [70]
and Wallace's "v-factor"). The results were not too encouraging.

A theoretical cup-drawing analysis for anisotropic sheet metal
intended to study the effect of the strain ratio R was published in
1964 by Moore and Wallace [66]. These authors used the basic aniso—
trOpic theory presented by Bourne and Hill [56] in their analysis and
assumed R to be constant and not a function of orientation in the plane
of the sheet. They also assumed a linear relation between stress and
strain to simplify their analysis which predicted the punch load at
instability.

In 1966, Chiang and Kobayashi [57] reported on their theoretical

analysis of the cup—drawing Operation using the incremental approach

15

with the aid of an IBM 7094 computer. Again, the theoretical base for
the analysis is Hill's work [27, 55, 56]. Work hardening was included
in their analysis by means of Ludwik's stress strain relation 6 = KEn
in place of the more approximate linear relationship used by Moore and
Wallace. AnisotrOpy was introduced in the form of the strain ratio R;
like Moore and Wallace, the variation of R, with orientation in the
plane, was neglected. This study predicted critical strains and critical
punch loads at instability, along with the limiting drawing ratio.

In 1966 Budiansky and Wang [67] also presented a theoretical
study of the cup—drawing Operation for anisotropic material. This
study is closely related to the work of Chiang and Kobayashi [57].
Frictional forces on the flange were neglected as was also done by
Chiang and Kobayashi. Again, the objective of the study was to predict
the limiting drawing ratio.

In 1967, Mir [69, 71] published his doctoral dissertation on
the cup-drawing process, with emphasis on the limiting drawing ratio.
Mir attempted to generalize the work of Moore and Wallace and that of
Chiang and Kobayashi by using polar anisotrOpy parameters; that is the
strain ratio R was considered to be a function of orientation. Experi-
mental work was conducted using aluminum, c0pper and 70/30 brass. The
author stated that the experimental results were in good agreement
with the predictions of punch load at instability and the strain distri-
bution in the cup. Three blankholder methods were used: spring-

actuated, rubber-pad—actuated, and constant-clearance blankholders. Mir

followed the example of Chiang and Kobayashi by neglecting frictional

l6

blankholder effects in his theoretical analysis. Mir did not include
the local thinning strains caused by bending and unbending at the die—
profile radius.

In 1968, Woo [68] extended the treatment of cup—drawing which
he had published in 1964 [63] by including normal anisotrOpy in the
form of the strain ratio R in his analysis. Planar anisotropy is not
considered; in fact, for the material used in his experimental work,

planar anisotrOpy was slight.
1.7 The Present Investigation

Sheet steel commonly develops four ears during the cupping
operation. These ears are either at 0° and 90° to the direction of
rolling, or else at 45° to the rolling direction. The aluminum-killed,
low-carbon steel used in this study exhibited ears at 0° and 90° to
the roll direction.

The general objective of this investigation was to study the
earing phenomenon theoretically using plasticity theory, and then to
compare the results of this theoretical analysis with the experimental
results of cupping tests. The theoretical analysis is an extension of
the work of Chung and Swift [21], and of Hill et a1. [27, 55, 56].

Commercially-produced A181 1006 steel, usually called aluminum-
killed steel, was used for the eXperimental work. The 0.035 inch thick
stock had been carefully sheared from the coil, such that the direction
of rolling remained self-evident. This permitted tensile coupons to
be cut at various orientations to the direction of rolling, thus per-

mitting an experimental determination of the yield stress as a function

17

of the orientation. Experimental methods were also required to deter-
mine the yield shear stress.

These eXperimentally evaluated test results permitted the ani-
sotropic yield function, for the plane stress case, to be determined.
In addition tensile test methods were used to determine the three para-
meter strain-hardening relation 5 = 50 + BEm.

The method of Chung and Swift [21] was extended to include
planar anisotrOpy, which permitted a theoretical study of the ear
formation. The anisotrOpic yield function for plane stress proposed
by Hill [27], in chapter 12, was used. The General Electric 265 computer
was utilized during this theoretical study to determine the stress and
strain history of elements in the flange during radial drawing. The
incremental approach was used. Since in the drawing Operation the
plastic strains are much larger than the elastic strains, the elastic
strains are neglected throughout the analysis. A rigid work—hardening
material is therefore assumed.

The experimental cup—drawing tests were performed using a
double-action draw die of the type shown in Figure 1.1-1, which was
mounted on the bed of a 150—ton, straight-sided, single—action Minster
press, equipped with an air cylinder to provide the necessary blank-
holding force. The die blankholder and the die ring were both machined
from tool steel, then hardened and ground. The punch was machined from
SAE 1020 steel without any additional heat treatment. The same stock,
from which the tensile coupons and the shear coupons were cut, was
used for the blanks. The blanks were 0.035 inch thick and 4.800 inch

diameter.

II. EXPERIMENTAL DETERMINATION OF THE TENSILE AND SHEAR YIELD STRENGTHS

2.1 Preliminary Remarks

In order to include anisotropy in the stress and strain field
calculations for elements in the flange of a partially-drawn cup, it
was decided to insert an anisotropic yield condition into the equilib-
rium equation. Hill [27] has suggested, in chapter 12 of his text,

that a suitable generalization of Mises yield ellipse would be of the

form
= ._ 2 _ 2 _ 2 2 2
2f(oij) F(oy oz) + G(oz ox) + H(ox oy) + 2LTyz + 2Mr2x +
2N12 = 1 (2J1-1)
xy

Since the plane stress assumption was used in this study, Equation

(2.1-l) reduced for the plane stress case to

= 2 _ 2 .2 = _
2f(oij) (G + H)ox 2Hoxoy + (H + F)oy + 2ery 1 '(2.1 2)

It was decided that Equation (2.1-2), or some approximation to it, would
7 be inserted into the equilibrium equation for an element in the flange
of the partially-drawn cup.

Equations (2.1-1) and (2.1-2) both assume orthotropic symmetry,
with three mutually-orthOgonal planes of symmetry in the metal. The
intersection of these planes of symmetry define the principal aniso-
tropic axes, where the x-direction is the direction of rolling, the
y-direction is the transverse direction in the plane of the sheet metal,‘
and the z-direction is the thickness direction (this is illustrated in
Figure 1.1-2). Corresponding to these directions, X0 represents the value

of Ox at initial yield due to uniaxial tension in the x-direction: Y0

18

19

represents the corresponding initial yield stress in the y-direction,
and 20 represents the initial yield stress in the z-direction. Simi-
larly, when a specimen yields in pure shear, the value of the initial
yield stress corresponding to Txy is called To.

There are four parameters (F, G, H, and N) in the plane—stress,
anisotrOpic yield condition, Equation (2.1-2), to be evaluated from
experimental data. The method used in this investigation can be called
the direct method, whereby the parameters F, G, and H are determined
eXperimentally by finding the tensile yield stress as a function of
orientation with respect to the direction of rolling. Similarly the
parameter N is determined by experimentally finding the shear yield
stress, To, in the direction of rolling, corresponding to Txy'

The objectives of this chapter then are two-fold. The first
objective is to present the method and the results of the experimental
study to determine the shear yield stress To. This is discussed in
section 2.3. The second objective is to present the method and the
results of the study to determine the tensile yield stress as a function
of orientation in the plane of the sheet for the A131 1006, aluminum-
killed, low-carbon sheet steel used in this investigation. This part

of the study is discussed in section 2.4.

2.2 Theory

If the anisotropic yield function for the plane stress case,
Equation (2.1-2), is Specialized for the case of a tensile test coupon

oriented in the rolling or x—direction, it reduces to

(G + H)o}2{ = 1 (2.2-1)

20

If the material is in the "as received" initial condition, then the
stress Ox will reach the initial yield stress X0 when yielding of the

tensile test coupon commences. Then Equation (2.2—l) is expressed as
__.= G + H (2.2-2)

Similarly it can be shown that a tensile coupon, oriented in the trans—
verse y-direction and loaded to the initial yield stress Y0, can be

mathematically modeled by specializing Equation (2.1-2) as

.2; g H + F (2.2-3)
Y
0

Additional equations can be found by specializing the aniso-
trOpic yield function (2.1-2) for tensile coupons oriented at an angle
H H

a measured clockwise from the direction of rolling (Figure 1.1—2

defines "a"). The transformation equations are

0 = 0 cos a o = 0 sin2 a r = 0 sin a cos a (2.2-4)
x Y xy

Here, 0 represents the yield stress in the a-direction. These trans—

formation equations can be inserted into the anisotropic yield function

of Equation (2.1-2) to produce
(G + H) cos” a — 2H sin2 a cos2 a + (H + F) sin1+ a + 2N sin2 a cos2 a =

--, (2.2—5)

or

F sin2 a + G cos2 a + H + (2N - F - G - 4H) sin2 a cosza =-l3 (2.2—6)
0

21

It appeared that the four parameters F, G, H, and N could be
determined by experimentally determining the yield stress for tensile
coupons oriented in four different directions. This would give four
equations to solve for the four unknowns. However, this turned out to
be impossible because the four equations were not independent. The actual
method used was to determine experimentally the initial value of TX
which resulted in a yield by shear, designated as To. Thus, the value of
N was independently determined by specializing Equation (2.1-2) for the

pure shear case to get

N = —1—— (2.2-7)

2]?2
0

After the value for N was determined, then in theory only three
equations were needed to solve for the remaining three parameters F, G,
and H. However, the direct method used to obtain the anisotropic param-
eters by directly measuring yield strengths is not so sensitive an
approach as the strain ratio method described on page 321 in chapter 12
of Hill [27]. Therefore, the actual procedure used was to experiment-
ally find the yield strength as a function of orientation, for nine
values of 6 equal to 0°, 30°, 40°, 42.5°, 45°, 47.5°, 50°, 60°, and 90°.
Equation (2.1-2) was specialized for each of the nine directions of "a"
listed above, and the nine corresponding yield strengths O, to produce

nine equations with three unknowns F, G, and H. Then a least—squares

method was used to find the "best" values of F, G, and H.

22

2.3 Experimental Determination of the Shear Yield Stress

 

A suitable method to experimentally determine the yield shear
stress for sheet material was found by studying the work of Yen [74],
published in 1960, and particularly the discussion by Bradley [76] of
the article by Yen. Bradley's comments include a study of the geometry
parameters of the coupon and their effect on the elastic shear stress
distribution in the stressed zone (between B and C in Figure 2.3-1). It
was decided to design the shear test coupon as shown in Figure 2.3—1,
because, for this design, Bradley has shown that the stress state in the
shear area is essentially pure shear, r = EI Other papers which con-
tributed to the develOpment of the single-shear specimen for testing
sheet material were presented by Penn and Clapper [72] in 1956, and by
Breindel, Seale, and Carlson [73] in 1958. Both of these earlier papers
were primarily concerned with ultimate shear strength, rather than the
yield shear stress.

Twelve single-shear coupons were cut from the A181 1006,
aluminum—killed, low-carbon sheet steel under investigation. These
coupons were oriented so that the length direction coincided with the
direction of rolling. The stressed area was measured for thickness to
the nearest ten—thousandth of an inch with a micrometer. The shear
length was measured to the nearest thousandth of an inch, with a Jones
and Lamson Optical Comparator and Measuring Machine, to give three sig-
nificant figures of accuracy when computing the area in shear. The cou—
pons were pulled on the Instron Tensile Testing Instrument, Type TT-C

using a crosshead speed of 0.05 inches per minute and a chart speed of

23

50. inches per minute. The ten-inch wide Instron chart paper was cali-
brated for a range from 0 to 200 pounds full scale reading. The cal-
ibration was checked before and after the series, as well as between
test runs, and held within 0.5 percent.

Figure 2.3-2 illustrates the general shape of the force vs
crosshead displacement curve from the Instron chart paper. It also
shows graphically the definition of yield stress used. The load-
extension curve has a reasonably-straight initial portion correspond—
ing to the elastic range; after the transition region, the load-extension
curve has a reasonably-straight zone corresponding to the plastic region.
The yield stress was arbitrarily defined to be at the junction point of
the transition zone with the plastic zone.

The shear test results are given in Table 2.3—1. The average
shear yield stress was computed to be 16,923 psi with a standard devi-

ation of 353 psi and a range of 1218 psi.

2.4 Experimental Determination of the Tensile Yield Stress as a Function

of Orientation

 

Inasmuch as the variation of the tensile yield stress with
orientation is very small for the AISI 1006, aluminum—killed, low-carbon
steel used in this investigation, an attempt was made to improve the
accuracy of the usual tensile test procedure. Nothing much could be done
to get better accuracy in the measurement of the stock thickness than
use of a micrometer caliper measuring to the nearest ten-thousandth of
an inch. However, the specimen width entered into the calculation of

the cross—sectional area and this could be better controlled by

24

 

 

 

 

 

 

 

 

 

 

I
/ I
I
I
0.153" DIA. (REAM)
—.26"
1" REMOVE THIS TRIANGULAR
115- , AREA T0 PERMIT
THICKNESS MEASUREMENT
‘- 2H
1H \

 

 

 

 

 

Figure 2.3-1 Half-Size Layout of the Single-Shear Test Coupon

25

mmmuum wamfi» mo aofiuflafimma Num.~ muswfim

HZmZMU<AmmHQ Q<mmlmmomo

mZON UHHm<Am

MZON ZOHHsz<mH

mmMMHm nquw

 

mZON UHHm<Am

  
      

 

¢ o<og

Table 2.3-1 Single-Shear Test Results

 

SHEAR AREA

SHEAR

 

 

 

 

 

 

 

 

 

 

 

 

 

385;: W 5:33.329!) YIEggsggms
INCHES)
1 0.008845 156. 17,637.
2 0.008845 149. 16,845.
3 0.008845 150. 16,958.
4 1, 0.00882 154. 17,460.
5 0.00882 150. 17,006.
6 0.00882 149. 16,893.
7 0.00882 152. 17,233.
8 0.00882 148. 16,780.
9 0.008795 147. 16,714.
10 0.00877 144. 16,419
11 0.008795 145. 16,486-
12 0.00877 146. 16,647

 

 

 

 

Average Shear Yield Stress = 16,923. psi.

Standard Deviation of Data = 353. psi.

Range of Data

= 1218 psi.

 

27

accurately machining the edges of the tensile test coupons to insure a
rectangular cross section. The Instron Tensile Testing Instrument,

type TT-C, which was used during these tests, had the capability of
suppressing the zero load and simultaneously increasing the scale factor
(pounds per inch) on the ten—inch wide chart paper.

With these possibilities in mind, the tensile test specimen was
standardized to be a rectangular shaped coupon approximately seven
inches long by 0.530 inches wide by 0.035 inch thick. The edges of these
coupons were machined in groups to give very uniform edges which were
square with the plane of the sheet metal and had excellent parallelism
along the lengthwof the coupon.

As discussed in section 2.2, it was decided to find values of
the tensile yield stress for coupons cut at various angles "a" to the
direction of rolling. The actual directions chosen were 0°, 30°, 40°,
42.5°, 45°, 47.5°, 50°, 60°, 90°. In order to get any statistical
evaluation of the results, it was decided to pull twelve coupons in
each of the nine directions.

Cutting each group of coupons, so that its orientation with
respect to the rolling direction was accurately maintained, was a ted-
ious job. A drafting machine was employed to lay out the coupons using
the edge of the coil stock as the reference line. A square shear was
used next to cut the coupons as accurately as possible with respect to
orientation. The final preparation was to take these rough blanks in
groups and to machine the edges. It was estimated that the orientation

accuracy was within one degree.

28

To get the required sensitivity to magnify the small differences
in yield stress between coupons cut at different orientations required
use of one of the "Special Operating Techniques" as discussed on page 41
in Instron's manual [75]. The technique used permitted calibration of
the recorder so that the pen travel from one side of the ten-inch chart
paper to the other corresponded to a change in load on the specimen from
400 pounds to 600 pounds, which meant that the zero was suppressed 400
pounds. Hence the scale factor was 20 pounds per inch, or 2 pounds per
tenth—inch. The actual yield load ranged from about 485 pounds to
about 545 pounds.

The system used was to calibrate the machine after an initial
warm-up period of at least an hour. Then a group of nine coupons were
run, one each at 0°, 30°, 40°, 42.5°, 45°, 47.5°, 50°, 60°, and 90°
to the roll direction in that order. After the first run, the calibra-
tion was rechecked before continuing with a second group in reverse order.
It was felt that this approach would distribute any error evenly over
the range of orientations. Actually the recalibrations indicated small
errors of less than 3 pounds. Since the yield load averaged over 500
pounds, the worst calibration error was 0.6%. After several of the runs,
the recalibration indicated no error.

The tensile yield stress was defined in the same way as the
shear yield stress, at the junction of the transition zone with the
plastic zone, as illustrated in Figure 2.3-2.

During the tensile testing program, the cross-head speed was

maintained at 0.2 inches per minute (four times as fast as that used

29

to determine the shear yield stress). The chart speed was set at 50.
inches per minute, the same as used in the single-shear tests.

The results of the tensile testing program are shown in Table
2.4-1. The bottom four rows of this table summarize the data statis-
tically by means of the arithmetic average, the standard deviation of
each column of data, the variance of each column of data (that is, the
square of the standard deviation), and the estimated true variance of
the p0pu1ation from which the sample of twelve in a particular column
was chosen. The statistics were obtained by using a standard computer
library program written in BASIC language and run on the General Electric
265 Time-Sharing Computer.

These data are used in chapter 3 to get the "best fit" for the

parameters F, G, and H of the anisotrOpic yield function.

30

 

owH.m~

 

 

ZOHHUmmHQ AAOM OH mmmmmmn

 

.Nwo.- .mmo.- Ham.o~ om~.mo .aoq.~ .amH.mo swa.mo .Hoqsh 66cmHu6> 6:09 .umm
.HqN.o~ .HaH.cm ~mo.mH ,msq.mo .Hmo.m~ .oms.o «No.5m .NmHso .ommwo 66¢6HH6> 6Ha56m
~.~sH H.~¢H “.mmH a.Hm~ ¢.HmH Ho.~m s~o.os~ w~.mm~ o~.Ho~ a6H06H>6a vumvamum
moH.H~ oHn.m~ mam.mm NHm.w~ mm~.w~ moq.m~ maH.w~ mmm.- mmm.H~ 6m6n6><
oom.a~ mmm.m~ owm.m~ OHm.m~ Nos.mm qu.m~ om~.w~ NmH.HN mas.nm NH
omq.- Hom.w~ sm~.m~ qu.m~ mse.wm som.m~ sma.- ~NN.HN me.H~ HH
sw~.- moq.m~ Nqs.w~ on.w~ mm~.w~ owm.wu auH.HN oms.sm mmm.o~ OH
smu.- mmm.m~ maq.w~ mmn.mm osm.m~ -H.¢~ mmq.- HHH.H~ a
on.HN smq.m~ mmq.m~ cwm.w~ mso.wm sms.w~ omm.m~ Hwa.o~ moH.H~ w
oHH.H~ mmm.m~ OHo.mN osm.m~ mms.w~ qu.m~ mmm.m~ MHm.HN ~H~.H~ n
oms.- Hum.w~ Hum.m~ mH~.wN mom.w~ Hus.w~ HHH.w~ oo~.- mmH.HN o
aNH.H~ Has.wm mmc.m~ on.m~ No“.m~ st.m~ Hoo.m~ «HH.- omo.w~ m
moH.H~ msm.m~ oom.mm wmm.mu mOH.a~ AHm.m~ wqm.w~ nmm.H~ ~w~.- q
mow.- qu.m~ on.w~ om~.w~ msm.m~ omo.m~ o-.m~ mNH.- N¢¢.HN m
msm.- Hmm.m~ Hmm.m~ om~.w~ «Hm.m~ NHq.w~ moq.w~ som.HN mos.- N
aNH.H~ Hmo.m~ wsm.w~ m~m.m~ ssm.m~ Ham.w~ Hmm.w~ mmo.HN aoH.H~ H
.oa .oo .om .m.sq .mq .m.~s .os .om .o .oz mmHmmm

 

doauwafiahoumn mmmuum waofiw oHHmGoH mnu mo muasmmm humEESm HI¢.N manme

 

III. THE DERIVED ANISOTROPIC YIELD FUNCTION

3.1 Preliminary Remarks

As discussed in section 2.1, it was decided to use the direct
method to determine the tensile parameters F, G, and H, and the shear
parameter N. The direct method implies that experimentally-determined
initial yield stresses will be used to evaluate the four anisotropic
parameters. These four parameters are needed for the plane stress
specialization of the anisotropic yield function of Equation (2.1-2),

repeated here for convenience.

(G + H) 0‘2 - 2H0 o + (H + F)02 + 2er = 1 (3.1-1)
x x y 7 xy

It must be recognized that this equation is valid only when the x-,
y-, and z-directions are the principal anisotrOpic axes as defined
by Figure 1.1-2.

Chapter two discussed the eXperimental procedures used to
determine the initial shear yield stress To (corresponding to Txy),
and the initial tensile yield stresses at nine selected values of
"a" to the rolling direction. Chapter three further develops this
anisotropic study. Section 3.2 discusses how the shear parameter N
was calculated, and section 3.3 discusses the "least squares" pro-
cedure used to determine the "best" values of the tensile parameters
F, G, and H. A discussion of the indirect method (strain-ratio

method) of computing the anisotropic parameters is presented in

section 3.4 as a check on the direct method used in the investigation.

31

32

In section 3.5, the anisotropic yield equation is transformed for
use at selected angles "a" to the rolling direction for elements

in the flange of the partially-drawn cup. The straight line approx—
imation to the anisotropic yield function in the fourth quadrant is

found for a = 0° and a = 45° in section 3.6.
3.2 Computation of the Shear Parameter N

The direct method was used to compute the shear parameter
N. The average value of To, corresponding to Txy for the orientation
of axes shown in Figure 1.1—2, was found to be 16,923 psi, as listed
in Table 2.3-1. The anisotropic yield equation for the plane stress
case was specialized for pure shear in Equation (2.2-7).

With Equation (2.2-7) and the average value of To from
Table 2.3-1, the value of the shear parameter N was computed as

follows:

N =-—l— = 1 = 1.745 x 10'9 (3.2-1)

2T3 2(16,923)2
Inasmuch as the load for initial shear yield, the thickness,
and the shear length were each measured to three significant figures,
the accuracy of the shear parameter N is also limited to three sig-

nificant figures.
3.3 Computation of the Tensile Parameters F, G, and H

An examination of the results of the tensile yield strength
study, as reported in Table 2.4-1, clearly shows the problem associ-
ated with the direct method of computing the tensile parameters:

The tensile yield strength changes only a small amount with a change

33

in orientation. For this reason, Hill [27] page 321 in Chapter 12,
suggests that a more sensitive measure of anisotrOpy is given by
the strain-ratio method, which will be discussed in section 3.4.

It is theoretically possible to compute F, G, and H by
choosing any three orientations of tensile coupons and their corres-
ponding initial yield stresses, as given in Table 2.4-1. As a pre—
liminary method, this approach was tried. Three pairs of (a, 00)
were inserted into Equation (2.2-6), which is repeated here for

convenience.

'3; = F sinza + G cosza + H + (2N - F — G — 4H) sinza cosza
“ (3.3-1)
Each pair resulted in one equation with three unknown parameters.
The values of the three parameters were then computed using the
three equations. However, the computed values of F, G, and H de-
pended upon which three pairs, from Table 2.4-1, were used. Hence,
it was decided to modify this approach.
F, G, and H were determined in this investigation using the

least squares method as described in section 5.6 of Wylie [78] be-

ginning on page 175. Equation (3.3-1) was rewritten in the form:

2
'3: = F(sin a)1+ + G(cos a)“ + H(cos Za)2 + N $§lgL2E2—'. (3.3—2)
0

d
Then, one term was transposed to get:

2
F(sin on)1+ + G(cos a)“ + H(cos 2002 = A. _ N(s:n 20)

0'

. (3.3-3)

2
a

34

The least squares method consisted of setting up a difference
equation by subtracting the right member of Equation (3.3—2) from
the left member. This difference was called 5a.

1 N(sin 20:)2
— + .
02 2

a
(3.3-4)

 

6a = F(sin a)“ + G(cos 01)l+ + H(cos 201)2 —

Each pair (a, on), from Table 2.4-1, gave a difference equation
which had to be minimized to get the "best" values of F, G, and H.
This was done by first getting an error function "E" which was

defined as the sum of the squared differences.

E = Z (0“)2
0.

 

2
E = z [F(sin a)” + G(cos a)“ + H(cos 2(1)2 - £3-+ N(sin 2&1]
a 0a
(3.3-5)
The error function was minimized in the usual way by setting the

partial derivatives of the function, with respect to the three

variables F, G, and H, each equal to zero.
——-= o ——-= o -—— = 0 (3.3—6)

This resulted in three equations to solve for the three

unknowns F, G, and H as follows:

 

2
0 = 2 (sin a)1+ [E(sin a)“ + G(cos a)” + H(COS 20)2 ' l;'+ N(sin 2a) ]

 

2
a 0a
(3.3-7)
2
0 = 2 (cos a)” [F(sin a)” + G(cos a)1+ + H(cos 201)2 — gj'+ N(sin 20) ]
a

(3.3—8)

35

o. z 6.. 2620.... 6» + .6... w + m... 262 -

O
G a

2

These are three linear equations for the "best—fit" values

of three unknown parameters, of the form

81F + b1G + ClH = d1

l
O.
N

(3.3—10)

a3F + b3G + c3H -

I
Q:
m

The coefficient evaluations were performed on the General
Electric 265 Time-Sharing Computer System, using the FORTRAN lan-

guage, to give

1.661F + 0.37350 + 1.176H 2.138 x 10"9

0.3735F + 1.6610 + 1.176H 2.224 x 10-9 (3.3-11)

1.176F + 1.1760 + 2.127s = 2.978 x 10-9

These simultaneous linear equations were then solved for F,
G, and H using the standard library BASIC language computer program
"SIMEQN" on the General Electric 265 Time-Sharing Computer. The

"best" values were

F = 6.94 x 10-10
0 = 7.60 x 10"10 (3.3-12)
H = 5.96 x 10"10

As was the case with "N," the accuracy of the "best—fit" tensile
parameters is limited to three significant figures. As the dis-
cussion of Figure 3.4-1 in the following section indicates, the

accuracy was actually less than this.

;L_+ N(sin 20:)2

J

36

3.4 An Appraisal of the Computed Anisotropic Parameters

 

There are at least two ways to judge the reasonableness
of the anisotropic parameters as computed by the direct method. One
way is to plot Equation (3.3—l), using the computed values of F, G,
and H, and to compare this curve with the results of Table 2.4-1
plotted on the same graph. A second way is to use the indirect
method, as described in Chapter 12 of Hill [27], to compute F, G,

and H from a few tests assuming that N = 1.745 x 10—9 as found in
section 3.2.. Both of these checks were made and are discussed in
this section. Equation (3.3-l) gives
0a = [F sin2 a + G coszci-Fli+(2N - F - G - 4H) sin2 a cos2 01]-1
(3.4-l)

With the values of N, F, G, and H from Equations (3.2—l) and
(3.3-12) a simple program in the BASIC language was used to compute
values of "0" at five-degree intervals of "a." The resulting
computer output is presented in Table 3.4-1 and plotted in Figure
3.4-1 along with the nine average values of the experimental data
from Table 2.4-1. From Figure 3.4-1, it is evident that the ex—
perimental data required a lot of "smoothing out" during the "least
squares" procedure.

The indirect method of computing the anisotrOpic parameters
requires a consideration of the plastic potential theory as proposed
by Mises [8, 9], and as given by Hill [27, 55] for the anisotropic

case. A rigid, work-hardening material is assumed; hence the plastic

strain increment is the total strain increment. Plastic potential

III I." l.IIIIKI.|¢¢| In. ll. l1. 1|
[i lll' [11' .l III III

37

theory assumes that the stress, strain-increment relation is de-

rivable from the yield function f(oi ) by the relationship

3

 

d6., = d1 3f (3.442)
13 30..
13

where f(oi ) is defined by Equation (2.1-2) repeated here for con-

1

venience:

Table 3.4—1 SIGPLT Computer Program Output

SIGPLT
A3 O 58 2714683
A3 5- 53 2717788
A' 108 53 2726986
A3 158 53 2741386
A3 208 33 2759688
A3 250 S8 2780189
_ A: 30; 5= 2800888
A3 358 58 2819684
A3 408 5: 2834587
A: as. s= 28442.3
A- so. 5: 2847887
A8 558 $3 284568
A. 608 = 2838381
A8 658 53 2827581
A8 708 53 2815088:
A: 755 53 2802987
A: 808 , = 2792982
A= 858 53 2786381
A: 908 = 2784001

TIME: 1 SECS.

Tensile Yield Strength

 

 

 

28800
28600 gpll?

28400 Zi%, 3‘\eg;\\k
28200 \q

28000 p ‘E

27800
27600 //§;/(

27200
CD/
27000

 

 

 

 

 

 

 

 

{u

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 10 20 30 40 50 60 70 80 90
Degrees to Roll Direction

I DATA FROM TABLE 2.4—1 (Experimental)
C) DATA FROM FIGURE 3.4-1 (Least Squares Fit)

Figure 3.4-1 Best-Fit Curve for Tensile Strength versus Orientation

39

a ._ 2 _ 2 _ 2 2
2f(oi ) F(oy oz) + G(oz ox) + H(ox oy) + 2LTyz +

J

l

ZMT2 + 2N12 . (3.4-3)
zx xy

Equations (3.4-2) and (3.4-3), give the following relationships:

dex = dA[(G + H) ox - Hoy - Goz]
day = dA[(H + F) oy - Foz — Hox] (3.4-4)
dsz = dA[(F + G) 02 - Gox - Foy]

de = dA[NT ] .
xY xy

It should be noted that dex + day + dez = 0, which is consistent
with the assumption of a rigid, work-hardening material.

For a tensile test coupon oriented in the x-direction,
which is the direction of rolling, oy = oz = Txy = 0, and Equations

(3.4—4) reduce to:
dex = dA(G + H)ox, day = -dA(H) ox, ds 2 = -dA(G) ox.
(3.4—5)

The ratio of width to thickness strain increments for this test

coupon at a = 0° then can be written:

dsw dc H
R0 33:: st = C— (3.4-6)

40

Similarly, a tensile test coupon oriented in the y-direction
(a = 90°) gives the following ratio for width to thickness strain

increments:

H
R90 - (1E - d8 = F . (3.4-7)

H
N

For the more general case of a tensile test coupon oriented
at the arbitrary angle "a" to the direction of rolling, the width
strain increment must be found from the general strain transformation

equation.

deij = Eik Ejm dekm (3.4-8)

Specializing Equation (3.4-8) for the width strain of a

test coupon oriented at the angle "a" results in

2

de = de sinza + ds cos a - 2de sin 0 cos a (3.4-9)
w x y xy

to produce a ratio of width to thickness strain increment as
follows:

ds de sinza + de cosza - 2de sin 0 cos a
w x y xy

a I d;;— = ds (3.4-10)

 

R
2

Equations (3.4-4) can be transformed to refer to a tensile
H 1'

coupon oriented at a to the direction of rolling by using stress

transformation equations

2

o = 0 cos a a = 0 sin2 a T = 0 sin 0 cos 0 (3.4-11)
x Y xY

(where o is the uniaxial stress acting on the test coupon) to get

the following stress strain increment relations:

41

dsx = dA[(G + H) 0 cos2 a — H s sin2 a]
day = dA[(H + F) 0 sin2 a - H 0 cos2 0] (3.4-12)
dsz = dA[—G 0 cos2 a - F 0 sin2 a]

de a dA[N a sin n cos 0]
xy

When Equations (3.4-12) are inserted into Equation (3.4—10),
the general strain ratio equation is obtained as follows:

d6

= EEK , (3.4-13)
t

= H + (2N - F - G - 4H) sin2 0 cos2 a
F sin2 a + G cos2 a

R
a

which reduces to Equations (3.4-6) and (3.4-7) for the special cases
of a = 0° and a = 90°.

Tests by several investigators, including Bramley and
Mellor [54, 77], Atkinson [2], and Lankford, Snyder and Bauscher [38]
indicate that, for low carbon steel, the width to thickness strain
ratio increments do not vary as the material strain hardens during
the tensile tests. This fact permits the strain ratios to be com-
puted at larger values of strains to get more accurate results and
permits finite strains to be used in place of strain increments in
equation (3.4-13).

During experimental determination of the data required for
strain-hardening information, discussed in Chapter 4, strain ratio
data were obtained for three coupons, one coupon at a = 0°, one at
a = 45°, and one at a - 90°. For the assumed volume constancy, the

strain ratio R is given by

[H
Width Strain _ ln w

Thickness Strain — 1 1w] '
n
I‘0‘”0

R = (3.4-l4)

42

In Equation (3.4-l4), "w" is the current coupon width, and "2" is
the current length between gage marks. Atkinson [2,3] recommends
that the strain ratio should be measured just prior to necking, at
a logarithmic length strain of about 0.20 for low carbon steel.
From measurements of the coupons, after each had been strained
approximately 20% in the length direction, the following strain

ratios were determined:

R0 = 1.63 8,5 = 1.22 R90 = 1.90 (3.4-15)

Using these values of the experimental strain ratios and
the value of N - 1.745 x 10-9 in Equation (3.4-l3), specialized
for a - 0°, 45°, and 90° resulted in the following three equations:

H

R0 = 1.63 8'6

= H + (2N - F - G - 4H)(cos 45°)2(sin 45°)2

 

R“5 = 1'22 F(sin 45°)2 + G(cos 45°)2 (3°4_16)
H
R90 = 1.90 ='§
whose solution gave
F = 4.68 x 10'10
0 = 5.46 x 10’10 (3.4-17)

H = 8.90 x 10’10

These results are based on very limited data. They were
calculated after the completion of the direct method calculations,
which were used for the analysis of the problem. Since Equations
(3.4-17) indicate greater anisotrOpy than the direct method Eq-
uations (3.3-12), it may be that the indirect method would be more

suitable to use in future research of this type.

43
3.5 Transformation of the Anisotropic Yield Function

The shear parameter N was determined in section 3.2, and
the tensile parameters F, G, and H were computed in section 3.3.
With these values, the plane stress, anisotropic yield function
of Equation (3.1-1) for the aluminum—killed steel used in this

study is
13.5762 - 11.920 0 + 12.962 + 34.9:2 . 1010 (3.5-1)
x xy y Ky

Equation (3.5-1) applies only for the coordinate axes as defined
in Figure 1.1-2. Stress transformation relationships must be used
for other axes.
During the cup-drawing process, the stress state for
a = 0°, 0 - 45°, or a - 90°, where "a" is the angle to the direction

of rolling, is:

or 0 0.1
0km = 0 06 0 (3.5-2)
0 0 0
L _J

 

 

since the radial and the tangential directions are principal dir—
ections for a = 0°, a = 45°, and a = 90°, and a plane stress condition
is assumed.

The stress transformation equation

7 . _
oij likljmokm. (3.5 3)

can be expanded as follows with the 0 components given by Equation

km
(3.5-2).

44

=2+2
Ox zxror £x8°e

a 2 2
0y Eyror + £y0°8 (3.5-4)

Txy = 2xrzyror + £xe£y8°0

With the definitions shown in Figure 1.1-2 and Figure

3.5-1, Equations (3.5-4) can be rewritten as:

2 2
0x - or cos a + 06 sin a

o - a sin2 a + 0 cos2 a (3.5-5)
Y r 0

Txy = (06 - or) sin n cos 0

Substituting these transformation equations into the aniso-
trOpic yield function of Equation (3.5-l) results in the following
equation:

a: (13.57 cosu a + 22.98 sin2 a cos2 a + 12.9 sinus) +
oroe(-ll.92 cos1+ a — 16.86 sin2 a cosza- 11.92 sin” a) +
as (12.90 cosl+ a + 22.98 sin2 n cos2 a + 13.57 sin” a) = 1010.

(3.5-6)
This equation has the form of a quadratic in or
2 2_ 10= _
blor + (bzoe) or + (b3o 6 10 ) 0 (3.5 7)

For a given angle "a," or can be found as a function of 06. For

a particular "a," Equation (3.5-7) can be plotted as an ellipse

in Or’ 0 stress plane to graphically show the yield function for

6
that orientation. Elements in the flange of a partially-drawn cup

are subjected to a non-negative radial stress and a non-positive

tangential stress; hence only the fourth quadrant of the or, 06

45

stress plane affects this investigation. Equation (3.5-7) was

solved for or using the quadratic formula.
The solution was obtained using the "YIELD" computer pro-
gram. The computer output is listed in Table 3.5—1 for a = 0°.

Figure 3.5-2 presents this data graphically for the fourth quadrant.

Table 3.5—1 Computer Output for YIELD Program at a = 0°

YIELD
ANGLE IN DEGREES T0 RDLL DIRECTION: 0
813 13-57 B2=~11-92 B3= 12-9
13-57 51'2-11-92 51*52+ 12-9 52'2 = 1
TANGENTIAL + RADIAL - RADIAL
YIELD 'YIELD YIELD
STRESS STRESS STRESS
-27842-3 5-39736 E-S -24456-9
-25058-1 5154-16 ~27165-4
~22273-8 9217-07 -28782-6
'19489-6 12632-1 ~29751-9
‘16705-4 15585-9 -30260-
'13921-2 18177-2 ~30405-7
'11136-9 20464-8 -30247-6
'8352-69 22485-9 -29823.
-5568-46 24264-3 -29155-7
-2784-23 25815- -28260-7
'5034058 E-S 2714603 '2714603
2784-23 28260-7 -25815-
5568046 '2915507 ‘2426403
8352-69 29823- -22485-9
11136-9 30247-6 -20464-8
13921-2 30405-7 -18177-2
16705-4 30260- -15585-9
19489-6 29751-9 -12632-1
22273-8 28782-6 -9217-07
2505801 2716504 '5154016
[27842.3 24456-9 -3-77815 E-4

46

90° a

 

 

Rolling Direction

N
ll

Transverse Direction

“<2
11

Thickness Direction

N
l

Radial Direction

'1
II

CD
ll

Tangential Direction

Figure 3.5-1 Definitions for Transformation of Axes

47

 

 

 

 

 

 

 

°0
10,000 20,000 30,000
i
5 f1
-10,000
-20 ,000 //
-30,000

 

 

 

 

 

 

 

 

 

Figure 3.5-2 Anisotropic Yield Function for a = 0°

48

3.6 A Linear Approximation to the Yield Function in the Fourth
Quadrant

The method used to find the "best" linear approximation
to the yield function in the fourth quadrant for selected values
of "a" will be discussed in conjunction with Figure 3.6-1. The
first step was to rotate the axes through an angle 9, such that
a straight line connecting points E and F in Figure 3.6-1 became
horizontal. Then the least squares method was used to fit a
straight line to the curve, after which the straight line was
rotated back through the same angle 6.

The necessity of rotating the axes before attempting a
least squares fit became apparent after observing results without
first rotating the axes. For a curve, symmetrical with respect to
the line 0G (whose slope is -l) in Figure 3.6-1, the slope of the
"best fit" line should be unity, but it is not. The usual least
squares method, which involves minimizing squared vertical devia-
tions at equal horizontal intervals, resulted in weighting that
portion of the curve from F to C more heavily than the part from
E to G. The rotation of axes method provided the necessary im-
provement in results.

The transformation equations for a rotation of axes counter-

clockwise as shown in Figure 3.6-1 through an angle 0 are

0 0: cos 6 - 0* sin 0

r 6
= * * _
00 or sin 6 + 06 cos 6 (3.6 l)

49

Yield Equation: A02 + Bo o + Co2 = D
r r 6 6

 

      
 

121/E. 0)

Yield Equation Graph

 

 

 

Figure 3.6-1 Transformation of Axes in Or’ 0 Stress Space

0

50

After Equations (3.6-l) are inserted into the anisotropic

yield equation of the form of (3.6-2)

2 2= _
Aor + Boroe + Coe D (3.6 2)

The equation can be simplified to get

(A sin2 0 - B sin 6 cos 6 + C cos2 0) 032 +
{[(cos2 6 - sin2 6) B + 2 sin 6 cos 6(C - A)] 0:} 03 +

[(A cos2 6 + B sinEBcos 6 + C sin2 6) 0:2 — D] = 0, (3.6-3)

which is a quadratic in 03. In the notation of Equation (3.6-2),
the coordinates of point E(\/§f, 0) are found by putting 06 = 0
in Equation (3.6-2). Similarly, by putting or = 0, the coord-
inates of point F are found to be (0, -\/%:). By rotating axes

through the angle 0, such that
F _

Land

6 = arctan = arctan fg' (3.6-4)

L-

that portion of the yield ellipse (in the fourth quadrant) between

{>IU

 

 

E and F becomes horizontal relative to the rotated axes.

The "YIELD 3" computer program, written in BASIC language,
both rotates the axes to make points A and B horizontal, and then
computes the coordinates of eleven points between A and B in the
rotated coordinate system. The YIELD 3 computer program output for
a = 0 is shown in Table 3.6-1.

A standard library program written in BASIC language called
POLFIT was used on the General Electric 265 computer to find the

"best" linear equation to fit the eleven points using the least

51

squares technique. For the computer output at a = 0° from the

YIELD 3 program, the "best fit" straight line had the equation

a; = 4.495 x 10"+ a; - 21,500 (3.6-5)

This is found on the computer output, Table 3.6-2 from the POLFIT
program.

The final step was to transform Equation (3.6-5) back to
the original coordinate system. The vector transformation equa-

tions used were

0* 0 cos 6 + 0 sin 6
r r 6

(3.6—6)

* _
06 or sin 6 + 06 cos 6

Inserting these vector transformation equations into the "best fit"

linear equation with respect to the transformed axes,
* = * _
oe Aor + B (3.6 7)

results in the "best fit" linear equation with respect to the orig-

inal or, o coordinate system.

6

 

 

6 [cos 6 - A sin 6 cs 6 - A sin

a a 17‘- cos 6 + sin 610 +l: B J (3.6-8)
r c 6

This calculation was accomplished using the BASIC language

program ROTATE.

52

Table 3.6-1 YIELD 3 Computer Program Output for a = 0°

N: 0798055

SINENJ= -716

CUSENJ= -6981

RAD STR CIR STR

’1993501 '1943607
'16046-5 '20729-2

'1215709 ‘2167609

‘8269-31 -22325-9

'4380-72 '22703-1

“492-133 ‘22823-5
3396-46 -22692-4
7285-05 '22306-6
11173-6 -21653-9
15062-2' '20710-7
18950-8 ‘19436-7

53

mmmm-ml o_-Vmom a-om¢~mn hoomvm_u w-Ommm~
momme-m: mom-_wb o-mov_ml h-o~homt N-mcom_
mmmNVF- vvm-mm~l v-vmv_ml m-mmo~mu o-m>~_—
mvosb-m wow-o_wl _-0mv_mn o-oommm: mo.mwm>
mmomm-m mm-vm—_o m-hmv_mn v-NOQNNI ov-bmmm
mhbm_-o m-mmm_a m-omv_mc m-mmwmmu MNH-NQVI
m~owm-m mh-Homut m-_om_Nu _-mo~mml mh-owmvu
Vvomw-m vow-Nch -mom_mu a-mmmmmu Hm-mmmml
~wooow- omo-m>~| w-QOWHN: m-ono_ml m-bm_m~c
bovmw-mu Nam-hhb o-oomHN: m-mmhomu m-ovoo_l

m_m©-mu Vo-_>om m-mom—NI h-omvm_u _-mmom_u
ummouhom kkHQ DJGUI> 4<3h0du> JGDHU¢1K

Vim oommv-v _
v-movumu o
mv-mmm_ u y mam wh¢zmhmm k8 mammm ohm
Hzmuommmmao . SKMH
.wmzomc

QZHQZMDmG ha Imama 2H m4<~?®2>4®m MIh mhuk ZNIH m2¢ uhuk
mm SH JdutnszQm wwmuwc hwmraJ wIH >uuom1m ah mmmD m2844<

7¢mu®m1 .mhznam <H¢Q oo~ ma xqt d 32¢ Huh mwrwmo Ihu__
MY< mumtuJ -dsrhwi J<~f82>Jsm J¢Z®3®Ihxs Zc uzumj ngdo
whdumd>mm Sh m4¢~2®2>4®1 mmrdUOMThmde mhmm £¢109wm mHIh

hakmam

no u o How Emuwoum Hquom vumvamum Eouw usauso Hmusmeoo Nlo.m mHan

54

The "best fit" linear equation with respect to the original

0 axes for a - 0° was

° 0

r,
06 I 1.027 0r - 30810 (3.6-9)
Equation (3.6-9) can be generalized as follows:

0 = N0 - 0* (3.6-10)

Equation (3.6-10) represents a linear approximation to the aniso-

tropic yield condition where 0r 3_0 3_0 As the material strain

6.
hardens, 0} will increase from its initial value 0: to its current
value 0}. Table 3.6-3 shows the "best fit" linear initial yield

equations for five orientations of the radial axis with respect to

the direction of rolling. For the isotropic case, the "best fit"

linear yield equation was computed to be

06 = or — 1.094 '50 (3.6-11)

instead of the more commonly used equation

0r - 69 = 1.1 30 (3.6-12)

Table 3.6-3 Linearized Anisotropic Yield Equations for Five

 

 

Orientations

0 Initial Linear Yield Equation
0° 06 8 1.02656 0r — 30810.

30° 06 = 1.01369 0r - 31610.
45° 0 = 0 - 31720.

6 r
60° 06 = 0.98624 0r - 31180.
90° 06 = 0.97412 0r — 30010.

 

 

 

 

55

The theoretical analysis of cup-drawing in Chapter 5 will
use these results along with experimentally determined strain—

hardening behavior.

IV. EXPERIMENTAL DETERMINATION OF THE STRAIN-HARDENING BEHAVIOR

4.1 Preliminary Remarks

As an element in the flange moves inward during the cup-
drawing Operation, it strain hardens. Strain hardening is indicated
by an increase in the magnitude of the yield strength during con-
tinued plastic deformation. There is no adequate theory available
for anisotropic strain—hardening behavior. Because it was desired
to include the effect of strain hardening in the investigation,
some measure of strain hardening had to be found.

Svensson [82] pointed out that three physical effects re-
sult from plastic deformation: strain hardening, the Bauschinger
effect, and the development of anisotropy. During the rolling mill
Operations the sheet steel receives a process anneal which virtually
eliminates the Bauschinger effect from the blanks prepared for the
cupping operation. This implies that the initial yield stresses
in tension and compression are equal for any coupon cut from the
sheet. During the cupping operation the additional hardening might
very well introduce a Bauschinger effect, so that subsequent test-
ing would exhibit unequal tensile and compressive yield stresses.
But since the stress components acting on a given element of the
flange change monotonically (that is, no unloading occurs during
the cup-drawing Operation), it was decided that the Bauschinger
effect during the cupping operation would not be considered.

In section 6.12 of his text, Fung [84] discussed several

proposed hardening rules. Isotropic hardening assumes that the

56

57

yield surface in stress space enlarges as the material strain
hardens while the shape of the original yield surface does not
change. Isotropic hardening ignores the Bauschinger effect, where-
as "kinematic hardening" as proposed by Prager [79] does not.

As Hill [27], page 332, has pointed out, the problem of
relating the parameters of an anisotrOpic yield function to the
strain history is extremely complicated. Hill suggested a pro—
cedure to follow in a metal which already has a pronounced pre—
ferred orientation when further deformation to be considered is
such that further changes in anisotropy are negligible. In effect
this procedure assumes that in the further hardening, the yield
surface enlarges without change of shape (as in isotrOpic harden-
ing), but that it preserves the initial anisotropy. This procedure
suggested by Hill is followed in this investigation, using Hill's
definitions of effective stress and strain together with Ludwik's
three-parameter stress-strain relation, as discussed by Hill [27]
on page 12. This stress-strain relation is consistent with the
assumption of a rigid, work—hardening material.

The strain-hardening assumption is presented in section
4.2. Section 4.3 discusses the experimental details of the tensile
testing, and section 4.4 presents the procedure used to determine
the actual measure of strain hardening used in the investigation.
Also presented in section 4.4 is some evidence that the procedure
suggested by Hill was reasonable for describing the strain-

hardening observations.

58
4.2 The Strain-Hardening Assumption

IsotrOpic hardening is usually described by one of two
methods. In either method, an effective stress 0'15 defined, a
scalar measure of the intensity of the combined stress state, which
may be interpreted as a characteristic dimension of the yield sur-
face in stress space. Then in the postyield isotropic hardening
under combined stress loading it is either assumed that 0'13 a
function of the total plastic work Wp or alternatively that 0'13
a function of an accumulated effective plastic strain J'EE ,
wheremc-l:p is the "effective plastic strain increment." When the
Mises yield condition is used, and the effective stress and plastic

strain increment are defined as the two invariant expressions

 

- -—- 3
= ' = - 1 ' _
1/3J2 2 Oij Oij (4.2 l)
p=\/%-depj deg , (4.2—2)
where J' =-l 0' 0' is the second invariant of the stress deviator

ijoij
0&1, the two alternative assumptions turn out to be equivalent,

since then it can be shown that

_ P - _'——P _
Wp - Joij deij — J 0 dc (4.2 3)

is a single-valued function of [02'1
The procedure of this section, following Hill [27], pages
332-334, is analogous to the isotropic hardening procedure based on

the Mises yield condition. An effective stress 0'15 defined, based

59

on the quadratic yield function of Chapter 3, in a manner anal—

ogous to the way the isotropic 0'13 based on the quadratic in-
variant Jé of the deviatoric stress. Since the anisotropic quad-
ratic yield function is not an invariant, however, the formula given
for 0 must always be evaluated with reference to the axes of ortho-
trOpic symmetry, which are assumed to be unchanged during the further
deformation process. An accumulated effective plastic strain will
also be defined in such a way that Wp = J 0 0;. .

In Chapter 3, the anisotrOpic parameters (F0, G0, H0, and No)
for the initial yield function were determined. The subscript "O" is
used here to designate initial values. Under the assumption that the
state of anisotrOpy remains constant during the cup-drawing Operation,
the yield stresses in the directions of the axes of symmetry will
increase in strict proportion to a single parameter expressing the
degree of strain hardening. If h = h(E) is a positive, monotonically-
increasing parameter expressing the amount of strain hardening, a
function of the effective strain E-to be defined below, starting at
unity, then the current yield stress "X" in the rolling direction can
be found by multiplying the initial yield stress "X0" in the rolling
direction by "h." At the same instant, the current yield stress in
the transverse direction is Y = h Y0, and the current yield stress
in the thickness direction is Z = hZO. Following Hill [27], page
332, a representative stress 0'15 defined, which will also increase
in strict proportion with h, that is 0'= h00.

Using the known relationships between the anisotropic param-

eters F, G, H, and N and the uniaxial yield stresses in the principal

60

anisotrOpic directions, one can find how these parameters change

with strain hardening. For example

2F=_]_‘.+_]_'.__]_'.=_.l__+__1____l__ (4.2-4)

2 2 2 2 2 2
Y Z X (hYo) (hzo) (hXO)

2F = —-:- [—12- + ~17: - 42:] = f [200] (4.2-5)
h Y0 20 x0 h
F0
or F = —3' (4-2-6)
h

Similarly, the other anisotrOpic parameters decrease in

strict proportion
G=_, H:-———, N=—,_. (482-7)

The anisotrOpic yield function, as prOposed by Hill, is

= _ 2 _ 2 _ 2 2 2
2f(0ij) F(0y oz) + G(0z 0x) + H(ox 0y) + 2LTyz + 2M1zx
2 _
2N1xy. (4.2 8)
The yield condition continues to be
2f(0 ) = l (4.2-9)

151
after deformation, with the increased yield stresses accounted for
by the decreasing anisotropic parameters.
For the isotrOpic case 0.1s the square root of the quad-
ratic yield function, multiplied by a numerical factor so that it
reduces to the usual uniaxial yield stress in the case of uniaxial

loading. For uniaxial loading in the x, y, or 2 directions, the

+

61

anisotropic quadratic yield function 2f(01j) reduces to

(0 + H)x2 = 2f(0ij)
(H + F)Y2 = 2f(01j) (4.2-10)
(F + 0)z2 - 2f(Oij) .

Evidently it is not possible to multiply the quadratic
yield function by a single numerical factor so that it would in
each case reduce to the square of the uniaxial yield stress. A
compromise representative dimension 0-of the yield surface is
defined by replacing X, Y, and Z, reSpectively by 0'in Equations

(4.2-10) and averaging the three to obtain

£(F+G+H)02=2f(0
3 .)
iJ
2f(0i )

—2=2__J_ _
or 0 2 F + G + H (4.2 11)

For the plane stress case, the anisotropic yield function

is

= 2 _ 2 2 = _
2f(01j) (G + H)0x 2H0x0y + (H + F)0y + ZNTxy 1 (4.2 12)

The anisotrOpic parameters F, G, H, and N thus appear linearly in
the numerator and denominator of Equation (4.2-ll). Hence, the
effective stress, as defined by Equation (4.2-11) can be written

as 1

2_ 2 2 —
(G0 + Ho)0x 2H00x02 + (H0 + F0)0y + 2NOT 2

3 xy (4.2-13)

0 = -

2 F0 + G0 + H

 

0

by using Equations (4.2—6), (4.2-7), and (4.2—12).

62

In order that the increment of plastic work per unit volume

can be computed as the product of the effective stress and the

effective strain increment (dW = Oijdeij - 0'de), Hill [27] defines

the effective strain increment for plane stress as

_ 2 _ 2 _ 2
F0(Godex Hodez) +.GO(H0dez Fodex) + H0(F0dex Codex)

 

EE'= A 2
(FoGo + 0080 + HOFO)

l.
2dy§ 2

+

where 2
A 3 (F0 + G0 + H0). (4.2-14a)

Consistent with the experimental evidence that the strain incre-
ments for aluminum-killed steel increase prOportionately during
tension testing in one direction [2, 38], Equation (4.2—l4) can be
integrated for the uniaxial tension test to get

_ 2 _ 2 _ 2
F0(G061A H022) + G0(Hoez Fosx) + HO(FOEx G082)

 

E=A +
2
(FOGO + GOH0 + HOFO)
.l
27$ 2
N 2. (4.2-15)
0

> where A is defined in Equation (4.2—14a).
Plastic potential theory furnishes the following relation—
ship between the plastic strain increments and the stress states,

where "d1" is a function to be determined.

 

de, = dx 3f (4.2-16)

1j aoij

63

For the plane stress case the plastic potential function is given
by Equation (4.2—12) and the strain increments can be found using

Equation (4.2-l6) as follows:
dex . d1 [(G + H)0x - Hoy]
d = dA H + F 0 - H0 4.2-l7
Ey [( ) y x] ( )
dc - d1 [—G0 - F0 ]
z x y

de - d1 NT
xy xY

It should be noted that de:x + day + dez = 0, which is the plastic
incompressibility assumption since the elastic strain increments
are neglected.

For uniaxial tension in the x-direction (the direction of

rolling) 0x = X, 0 = 0 = T y = 0, and Equations (4.2—l7) imply

e : s : e = de : d6 : de = (G + H) : (-H) : (—G)
y z

(4.2-l8)
Specializing Equations (4.2—13) and (4.2-15) for uniaxial tension in

the x—direction gives

 

 

; =/l (G + Hi ()9 (4.2-19)

 

(4.2-20)

 

Ml
II
a}
f"
'11
05+
+05
01+
:13
b1
A
0)
>4
V

64

Similarly for uniaxial tension in the y—direction (trans-

verse tO the direction of rolling),

 

 

 

 

O = Y, O = 0 = T = 0 (4.2—21)
y x z xy
Ex 6y £2 = ds de de = (-H): (F + H) : (-F)
(4.2—22)
3 = g (F iHG++FHL) (Y) (4.2-23)
— _ _2_ (F + 0 + H) _
e — 3 F + H (5y) . (4.2 24)

If the strain—hardening assumption is correct, then it is
possible to use tensile test coupons cut at any orientation to
experimentally determine the strain-hardening equation. The effect-
ive stress, effective strain equation derived from data taken from
coupons parallel to the direction of rolling should be identical
with the equation based on coupon data for an arbitrary orientation.
As a check on the reasonableness of the assumption, several experi—
mental points from each test are shown on the best fit plot in
section 4.4.

The effective stress, as defined by Equation (4.2—13), can
be transformed for the case of the tensile coupon oriented at the
angle "a" to the roll direction using the stress transformation
equations
2

0x = 00 cos a 0 = 0 sin a T = oasin 0 cos 0 (4.2—25)

y 0 XY

65

1

__ 3 F sin1+ a + G cos” a + H(cos 20)2 + (%)(sin 20)
°= '2 F+G+H

2 2
0
The effective strain increment equation can be similarly
transformed. The strain transformation equations for a tensile
test coupon oriented at the angle "a" to the direction of rolling

are

de cos2

2

ds 0 + de sin2 a
x w

ds

de sin2 a + de cos2 0 (4.2-27)
y 2 p w

dsxy a dny - sin 0 cos a(dsw - deg)

daz = dst.

Equations (4.2-27) can be integrated for the uniaxial case to get

2 2

e = 8 cos a + s sin a

x 1 w

s = a sin2 a + 6 cos2 0 (4.2-28)
y 2 w

exy = ny = Sln 0 cos 0(6W - 22)

Ez=°t

With Equations (4.2-28) and (4.2-15), the effective strain
can be found for deformation of a tensile coupon oriented at the
arbitrary angle "a" to the roll direction. The effective stress
corresponding to this deformation can be computed with Equation
(4.2-26). Thus the effective stress and strain coordinates can be

computed from the results of tensile testing coupons at any orienta-

tion "a."

66

4.3 Tensile Test Procedures

 

Eleven tensile test coupons were cut, at selected orienta—
tions, from the same sheet steel from which blanks for the cupping
tests had been prepared. Three coupons each were oriented at 0°,
45°, and 90° to the direction of rolling; the other two tensile
coupons were oriented at 60° to the roll direction. The approxi-
mate dimensions of these rectangular tensile coupons were 0.035
inch thick, 0.571 inch wide, and 7 inches long. Two-inch gage
marks were machine inscribed on the test coupons, and the coupon
edges were carefully machined to provide square edges which were
parallel one to the other. The tensile tests were performed on an
Instron Tensile Testing Instrument, type TT-C using a cross-head
Speed of 0.2 inches per minute.

Before the test, eight pairs of dividers were carefully set
to the following dimensions, respectively: 2.10, 2.15, 2.20, 2.25,
2.30, 2.35, 2.40, and 2.45 inches. After the coupon was placed in
the jaws of the Instron, the machine was actuated, and the deformation
recorded by a three-man team. The divider man noted when the gage
marks had stretched to the 2.10 inch setting of the first pair of
dividers. At that instant, the second man measured the current coupon
width, using a one-inch micrometer caliper, to the nearest thousandth
of an inch, while the third man marked the graph of the load curve by

momentarily lifting the recording pen on the Instron chart. This

67

procedure was repeated using the eight dividers in turn to get eight
experimental points of true stress and logarithmic strain for each
tensile test coupon.

The original volume of metal between the scribed lines on
the tensile coupon was computed as V0 = lowoto, where 20 is the
original gage length, We is the original coupon width, and to is
the original thickness. Assuming volume constancy, the volume of
metal "V" at any other length "2" is equal to the original volume
V0. This assumption permitted computation of the instantaneous
cross-sectional area "A" at each load reading "P," and hence the

true stress:

V0 = 20W0t0= V = 2W1: = 2A (4-3‘2)
- 1 _ _,Q,__
8 o A - V0 (4.3-3)
P _ fl _
and = A V0 . (4.3 4)

The logarithmic strain was computed from

s, = 2n (lg-3) . (4.3-5)

The effective stress was computed by using Equation (4.3—4)
followed by Equation (4.2-26). The corresponding effective strain
was computed by using Equations (4.2-28) first, and then inserting
the values obtained into Equation (4.2-15). The actual calculations

were performed on the General Electric 265 Time-Sharing Computer

68

using the computer program TENSIL written in the FORTRAN language.
Typical output for the three tensile coupons at a = 0° are given in

Table 4.3—1.

Table 4.3—1 Computer Output from the TENSIL Computer Program

for Three Coupons at a = 0°

fiaVSIL

EFF‘esinhkfis SITuAIN
38714130 .00963
42867- -O7391
46084- -09780
4102660 012313651
49921- -14948
51895- -16561
534240 -1n5h1
511.7811:- - 9069/:

2‘51‘ 1' :9 1:12.35 111711»)
buff/($- -(1/:>~;'>;)
42743- .07391
45841- oUVYOU

/HX£HI-
jUQUo-
51627-

salini-

12108
10305
1(>f:15
19004

C
if, S 632'; 8 o '5' 2' I'.) " 1"
LE? HEARS; ifmelg
.595153- oLRNIAB
21,31163- -i)'//.;H‘J
, «1.5-12>, - - -‘"‘: 9 >5 "/ lg
”05252- o1.?15x5
:y)3973 -1¢Lsav
DI (0‘73. 010416)“)
OsaYnu .1 (£43
I)" '2 .L . o 02:111777"

69

4.4 Ludwik's Three—Parameter Stress—Strain Equation

 

Ludwik's three—parameter equation [85] can be used to get
a reasonable mathematical model of the effective stress—strain
curve during plastic deformation. This equation is discussed on
page 14 of Johnson and Mellor [80], page 12 of Hill [27], and page

20 of Mendelson [83] and has the form

6' = A + B? m. (4.4-1)
The three parameters A, B, and m must be computed to give the
"best" fit for the experimental data. From Equation (4.4-l) it
is evident that "A" is the initial effective yield stress.

For the experimental data available, a suitable approxi-
mation was obtained by using Equation (4.4—1) with A - 0b, which is
the initial effective yield stress computed by inserting into
Equation (4.2-26) the average yield strength given in Table 2.4-1
for each of the nine orientations. These nine results were then
averaged to determine Eb = 27050. psi. The actual calculations
were performed onfthe General Electric 265 Time—Sharing Computer
using the FORTRAN program "AVESIG."

The two remaining parameters B and m were then calculated
using the "least squares" technique. Equation (4.4—1) was re—
arranged by transposing 0b and then taking the logarithm of each

member to get

2n(0 - 00) = in B + m 2n 2.. (4.4-2)
Equation (4.4-2) indicates that after the eXperimental true

stress and logarithmic strain data are converted to effective stress

70

”0" and effective strain "E“ for tensile coupons at any orientation

0 then a plot of reduced stress "0'— 05' vs effective strain
should plot as a straight line on log log graph paper, if Equation
(4.4—l) is applicable.

For each of the eleven tensile tests described in section

4.3, eight values of effective stress "0“ and effective strain

"E“ were computed as illustrated in Table 4.3—1. For each pair

(0; 23 a second pair (0 - 0b, 2) was found; the coordinates of the
second pair were called reduced stress and effective strain. When
the reduced stress and effective strain coordinates, based on the
experimental data discussed in section 4.3, were plotted on log log
graph paper, the data from all eleven tensile tests reasonably
approximated a straight line with the exception of the coordinate
for the smallest strain value of each test coupon. A similar finding
is discussed on page 86 of the text by Thomsen, Yang and Kobayashi
[81] for the two-parameter Ludwik equation 0I= C: n. It was decided
to exclude this small strain coordinate from the calculations for
"best" fit.

Figure 4.4-1 presents the data from three representative
tensile coupons: one at a = 0°, the second at a = 45°, and the
third at a = 90°. This graph of reduced effective stress vs
effective strain includes the "best fit" line as a solid line.

The dashed lines give an indication of the scatter of the data

(with the exception of a few data at low strain values).

71

Using the same "least squares" procedure as discussed in
section 3.3, a difference equation was first written for each ex-

perimental point based on Equation (4.4—2).

61 = (in B + m In 21) — in (01 - 00) (4.4—3)
The error equation was next written.

E = 28; = £[(9.n B + man '51) — 1:161 - 30)]2 (4.4-4)

The error equation can be considered to be a function of two
variables

E = E [in B, m] (4.4-5)

The "best" values of these two variables are found by differen-
tiating the function with respect to each of these two variables
and setting these derivatives equal to zero. This procedure re—
sulted in the following two equations.

6E

SIEETB).= 0 = £2[£n B + m zn'zi) — 2n(0; - 36)] (4.4-6)

3% = 0 = £2[(9.n B + m 9.6 21) - 2:161 — 30)] (in E1) (4.4-7)

These two equations were then solved for the values of the unknowns
B and m.

Mathematically, the best values of in B and m were found
by this procedure rather than B and m. This problem is discussed
by Wylie [78] on pages 186-191 of his text; the procedure is reason-
able unless experimental strain values close to unity are used.

The calculations to determine the parameters B and m using

the "least squares" technique were performed on the General Electric

72

265 Time-Sharing Computer using the program LUDWIK written in the
FORTRAN language. The strain-hardening equation suitable for the

particular aluminum-killed, low-carbon steel used in this investi-

gation was

_ _ 0.518
0 = 27050. + 60700. e (4.4-8)

73

maoaumucmfiuo sense you

cfimuum O>wuoommm

uoam afimuumlmmmuum m>fiuommmm qu.¢ muawfim

 

 

 

 

 

 

 

 

 

 

 

 

 

o.H w. o. c. m. cm. we. 90. co.
i
.8 u 8 m 38 d
a“
.me u e H meme nu
no u a m mama nu nv\\hHUHHMHu\\\\\.
Y) \
\3
W.
1 \
1 V
\
0\
f \,
1 \\
\\

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

OH

ma

0N

mm

0m

06

om

00

oh

('ISd J0 SPUESHOLLL) SSBJJS BAIJDBJJH pBOUPBH

V. THEORETICAL ANALYSIS OF THE CUP-DRAWING PROCESS

5.1 Preliminary Remarks

In order to calculate the stress and strain history of an
element in the flange of a partially-drawn cup, a mathematical model
must be introduced to represent the actual physical model. In sec—
tions 1.4 and 1.6 of this thesis, previous work in this regard is
mentioned including the theoretical investigations of Chung and
Swift [21] and of Hill [27].

The analysis presented in this chapter assumes that the plane
stress condition prevails for elements in the flange. A blankholding
force of 3980 pounds was required during the experimental draw oper-
ation to prevent the formatiOn of wrinkles. This blankholder force
correSponded to an average pressure of 355 psi on the flange surface
at the beginning of the draw, and an average pressure of 1530 psi on
the flange surface when the partially-drawn cup was 1.12 inches deep.
Since this stress is small compared to the yield stress for the
aluminum-killed steel used for the experimental investigation, the
plane stress assumption was used. These average figures are based
on a uniform pressure distribution over the flange. Since, at any
stage of the draw except the beginning, the rim thickness is greater
than the thickness of interior elements, the greater part of the
‘blankholding force acts at the rim, where the condition is in fact
not plane stress. (See the results of the investigation of the
distribution of blankholding force over the flange, published in

1964 by Woo [63].)

74

75

Because the blankholding force "H" is considered to be con—
centrated at the rim, the plane stress assumption is even more reason—
able in the interior than the analysis based on a uniform distribution
indicates. The friction at the rim then furnishes a radial force,
which is introduced into the plane stress analysis as a radial boundary
stress 0b.

Equilibrium of forces in the radial direction at the rim,

where the current radius is "b" and the thickness is "tb" then yields

(20b tb) 0b = 2uH (5.1-1)
0 - JE— (5 1-2)
b Tbtb °

where u is the coefficient of friction.

The coefficient of friction between the blankholder and the
cup flange and also between the die face and the flange of the par-
tially-drawn cup had to be used in the calculations. It was decided
to use a coefficient of friction u = 0.06 based on experimental work
by Swift reported in Table 3, page 359 of his 1948 publication [19]
for mild steel of comparable thickness with good lubrication. In
Figure 35, page 216, of their 1951 publication Chung and Swift [21]
compare radial strain results for three coefficients of friction
u = 0, u = 0.06, and u = 0.128; they used u = 0.06 in their cup-
drawing calculations.

Another assumption implicit in this analysis is that the
punch force acting on the partially-drawn cup is balanced by a line
distribution of concentrated force exerted by the die ring at the

die—profile radius OD. See Figure 1.1-1. A more realistic

76

assumption would be that this force exerted by the die ring on the
partially-drawn cup is distributed over some area of the flange;
however, it was decided to use the more convenient assumption of a
concentrated force.

The three-parameter Ludwik equation discussed in Chapter 4
was used as the measure of strain hardening for both the isotrOpic
and the anisotropic analysis. This is given as Equation (4.4-8) for
the aluminum-killed steel investigated and more generally as Equa-
tion (4.4-l).

It is the purpose of this chapter to develop the approximate
theory used and to present some of the results for the stress and
strain field calculations for elements of the cup flange at any stage
of the draw. In section 5.2, the yield condition is discussed.
Section 5.3 introduces the equilibrium equation and the stress anal—
ysis theory, and section 5.4 presents the strain analysis theory.
The stress and strain analysis theory is specialized for a rim
element in section 5.5 along with the associated computer analysis.
Section 5.6 introduces the computer analysis for interior points.
Results of the strain analysis are presented in graphical form in
section 5.6. The notation used in this chapter is given in Figure

1.1-1 and Table 5.1-1.

77

Table 5.1—1 Notation for Chapter 5

Current radius to the rim of the partially—drawn cup

Original radius of the rim; Radius of the flat blank from
which the cup was drawn

Variable radius to an element in the flange
Current radius of the element being followed in the flange

Original radius of the element being followed as measured
on the flat blank

Current thickness and original thickness of the element
being followed

0
Current thickness of the flange metal at the rim

Mean thickness of the metal between the rim and the element
under consideration

Blankholder force in pounds

Coefficient of friction

Radial stress and strain components

Circumferential or tangential stress and strain components
Component of stress in the thickness direction

Strain component in the thickness direction

Radial stress component at the rim of a partially-drawn cup
Effective or equivalent stress and strain

Initial effective stress

Material constants in the three—parameter Ludwik stress-
equation.3 = Eb + BEm

Initial effective stress in the linear approximation to
Hill's anisotrOpic yield condition 06 = Nbr - 0*

Effective stress after strain hardening in the linear approxi-
mation to Hill's anisotrOpic yield condition 0* = 0-* (-)
C’0

78
5.2 The Yield Condition

An element in the flange of a partially-drawn cup is char-

acterized in this analysis by the plane stress condition 0 3_0 =

r 2

0 3_0 In the or, 0 stress plane (for oz = 0) the yield condition

6' 6
may be represented by Mises' yield ellipse for isotropic metals or

by Hill's anisotrOpic yield ellipse for anisotropic metals. For the
anisotropic case different ellipses are obtained for different angles
a; see section 3.5. Only one quadrant of the Or’ 06 plane represents
possible stress states for an element in the flange; therefore, it is
possible to use one linear equation to approximate each ellipse. See
Figure 5.2-1 for the isotrOpic case. Because of computational ad-
vantages, it was decided to use linear approximations to the yield
conditions for both the isotropic and the anisotropic cases.

The linear approximation of the initial anisOtropic yield
condition is discussed inChapter 3. For an element along the dir-
ectiOn of rolling the linear approximation to the initial anisotropic
yield condition is given by Equation (3.6-9). The linear approxi-
mations for certain other radial directions are given in Table 3.6—3.
The general form of the linear approximation to the yield condition

is

0 = N0 - 0* (5.2-l)

where 08 will increase in direct proportion to 0 with the strain
hardening, starting at the initial value 03 = 30810 psi for a = 0°

and at 0: = 31720 psi for a = 45°:

3* = 3.3 (3;) (5.2—2)

°0

79

 

 

 

 

Tresca

 

 

' . Mises

Modified Tresca

 

 

 

 

Figure 5.2—1 Yield Conditions for the Isotropic Case

80

On page 284 of his text [27], Hill remarks that the effective
strain E for an element in the cup flange is never more than 3% greater
than the absolute value of the tangential strain 56' Using this approxi-
mation gives

r r
e ; Is = - [ d5 = _ [ 23-: ln-—2 (5.2-3)
0 e r r

Jro
This permits the three-parameter strain—hardening Equation (4.4-1) to
be written in the alternative form
- - -m - r0 m
o = 00 + B5 = 00 + B (1n 7) (5.2-4)
Inserting Equations (5.2-2) and (5.2—4) into the yield Equation (5.2-1)
results in the following alternative forms of the linear approximation,

including strain hardening, of the anisotropic yield condition

_ _ _ _ (5.2-5)
0e - N0r - (03/00)0
_ 36) to m (5.2-6)
06 I Nor - (5‘ [0'0 '1' B(ln 7)]
0
06 = Nor — ‘3 (1 + E—E In) . (5.2-7)
0

Equation (5.2—7) is of the form to show the factor h = h(€ij)
discussed in the early part of section 4.2. The function h = 0/00 =
1.4-::- is the parameter expressing the amount of strain hardening,
starting at h = 1. The value of N also depends on a (see Table 3.6-3),

but the hardening function constants are assumed independent of a, as

discussed in Chapter 4.

81

As illustrated in Figure 5.2—1 for the isotropic case, a

modified Tresca yield equation has the form
0 — 0 = L0 . (5.2-8)

If L = 1, then Equation (5.2-8) reverts back to the standard Tresca
yield condition, which is a hexagon inscribed within the Mises yield
ellipse. Many authors find it convenient to let L = 1.1 to get a
better approximation to the Mises yield ellipse in the fourth quad-
rant; however, Equation (3.6-11) gives L - 1.094 as the "best-fit"
value. Strain hardening is included in the yield Equation (5.2-8)
since 0 is the effective stress, which increases with strain hard-
ening as given by Ludwik's three—parameter Equation (4.4-l). Insert-

ing Equation (4.4-l) into Equation (5.2-8) gives the alternative

forms
06 = 0r - L0 (5.2-9)
— —m
06 = 0r - L(00 + Be ) (5.2-10)
= - I: <1 +931“) <5 2-11>
0e 0r 00 __ e . .

°0

Replacing the effective strain E.by Hill's approximation, Equation
(5.2-3), gives

r
_. o m
06 — 0r - L[00 + B(ln 17)] , _* (5.2-12)
_ 0
a special case of Equation (5.2-6) with N = l and :f-= L.
0
0

82

5.3 Stress Analysis Theory for the Flange

The differential equation of force equilibrium in the radial
direction for an element in the flange can be found as follows (see

Figure 5.3-1).
ZFr = 0 (5.3-1)

0 - (0r + dor)(r + dr)(d6)(t + dt) - 0r(rd6)(dt)

1 d6
- 2[0e dr(t + 2 dt)]( 2) (5.3-2)
Neglecting higher order terms and rearranging gives

d(0rt)

O g dr

+ (0r - 06) E-. (5.3-3)

The equilibrium Equation (5.3-3) and the plane stress
assumption assume that the element thickness may change as the cup
is progressively drawn. Chung and Swift [21] point out that although
the flange thickness generally increases during radial drawing, the
actual difference in thickness across the flat flange at any instant
is small. For a drawing ratio of less than 2, the maximum difference
is about 5%. Hence Hill [27] on page 285 of his text suggests that
if uniform metal thickness, at any instant, is assumed for the flange,
the maximum error in the radial stress 0r will be 5%. Assuming uni-
form thickness across the flange at any instant means that dt = 0 in
the equilibrium Equation (5.3-3), which simplifies to

d0r 0r - 0e
dr + r = 0. (5.3—4)

 

83

 

 

d6

NMH

 

1 t)
01’“
0 agar) '1

Figure 5.3-1 Notation Used in Force Equilibrium for 3 Flange Element

84

Rewriting equation (5.3—4) to solve for d0r results in

dr
d0r = r (06 - 0r) (5.3-5)

Inserting the yield condition Equation (5.2-6) into equation (5.3—5)

and integrating from the rim to the element being followed results

in

or' __ r'dr ._ B r' r m d __ r' 0r

J d0r - -03 J -1? - 03 (27% J (In -39 —£-+ (N - 1) J (-—) dr.
r r r

0b b 00 b b

By introducing Equation (5.1—2) for Oh, Equation (5.3-6) can be

reduced to

ER - b - B J r0 m dr - Or

8 + * — + — — — _ .—

0r, "btb 00 1n r' 03 ( ) r,(ln r ) r -+ (l N) J ,(r ) dr.
00 r

The fourth term in the right-hand member of equation (5.3—7)
requires a knowledge of how the radial stress 01. varies with r. The
stress and strain history for an element at the rim as the cup is
progressively drawn must be computed first. Next the stress and
strain history for an element close to the rim is followed. By this
procedure, it is possible to progressively formulate the function
0r = 0r(r).

For the isotrOpic case N.= l and the fourth term of Equation
(5.3-7) vanishes; also, 0% = L00 for an isotropic material, which
alters Equation (5.3-7) somewhat.

The third term of Equation (5.3-7) has an integral that re-

quires interpretation. The integration variable "r" varies from r',

85

where the radial stress is desired, to the current rim radius b of the
partially-drawn cup, and r0 denotes the initial radius of the element

now at r. It is evident that r is not a constant, but rather r

O = ro(r).

0
Hence before this integral can be numerically integrated, some relation-

ship between ro and r must be determined.

In this analysis, it was assumed that an element in a pie-
shaped region of the blank moves only radially inward, which seems to
be a reasonable first-order approximation. Referring to Figure 5.3-2
and assuming that the volume of metal between the element being followed
and the current rim at any instant is equal to the volume between the

same element and the rim at the beginning of the draw operation, we

obtain
«(63 - r3) t0 = «(62 - r2)tm (5.3-8)
or
r3 tun+1052 b2 t‘“ ‘ (539)
"""'"" "" 0‘ 0—9 -‘
r2 to r2 t0

where tm is the mean thickness of the flange metal between the rim

and the element being followed.

 

Hence
r t t
._2.=.l .11 .1. 2 _ 2.49 _
1n r 2 1n L0 + (b0 b0 toE‘ (5.3 10)
r
and
j: 2 2 to
r = b - (bo - r0) f-I (5.3-ll)
m

Equation (5.3-ll) gives the current position of an element being

followed in terms of its initial position r0.

'86

 

 

 

Figure 5.3-2 Terminology for a Partially-Drawn Cup

87

Equation (5.3-10) can now be inserted into equation (5.3—7)

to get
__ __ b 0
0,=-JJ-I'-I-+0*ln-b—,+(l-N)[ (-£)dr

I

r 1rbtb 0 r Jr r

._ b t t m
+ O*(‘§\)J l 1n JR + i— (b2 - b2 1'1) 95 (5.3-12)

00 r r 0

In the integration of Equation (5.3-12) known values of b0 and
to on the flat blank will be used along with the other known values
of u, H, 3:, 30’ B, and N; Then it will be assumed that the blank has
been partially drawn to a somewhat smaller rim radius b. Some method
must be used to estimate the rim thickness tb as well as the average
thickness tm of the flange between the element being considered at
radius r' and the rim at radius b. After 0r has been calculated for an

element at radius r', then 06 can be computed using the yield Equation

(5.2-5) which is repeated here for convenience

06 = NOr - LO (5.3-l3)

where L = 03/05 (5.3-l4)

5.4 Strain Analysis Theory for the Flange

Equation (2.1-l), which is the anisotrOpic yield function in
terms of the principal anisotrOpic directions (shown in Figure 1.1-2),
can be transformed from x, y, z coordinates into r, 6, z coordinates
using the transformation Equation (3.5-3). Since the radial, the
circumferential and the thickness directions are principal stress

directions for 0 = 0°, 0 = 45°, and a = 90° (which implies

88
T = T = T = 0), the transformed anisotropic yield function can
r6 26 zr
be written as

2

2f = F(0r sinza + 06 cosza - oz)2 + G(0z - 0r cos a — 0 sinza )2 +

6

H[0r(cosza- sinza) + 06(sin20 - c0320)]2 + 2N(0e - 0r)2 sinza c0520

(5.4—l)
With the plastic potential flow rule
de - d). Ji- , (5.4-2)
ij 60ij

the strain increments in the circumferential and the thickness dir-
ections can be derived from Equation (5.4-l), when the elastic strain

increments are neglected.

: = _ 2 2 _ - 2 _ 2
dct - dez dA[ F(0r sin a + 06 cos 0 oz) + G(0z 0r cos a 0esin 0)]
dee = dAlF cosza(0r sinza + oecosza - oz) + G sin201(0r cosza + 06 sinza -

2 _ _ 2 2 _
oz) + H(cos 20) (06 or) + 2N(0e 0r) sin 0 cos a] (5.4 4)

After letting 02 = 0, consistent with the plane stress assumption,
the ratio of the thickness strain increment to the circumferential strain
increment can be specialized for the two directions a = 0° and a = 45°

to get

dct F0e + Gor

a (5.4-5)
de6 H0r - (F+H)0e

for a = 0°,

‘89
and

dct -2(F+G)(0r +06)

dee (F+0-2N)or + (F+0+2N)6e

for a = 45°.

(5.4-6)

 

Equations (5.4-5) and (5.4-6) reduce, for the isotropic case (where
3F - 30 - 3H s L - M . N), to
de 0 + 0
'6'? = 32:75. (5-4‘”
6 r 6

Inserting the experimentally-determined values of the aniso—
tropic parameters F, G, H, and N reported in Chapter 3 as Equation
(3.2-1) and Equation (3.3-12) produces the following specializations

Of the strain-increment ratios.

For a = 0°
6.94 06 + 7.60 0r
d6t ‘ 5.96 or - 12.90 06 dee’ (5'4‘8)

and for a = 45°

 

 

-2(14.54)(or + °0) 7
d6t = -20.36 or + 49.44 0 dee' (5'4‘9)

6.1

 

The tangential stress 0 in Equations (5.4-8) and (5.4-9)

6
can be eliminated by inserting Equation (5.2-5) and using the appro-
priate values of N and 0% from Table 3.6—3 or Equation (3.6—ll) along

with the value of 36 = 27,050 from Equation (4.4—8) to get

14.730 - 7.905 3 dr
de = r -—- (5.4-10)

t 14.693 - 7.288 °r r

 

for the direction a = 0°,

90

 

 

d6 [58.163r - 34.1 0] 5.1.; (5.441)
t 57.97 '3 - 29.086r r
for the direction a = 45°, and
d; [2 0r - 1;09‘4 0 ] £1}; (5.4_12)
2.188 '3 - 0 r
yr
for the isotrOpic case.
where
_. _. r0 m 1.000518
0 = 0 + B (1n -- = 27,050 + 60,700 (1n -—)
0 r 1' (5.4-13)
The increment of tangential strain is
dr
dse = r (5.4-l4)

The strain history of an element is studied, as that element moves
radially inward. Hence "dr" must be interpreted in Equation (5.4-l4)

as the incremental distance travelled by the element currently at the

radius "r." This is in contrast to the use of "dr" in the equilibrium

equation, Equation (5.3-4), where dr represents an increment of length

over which the tangential stress 06 acts.

Integrating Equation (5.4-l4) produces

r0
86 = -ln :7 (5.4-15)

which gives the tangential strain for an element which was originally

at radius r0 and finally at radius r'.

91

The radial strain increment for an element can be found from
the volume constancy assumption
det + der + (166 = 0 (5.4—16)

after the other two strain increments have been computed.

The definition of thickness strain increment is given by

_.2£ _
which can be integrated to get
t' '
at = 1n ?0- (5.4-18)
E:1:
t, 8 toe O (504‘19)

Equation (5.4-l9) can be used to find the thickness t' of an element

which has experienced a thickness strain a

t

5.5 Stress and Strain Analysis_for a Rimgglement

 

The analysis to determine the stress and strain field for the
flange of a partially-drawn cup must start with a rim element having

the following specifications.

r0 = b0

r = b

dr = db (5.5—1)
0r = 0b

Several equations were needed for the computational work.

The thickness strain increment Equations (5.4-10) and (5.4-ll) were

92

specialized for a rim element to

14.73 0 - 7.905 2?
det' L _b :1 51-:— (5.5-2)

 

for a = 0, and

d6 8 58.16 Ob - 34.:0 1 db (5.54)
t 57.97 3' - 29.08 0b_| b

for a = 45°.

Equations (5.5-2) and (5.5-3) were integrated incrementally by permitting
the rim radius to move in a fraction of an inch at a time. For each
increment of rim displacement "db," the thickness strain increment was
computed. The total thickness strain for the rim element located at
a current rim radius "b" was found by summing the incremental thick-
ness strains. Any desired degree of accuracy was possible depending
on the number of increments used. The effect of varying the increment
size is discussed in section 5.7. The increment size finally used
for the rim analysis was 0.001 inch.

In order to compute an increment of thickness strain using
Equations (5.5-2) and (5.5—3), values of the effective stress 0 and the
rim radial stress 0b were needed for each current rim position b. The

effective stress was computed using the strain-hardening Equation

(5.4-l3) specialized for a rim element.

-' -' b0 m
0 = 00 + B (1n b—-) (5.5-4)

93

The value of the rim radial stress needed in Equations (5.5-2) and
(5.5-3) was evaluated from Equation (5.1-2), repeated here for con-

venience.

. _H§L. _
0b Hbt (5.5 5)
b
The rim thickness tb in Equation (5.5-5) was determined by special-

izing Equation (5.4—19) for a rim element.

tb g to8 (5.5-6)
Initially the rim thickness strain at was put equal to zero, which
implies an initial rim thickness of to.

The tangential strain increment, defined by Equation (5.4-l4),

can be written for a rim element as

de = -—-. (5.5-7)

The radial strain at the rim of a partially-drawn cup was determined
by numerically summing the radial strain increments. The volume

constancy, neglecting elastic strains, demands that

der = -det - dee (5.5-9)

The tangential stress 06 at the rim was computed by specializing the

yield equation, Equation (5.3-l3), for a rim element.

06 = Nob — L0 (5.5—10)

94

The flow chart for the stress and strain analysis of a rim
element is given in Figure 5.5-1. This flow chart was used to design
the computer programs for the rim element in the two directions in-
vestigated, a = 0° and a = 45°, for the anisotropic case.

The computer programs were written in the BASIC language and
run on the General Electric 265 Time-Sharing Computer. The ANI Al
computer program was specialized for a rim element oriented along the
direction of rolling (at a = 0°). This program is shown as Figure

5.5-2. Partial output for this program is given in Table 5.5-1.
5.6 Stress and Strain Analysis for Interior Elements

The analysis for interior elements, that is elements with
initial radius r0 < 2.4 inch, parallels that for the rim element. The
stress and strain analysis for the rim preceded the analysis for in—
terior elements, since some of the rim element results were used in
the computations for interior elements.

FOur equations that were needed for section 5.3, starting
with Equation (5.3—ll) are repeated here.

b 0

 

=_1£L - .12. _— .1
0r. Hbt + 03 1n r' + (l N) J ' r dr +
b r
‘b t t m _
L B J szln [253+i- <63 - b2 2.12)] 51f- (5.6—1)
r' 0 r2 0
06 = Nor - Lo (5.6-2)
="*_ .6-
L 00/00 (5 3)
\/2 2 2 t
r = b - (b0 - r0) E—- (5.6-4)

 

95

 

 

H = 3980. , , B = 60700.
0 = 0.06 - _

t0 3 0.035 60 — 27050.
b0 3 2 4 m = 0.518

 

 

 

 

 

 

 

 

 

0 = uH/(nth)

 

 

 

 

06 = N0b - L0

Equation (5.5-2) or
Aer = Equation (5.5-3)

 

 

l

 

£6 = —1n(b0/X)

A2 = -A2 - AX/X
1' t

 

 

 

 

 

 

 

 

to

 

 

l

 

 

 

 

3 g

 

=te°t

00 + B ln(bO/X)

 

 

 

NO ‘

 

I'+ I + l

 

YES

 

PRINT
8r’ 86’ E:t

°b’ °e

 

 

 

I + 1 ‘
* YES ( STOP >
0 YES

NO

 

X + X + AX

2 +2 + As
r r

 

 

r
s +—e + At
t t t

 

 

 

Figure 5.5-1 Flow Chart for the Stress and Strain

Analysis of a Rim Element

ANI

100
110
120
122
125

126,

130
140
150
16’)
170
180
190
195
200
210
220
2.30
240
260
270
280
290
300
3.31
302
304
305
310
320
321
330
340
350
363
365
370
375
500
505
510
520
600

96
A1

LET TO=-J35

LET T330

LET Z=O

LET L=30310/27053

LET L1=1-027

LET N=‘0031

FUR K1=30'TJ -499*30 STEP N
LET T5=TO¥EKPCT3)

LET T1='LQG(30/K1)

605J3 503

LET T7=(14-73*55-7-905*L3)/(14-69*L3‘7-238*55)
LET T8=N*T7/K1

LET T9=-T3-N/X1

LET $1=L1*55-L4

LET =-0001

FQQ 1:204 T0 10199 STEP '01

IF A35(K1‘I)<E THEN 300

NEXT I

LET T3=T3*T8

LET Z=Z+T9

NEXT K1

STOP

PRINT "KIA RAD="KI

PRINT "3/30="K1/2-4

LET T2='T1‘T3

LET P1=(T1'T2)'2+(T2'T3)'2+(T3'T1)12
LET T=SQNCP1*2/9) .
PRINT "THICK="T5

PRINT "RAD STR="SS

PRINT "RAD STR/RAD="SS/K1

PQINT "TAN STQ="51

PRINT "THICK STN="T3

PRINT "TAN STN="T1

PQINT "RAD STN="Z

PRINT "EFF 5TN="T

PRINT -

G0 T0 260

LET L3=27050+60700*(LQG(30/K1))t-518
LET L4=L*L3

LET SS:-06*3980/(3-14159*K1*TS)
RETURN

END

Figure 5.5-2 Computer Program for the Stress and Strain

Analysis of a Rim Element at a = 0°

97

TABLE 5.5-l Partial Computer Output for the Stress and Strain
Analysis of a Rim Element at a = 0°

ANI A]

r-{IM RAD: 2.4 RIM RAD-'- 20

9/80= 1 8/so= .833333

THICK: .035 THICK= 3.84916 E-e
RAD SIR: 904.91 RAD STR= 987-39

RAD SIR/RAD: 377.046 RAD STRIRAD= 493-695
TAN 5TR=-29880.7 TAN STR=-58426-4
THICK SIN: 0 THICK SIN: .095093
TAN STN= 0 TAN STN=--182322

RAD SIN: 0 RAD STN= 8-71869 E-2
EFF STN= 0 EFF STN= -182378

RIM RAD: 2.3 RIA RAD: 1.9

3/30: .958333 B/BO= .791666

THICK: 3-57806 5-2 THICK= 3-95407 E-2
RAD SIR: 923-654 RAD STR= 1011-78

RAD SIR/RAD: 401-589 “60 37R/RAD= 532-517
TAN 5TR=-43336.6 TAN STR=-62324.3
THICK sr~= 2.20579 3-2 THICK SIN: .121982
TAN sr~=-4.25597 E-2 TAN 3TN='-233615

RAD SIN: 2.04927 E-a RAD STN= .111573

EFF STN: 4025692 5-2 EFF STN= -233692

R1» RAD: 2.2 RIH RAD: 1-8

9/90= .916667 also: .75

THICK: 3-66194 5-2 THICK8 4-06785 E'2
RAD srg: 943.519 RAD SIR: 1038-12

RAD SIR/RAD: 428-872 860 5TR/RAD= 576-734
'TAN STR=-49358. TAN STR=-66004-I
THICK SIN: 4.52312 E-2 THICK STN= .150351
TAN STN=-3.70116 5-2 TAN STN=--287683

RAD SIN: 4.17612 E-2 RAD STN= ~137262

EFF STN= 8.70344 E-2 EFF STN= -287781

RIH RAD: 2.1 RIM RAD: 1.7

BIBO= .875 BIBO= .708333

THICK: 3075208 5-2 THICK= 4-19178 E'2
RAD STR: 964,703 RAD 5TH: 1066-69

RAD SIR/RAD: 459,333 RAD STRIRAD= 627-464
TAN STRz-54gg4,2 TAN STR=-69543-5
THICK 570: 6-95471 E-2 rHICK 5TN= ~180362
TAN STN=--133532 TAN STN=--344841

RAD SIN: 6-39546 E-2 RAD STN= .164393

EFF SIN: .13357 EFF SIN: -344963

98

Both Equations (5.6-l) and (5.6—4) require the mean thickness
"tm" between the rim and the element being followed. Equation (5.6-l)
requires in the third term a knowledge of how (EE) varies between the
rim and the element being followed. Both of these requirements dictate
that the analysis proceed from the rim inward in order to build up a
record of the thickness and the radial stress between the rim and the
element being followed. Computer print-out of the thickness and radial
stress for elements originally at 0.1 inch increments from the rim
were used to evaluate the mean thickness and the third term of Equa-
tion (5.6-1). Since the thickness and the radial stress of the element
being followed must be included in these calculations, three iterations
were usually required to accurately determine the stress and strain
history for each element followed. When the thickness change between
successive iterations was less than 3 x 10-6 inch, the accuracy was
considered satisfactory.

The following equations from section 5.4 were used. They

include some of the equations starting with Equation (5.4-10).

14.736r — 7.905 3 dr
dc = T (5.6-5)

t 14.69 6' - 7.288 0

 

for a = 0°,

58.16 or - 34.1 E
de = .1 d

57.97 3' - 29.08 or

 

(5.6-6)

H

for a = 45°

_ _. r0 m r0 0°518
O = 00 + B(ln '—r') = 27050 + 607000.11 T) (5.6-7)

99

£6 = -1n -% (5.6-8)
dcr = -det - dee . (5.6-9)
Et

0

The design of the computer programs for the stress and strain
analysis of interior elements of the flange is presented as a flow
chart, Figure 5.6—1. In particular, this flow chart was the basis for
the computer program ANI C7 which followed the stress and strain history
of the element at a - 0° whose original radius was r0 = 2.3 inches.
The appropriate yield equation, Equation (3.6-9), for a a 0 was included
by the READ statement where values of N and 03 are required. Since the
rim thickness tb is needed as indicated in the flow chart, a series
of second-degree "best-fit" equations were prepared from the computer
output for the rim analysis at a = 0°; the library computer program
POLFIT provided these equations for the rim thickness as a function
of the current rim radius "b." For the first iteration, it was satis—
factory to assume that the mean thickness between the rim and the
element identified as r0 = 2.3 was equal to the rim thickness tb. For
this first iteration it was also satisfactory to assume that J (EE) dr =
0- The computer output from this first iteration gave values for the

thickness and the ratio (or/r) for the element r = 2.3 during the cup—

0
drawing process.

Two auxiliary computer programs, AVTHIK and AREA, were designed
to aid in further iterations to improve the accuracy of the initial

output from the main program ANI C7. AVTHIK computed the arithmetic

average of the element thicknesses between the rim and the element

 

 

H = 3980.
( START ;:>- u - 0.06

 
 
      
   
   

ESTIMATE

t
m

Arith.Ave.

ESTIMATE

t0 = 0.035

 

 

 

 

Compute r'
using

 

   

Eq. (5.6-4)

 

 

100

 

B = 60700.
50‘ 27050.
m = 0.518

 

 

 

 

 

b + b0

4.

 

 

 

 

 

 

NO
IS

« b < 1.2

 

 

 

- = _ '
0 00 + B ln(rO/r )

 

 

1

 

b

r!

 

0 = uH/(nb tb)

S6=0* 1n(9—J + (l—N)S4

 

 

 

STOP >

 

 

 

 

 

 

 

r + r'

b + b + Ab

 

 

YES

 

 

 

 

0 =0 +S6+S*
r b

1‘0

ee=-ln(;7J

 

 

 

 

 

 

 

 

 

 

 

__..A6 2 [Equation (5.6-5)]
t or (5.6-6)

 

 

 

 

 

 

 

Figure 5.6-1 Flow Chart for the Stress and Strain Analysis

of an Interior Element

101

being followed for selected values of the rim radius b. The POLFIT
program then converted these output pairs (tm, b) into suitable
second-degree best-fit equations of mean thickness as a function of
the rim radius. The computer program AREA evaluated 84 = J (S£)dr
from the rim radius "b" to the element being followed (using ghe trap—
ezoidal rule) at selected values of the rim radius. Then the POLFIT
program was again used to find suitable second-degree best-fit equa-

\ 0
tion of the integral S4 = J (—E) dr as a function of the rim radius.

r

These best-fit equations for tm - tm(b) and S4 - 84(b) were
then inserted into the main computer program ANI C7 for the second
iteration. Iterations continued until there was negligible thickness
change between successive iterations; normally this required three
iterations.

The main computer program ANI C7 along with the auxiliary
programs AVTHIK and AREA are included in the appendix. Partial com—
puter output for the final iteration of ANI C7 is shown in Table 5.6—1.

The increment size used for the analysis of interior elements was

0.005 inch.
5.7 Results

The results of the computer analysis for the rim element at
a = 0° are summarized in Figure 5.7-1 which presents the strain history
for this rim element as it moves inward during the cup-drawing process.
Logarithmic strain is plotted against rim position which is made
dimensionless by dividing the current rim radius "b" by the original

rim radius "b0.' The stress analysis results are not presented, since

they can not be compared to any experimental results.

102

Table 5.6-1 Partial Computer Output for ANI C7

ANI C7

CURR RAD= 2.3 CURR RAD= 1.99278
52/R1= 962.883 52/R1= 1945.32

RIM RAD= 2.4 RIM RAD: 2.1

R/BO= .958333 R/BO= .830324
THICK= .035 THICK= 3.74494 E-2
RAD STR= 2214.63 RAD STR= 3876.59
CIR STR=~28545. CIR STR=-S2108.6
CIR STN=-3.44589 E-8 CIR STN=-.14338
THICK STN= D THICK SIN: 6.76433 E-2
RAD STN= 3.44589 E-g RAD SIN: 7.57367 E-2
EFF STN= 3.97898 E-8 EFF SIN: .143456
CURR RAD= 2.19772 CURR RAD: 1.89001
$2/R1= 1340.7 52/R1= 2315.68

RIM RAD= 2.3 RIM RAD= 2.

R’BO‘ 0915715 RIBO= 0787506
THICK= 3.57336 E-2 THICK= 3.84049 E-2
RAD STR= 2946.48 RAD STR= 4376.67
CIR STR=-41732.2 CIR STR=-56064.3
CIR STN=-4.S4908 E-2 CIR STN=-.l96325
THICK STN= 2.07434 E-2 THICK STN= .092837
RAD STN= 2.47475 E-2 RAD SIN: .103488
EFF STN= 4.55495 E-2 EFF STN= .196421
CURR RAD= 2.09533 CURR RAD= 1.78702
$2/Rl= 1627.81 S2/Rl= 2759.76

RIM RAD= 2.2 RIM RAD= 1.9

RIBO= .873056 RIBO= .74459
THICK= 3.65611 E-2 THICK= 3.94356 E-2
RAD STR= 3410.82 ’ RAD STR= 4931.74
CIR STR=-47S30.8 ' CIR STR=-S9626.5
CIR STN=-9.31958 E-2 CIR STN=-.252362
THICK $TN= 4.36377 E-2 THICK SIN: .11932
RAD STN= .049558 RAD STN= .133043

EFF STN= 9.32584 E-2 EFF STN= .252437

103

It is interesting to evaluate the effect of increment size
on the computer output. As the incremental rim displacement "Ab"
gets smaller, one expects an increase in computing time with improved
accuracy up to a certain point where computer round-off errors,inter—
fere. Table 5.7-1 shows the effect of increment size on the rim
thickness. While computer round-off errors do not appear to affect
the output, it was decided that an increment size of 0.001 inch was a
reasonable compromise between cost and accuracy.

It was expected that the theoretical analysis would show a
distinct difference in strains between the direction of rolling where
a = 0° and the radial direction where a = 45°. This difference can
be seen in Figures 5.7-2 and 5.7—3 where the thickness strain and the
radial strain histories of these two rim elements are compared. The
thickness strain curve for the a = 45° direction is above the curve
for the a = 0° direction indicating a greater degree of thickening
along the 45° direction. In addition, Figure 5.7-3 indicates that
the radial strain along the a = 45° direction is less than along the
a = 0° direction. Both of these facts are consistent with the experi—
mental evidence of ears in the direction of rolling.

Table 5.7—1 The Effect of Increment Size on
the Computed Rim Thickness for a = 0°

 

 

Increment Computed Rim Thickness Required Computer
Size at b/bo = 0.583 Time

0.1 .0460524 2 secs

0.01 .0463866 4 secs
0.001 .0464206 27 secs

0.0001 .0464240 4 mins 19 secs

 

 

 

 

 

Logarithmic Strain

.30

.20

.10

-.10

104

[J Radial Strain
() Thickness Strain
A Tangential Strain

H
H

"177::5 3;“. .3 .9 ‘
Currant Rm Faxiflaq ’ b/bo

y... ....- A- _—-- ..-. .. .2-.. ..--.-.. _. V. n...” _ ,_

p. .._., w —. , .5-“ ‘—- ._-~— ... -._. ....—.- _- -

 

Figure 5.7-1 Computed Strain History of a Rim Element at a = 0°

During the Cupping Operation

Thickness Logarithmic Strain

105

' -

lb“...

p --.-;;;—::.'u.:scra1n .. a = 0°

6

0-30“T.'7.‘“2‘“" .. ..'...-.:_._'----:.;;°i.li;;.+o;zgW4: a 9 45°

. 10.....--

 

! . I ,
4 I. ‘
o f . g.
- ‘. 0-. “4*“ L.
f .' {.A --o I ‘
l .
19
_.
b
— a" a” .— ‘v. a. “50.1 c a “a -o
- ' .
‘ t
p , ) . I
q A I V . '
1 - ..-. .._a n. .. L... .. s —-
-. .' Q
. ' .
I. . x
A l . A I
C

 
   
    
 

,- ---_1 -_...-......--- ._-L - , ,‘ . -.
. .. - J
- . . . . ..s
‘ _ . . 4 . . .
...~.-’ -'--v-u."o- .,-.--.— . . ”-14....
‘ ...—.J - ,. . . ;

  

 

Figure

 

Current Rim Position b/b0

5.7—2 Comparison of Computed Rim Thickness Strain at

a = 0° and a = 45°

Radial Logarithmic Strain

0.30

106

[3 Strain at a 0°
0 Strain a = 45°

 

 

 

Current Rim Position b/b0

Figure 5.7—3 Comparison of Computed Rim Radial Strain at

a = 0° and a = 45°

107

One simplifying assumption used in this investigation and dis-
cussed in section 5.3 was that all elements within a particular pie—
shaped region of the blank remain within that sector during the cup-
drawing Operation; another way of stating this assumption is that any
element moves radially inward during the draw. This assumption meant
that no differences in tangential (circumferential) strains would appear
when tangential strain is plotted as a function of rim position b/bo.

All three strain components for a = 0° were plotted on one
graph, Figure 5.7-1, to permit certain comparisons to be easily made.
For example radial or proportional straining exists for the rim element
if the strain ratios for the rim element remain constant during the
draw Operation. This theoretical analysis supports the contention of
radial loading. It can be seen from Figure 5.7-1 that the thickness
and radial strain components for the rim element are predicted to be
approximately equal during the draw by this theory and, therefore, each
is about one-half of the magnitude of the tangential strain. Strain
ratios (er/se and et/ee) from the computer output for the rim analysis
are reported in Table 5.7-2; these results also indicate prOportional
straining.

The results of the analysis for interior elements were com-
bined with the rim analysis and presented in Figures 5.7—4 through
5.7-8. Each of the three strain components is presented on a sep-
arate graph. To follow the strain history for a particular element
during the cup-drawing operation, one must follow a particular solid
line; for example the element originally at r0 = 2.3 inches in the

flat blank is identified by data points plotted as small circles and

108

intersecting the abscissa "r/bo" axis at the point rO/b0 = 2.3/2.4 =
0.958. As the draw progresses, this element moves inward, resulting
in an increase in the appropriate strain magnitude from zero to its
current strain at r/bo following the solid line. Since the computer
output was in the form of strain for any element at selected values

of rim radius, it was possible to connect appropriate data points on
these three graphs with dashed lines which represent the strain across
the flange at some particular value of the rim radius. From Figure
5.7*4 or Figure 5.7-5 it is seen that the dashed lines are almost
horizontal, although the thickness at the rim is slightly greater than
the thickness of interior elements at any stage of the draw. These
dashed lines support the assumption that, while the thickness of any
element increases as the draw progresses, the thickness does not vary
appreciably across the flange at any particular stage of the draw

and can be considered constant as a first approximation.

Table 5.7-2 Computed Strain Ratios for the
Rim Element vs. Rim Position

 

 

 

 

 

 

 

b/bo et/ee sr/ee
a = 0° a = 45° a = 0° 0 = 45°
0.958 0.518 0.569 0.482 0.431
.875 .521 .572 .479 .428
5.750 .523 .573 .477 .426
.625 .524 .574 .476 .425
.500 .524 .575 .475 .425

 

 

 

 

 

 

 

Thickness Logarithmic Strain

109 _

r
- -- - _ -.
" f
1 1
- ‘c"*‘w » 4 .vmunm *--y——-.--»»~. - - ---—o‘~—-- - b n ..
.£ I . .
‘ C

-.,: . ... ; TO: ."ro s 22.3
11-14-- L1-

‘ie:¥O§<*‘~f‘g *fZé-T

..o ~-o..-——-..-

 

. . . o
o - .

* V "in? a; 1,9

.ua .0- .4

O 30 , Fl — - -..w 0...... ..: E. - h--”¥f0~ia;f-‘i:j7‘ '

' .
-A I ~ . O
-wnrk .a-o—oc“ V. 9- ~- -l.:..-T~A: A --d.?.‘—:-A. - .‘ '.~ — :— ’ - - a.-- 1‘ .
, . .
. - ~ . ’I . l 40 . . <
. v, .. . .
, |.

. --"—.-v -v. . ..
A .

a
h

‘
o x .' C C
i - ‘ 4 o . >o—-‘-‘~ ,....
- a ‘ .
J . '
. I ‘.
. A ' .—.— » ‘-.afi - .
~- 0......— 1... x - I. cool- «.
’ ‘1

. 10 4 W" f‘-‘ w 7.7 —-«~r.-:U-I.D ‘O- ‘ ..-0-. .
a .

 

3
z
I
1..
I 1
9 q ‘
. '12
. 1...;
‘1? n
1;; *
A 7}
a "J ‘
o.
l
4o H
I
M

I
0
'll

.6 .7 .8 .9 1.0
Current Element Position r/b

Figure 5.7-4 Computed Thickness Strain for Flange Elements

at a = 0°

Thickness Logarithmic Strain

 

11“.; m7“
..- ~; in
—'“Y

 

.“‘r *u-"*‘1';"
-d- 4.. , . -¢- -Q .4-
1.14:..1.111+.L- It

0'.-

110

--o-¢¢ '1<$-- 0‘
T;— -. _.I'. ,.
c y. -3. ..A._

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

« u I.-.

 

44 - ‘ s — -
..‘ .' w I
- 4 u.
1-.. .. _ a .
;
I
...—-~..... . .._. —

Q ,: .2.“

- “a. . «---1.J.¢H.. 5.“...
' ‘_..f,... ' . I .I I r‘....
I I. . _, . . ‘ ‘ .
: ~ $4. 7' ; . .~

9.

U

 

1.0
Current Element Position r/b0

Figure 5.7-5 Computed Thickness Strain for Flange Elements

at a = 45°

Radial Logarithmic Strain

 

111

 

 

Computed Element Position r/bo

Figure 5.7-6 Computed Radial Strain for Flange Elements

at a = 0°

112

E] r0 = 2.4
0.". r0 = 2.3
‘0’ row: 2.1
‘7 r = 1.9
.30 _, 0.. ,
[5 r0 = 1.7

 

Radial Logarithmic Strain

 

 

 

Current Element Position r/b0

Figure 5.7—7 Computed Radial Strain for Flange Elements
at a = 45°

Circumferential Logarithmic Strain

 

113

__' . , U" r0"'='2.4
IO :0 = 2.3
' "0'" 7:30. B 52.1
v- r6513
- ‘6“,‘1-0 =."1,'7 ..

 

 

 

Current Element Position r/bo

Figure 5.7—8 Computed Circumferential Strain for Flange
Elements

VI. CUP-DRAWING EXPERIMENTS
6.1 Preliminary Remarks

Many cup—drawing eXperiments have been performed and re-
ported in the literature. Some of these eXperiments were intended
primarily to analyze and thus better understand the drawing process,
while others were performed to determine the accuracy of certain
theories. The series of tests reported here were intended to be
used for comparison with results of the theoretical investigation.

After several preliminary cupping tests, it was decided
that blanks of 4.800 inch diameter and 0.035 inch thick would be
used. The blanks were cut from commercially-produced, aluminum—
killed, low-carbon steel stock which had been carefully sheared
from the coil, so that the rolling direction remained known. The
blanks were reduced 46 percent in diameter (which is a safe maximum)
during the single draw Operation.

The die, Figure 1.1-1, was built to use a cylindrical punch
2.480 inches diameter, with a punch-profile radius of %-inch. The
die ring had a cylindrical hole 2.587 inches in diameter with a die-
profile radius of %g'inch. The ratio Of die-profile radius to sheet
metal thickness was 0.188/.035 = 5.4. The die ring and the blank-
holder were machined from tool steel, then hardened and ground. The
punch was machined from SAE 1020 steel, but was neither hardened
nor ground.

The draw Operations were performed using a double—action draw

die mounted on the bed of a 150 ton, straight-sided, single-action

114

115

Minster press, model number SC2—150-42-40—H. The press Speed used
was 60 strokes per minute and the press stroke was 4 inches. The
Minster press was equipped with a die cushion (a pneumatic cylinder
attached to the press bed) to Operate the blankholder of the die.
The blankholder force must be sufficient to prevent the formation
Of wrinkles on the flange of the cup during the draw. Preliminary
experimental work indicated that an air pressure of 13 psi in the
die cushion was the minimum to consistently produce wrinkle—free

cups. This corresponded to a blankholding force of 3980 pounds.
6.2 Producing a Polar-Grid Pattern on Sheet-Metal Blanks

In order to experimentally measure the deformation and the
strain associated with cupping of sheet steel, it is necessary to
apply a suitably-chosen network of lines to the surface of the sheet
metal blanks. Then, after cupping these blanks, the associated
deformation and strain can be computed from suitable measurements.

Many methods of applying a network of lines to sheet metal
blanks have been tried and compared [86-90]. The electrochemical
method was chosen for this work because it conveniently provides
for suitable deformation measurements with minimal effect on the
draw process. A ten-inch-square fibrous stencil, a felt pad, the
sheet metal blank, and a power unit supplying a 15 volt A.C. current
were used to etch a polar-grid pattern onto the sheet metal. The
stencil has a polar-grid pattern which is electrically—conducting,
while the remaining area of the stencil is non-conducting. The

stencils, the power unit, the felt pad and the associated chemicals

116

were purchased from the Electromark Corporation of Cleveland, Ohio.

Since it is essential that the rolling direction be carefully desig-

H II
nated on the blanks, 5%- x 54' rectangular-shaped coupons were square—

sheared from the aluminum-killed sheet steel stock so that the longer
5% inch side was parallel to the direction of rolling.
The polar-grid stencils, which are 10 inch square as pur—

chased, were carefully cut to 5% inch width so that the roll dir-

ection could be easily distinguished and marked after the etching
Operation. It was estimated that the roll direction on the blanks
is known within an error of 2“; the possible error is principally a

result of the square shear operation and the etching Operation.

H II
The process of electro-etching the 5%-x 5%- sheet coupons

as standardized for this series was as follows:

1. Clean both surfaces carefully using warm water and
Oil immersion cleaner.

2. Rinse with warm water. Dry with clean towel.

3. Clean the surface with vythene degreaser using safety-
glass cleaning tissues.

4. Lay sheet coupon on a flat piece of plywood so that
one electrode of the power unit can be attached to a
corner of the coupon.

5. Overlay the coupon with the electrolyte-soaked
stencil. Position carefully for alignment.

6. Turn the power unit to the A.C. setting, using the
maximum time-setting.

7. Apply the roller—type bench-mark with attached felt
pad (soaked with electrolyte, and connected to the
second terminal of the power unit). The actual
electro-etching takes about 8-10 seconds as the bench-
mark is rolled over the stencil and coupon completing
the A.C. circuit.

117

8. Remove stencil and electrode from the sheet metal
coupon. Rinse with warm water. Dry with clean
towels.

9. Apply polarized Oil to both sides of the coupon to
avoid oxidation of the coupons.

10. Mark each coupon to show the roll direction.

After the rectangular coupons had been electro—etched,
circular blanks had to be cut such that the center of the polar grid
coincided with the center of the circular blank. Later, during the
cupping Operation, the blank had to be centrally positioned on the
punch so that a symmetrically-drawn cup was produced.

During the preliminary cupping experiments, many difficul-
ties arose, one Of which was maintaining coincidence of the blank
and the punch centers during the draw operation. A second diffi-
culty was in producing a blank edge which was square with the blank
surface. Both of these difficulties were resolved by accurately
reaming a 0.125 inch diameter hole at the center of the polar grid,
and then turning a group of these blanks on the lathe after first
mounting them on an eighth—inch diameter spindle. Since the strain
in the bottom of the cup was not being investigated, this procedure

proved to be a convenient compromise.

 

6.3 Measurement of Grid Spacing

After the 4.800 inch diameter blanks had been prepared
(with the polar grid pattern applied), suitable measurements were
made to facilitate the required deformation and strain calculations
at chosen angles to the direction of rolling. Since the aniso-

trOpic computer analysis was carried out at 0° and at 45° to the

118

direction of rolling, the only eXperimental verification needed
was along these two directions.

Because of the symmetry in the sheet metal, the 0° dir-
ection and the 180° direction are both in the roll direction.
Since both are equivalent to the roll direction, each blank was
studied to choose the particular direction most convenient from a
measurement point-Of-view. Again, because of symmetry, any one of
four directions (45°, 135°, 225°, and 315°) can be chosen for
measurement at 45° to the roll direction; again a choice was made
based on ease Of measurement for each blank.

The experimental procedure for measuring grid distances
on the blanks (and also on the flat portion of the cup flange after
drawing) utilized a Jones and Lamson Optical Comparator and Measur—
ing Machine, Model FC-l4. This machine incorporates micrometer—
measuring equipment to indicate table displacements; the least count
of the micrometers is 0.0001 inch. The comparator magnified the
light reflected from the grid surface fifty times, which resulted
in actual line widths of 0.005 inch appearing on the screen as .250
inch. Every fifth grid line is double width, approximately 0.010
inch wide, and hence is magnified to appear .500 inch wide on the
screen.

The grid distances were measured from the center line of
one grid line to the center line of the neighboring grid line.
This meant that the hair-line on the screen had to be lined up to

coincide with the center of the magnified grid lines. Experience

119

indicated that measurement errors were caused by (l) the variation
of quality of the etched lines, and (2) variation in judging the
location of the center of the magnified grid line. After locating
a particular grid centerline and taking a micrometer reading
(least count of 0.0001 inch), I have returned to the same grid
centerline within 30 seconds with a typical error of 0.0003 inch.
A measure of the variation of these grid distances was determined
by measuring each chordal grid distance four times and then calcu-
lating the average and range of these four readings; each radial
grid distance was measured five times, after which the average

and range were computed.

According to Keeler [91], the accuracy of the applied
electrochemically-etched grid is equal to or better than a grid
obtained by scribing. Keeler further states that 0.1 inch dia-
meter grid circles can be produced with a diameter accuracy of 1%
without the stress concentration factors associated with a scribed
grid pattern. Pearce and Drinkwater [92] consider this question
of accuracy. They conclude that an accuracy of 12% is reasonable
to expect using a paper stencil. It should be noted that both of
these references discussed the accuracy of producing a grid of
specified dimensions. If the desired grid Spacing were 0.1 inch,
then the actual grid spacing might be as low as 0.098 inch or as
high as 0.102 inch assuming 12% accuracy. The radial increment on
the polar grid pattern used in the experimental cupping tests

reported in this thesis was supposed to be 0.1 inch. Since the

120

sample averages of the radial readings are indicative of the actual
radial grid dimensions, a quick run-down of the experimental find—
ings indicates only a very few radial grid readings outside the
expected i2% accuracy.

The experimental data collected in the series of chordal
measurements indicated an average range, for the 156 chordal samples
of four readings in each sample, of .000641 inch. Using Table I
on page 155 of Moroney [93] or Table D on page 614 of Duncan [94],
the factor d = 2.059 for a sample size n = 4 can be used to estimate
the standard deviation for individuals from the average range.

O - R7d - (.00064l)/(2.059) - .000312 inch is the estimated value
for the standard deviation of the population of individuals from
which the samples of four were taken.

Using this computed estimate of the standard deviation
for individuals, the Student "t" test can be used to estimate the
maximum expected difference between the sample mean and the "true"
p0pu1ation mean for any desired confidence limits. This is a
significant calculation since the sample means were used for the
deformation and the strain computations. 95% confidence limits
were used, which implies that only once in twenty times one would
expect a larger difference in the means. Figure 81, page 230, of
Moroney [93] showed a value of t = 3.3 for three degrees of freedom
(one less than the sample size). This information estimated the

maximum eXpected difference in the means to be

|§ - I] = (two/JR) = (3.3)(.ooo312//Z) = .00052 inch.

121

Applying this same procedure to the 122 samples of radial
measurements (sample size was 5), the estimated standard deviation
for individuals was 0 ='R/d = .00034. The maximum eXpected differ—

ence of the means was then computed to be
If - El = (tug/fr?) = (3.3)(.00034//§) = .0005 inch

for the samples of radial measurements.

The Student "t" test indicated that the sample averages
approximated the true grid dimensions within :,0005. Since the
sample averages for the radial dimensions varied from .0979 inch
up to a maximum of .1014 inch, it was obvious that the experi~
mental measure of the radial grid dimensions on the sheet metal
blank was necessary for the sake of improved accuracy, rather than

assuming them to be 0.1 inch.
6.4 Exgerimental Cup Drawing

In order to verify the computer-aided theoretical analysis
of the cup-drawing process, it was decided to take a series of blanks
with suitably etched polar grid patterns and partially draw each
blank a different amount. Since the computer print—out from the
theoretical analysis gave stress and strain results for radial dis—
placements of the rim at 0.1 inch increments, it seemed desirable
to get corresponding experimental strain data. However, the earing
phenomenon resulted in rim elements at 45° to the roll direction
moving in faster than elements at 0° and 90°. The cupping tests
were planned so that the average of the rim displacements at 0°

and 45° to the roll direction would correspond to the values of

122

the computer print-out from the theoretical analysis. A numerical
indicator, mounted on the press ram, showed the distance from a
zero reference up to the slide face at bottom dead center position
of the crank. The least count of the indicator is 0.001 inch and
it was possible to reset the slide to a predetermined reading within
31.002 inch without difficulty. A series of partially-drawn cups
were produced during the preliminary investigation using the stan-
dard 4.800 inch diameter blanks and the standardized draw die, so
that the depth Of draw varied from a fully-drawn cup down to a
partially-drawn cup where the punch had barely deformed the blank.
In each case the indicator reading was noted as well as the average
rim diameter of the cup. The diameter of the rim at 0° to the
rolling direction was averaged with the diameter of the rim at

45° to the rolling direction to get the average rim diameter. A
computer program then fitted a second degree equation to this data
with an index of determination of 0.99897. A second computer pro—
gram then predicted the ram indicator setting for pre—selected
average rim diameters of the partially—drawn cups. This procedure
proved to be convenient and useful.

Another concern during the draw operation was the type and
method of lubrication. The principal requirement for the work re—
ported here was that the etched lines remain clearly visible after
drawing. Lubrication literature is very extensive in the field of
metal working Operations. Lloyd [95] reported in part 2 of a five—
part article that one of the earliest examples of dry lubrication

was the use of plastic polymer films. Wilson [96] reported the

123

results of a series of deep-drawing tests using several different
lubricants. Rao [97] reported on the use of polyethylene for
lubrication during sheet-metal drawing. He discussed alternative
ways of applying the lubricant to the sheet metal.

After considerable preliminary testing, it was found that
0.002-inch-thick polyethylene film provided excellent protection
for the etched grid lines. A sandwich was produced by placing a
sheet metal blank between two pieces of the polyethylene film and
heat sealing the edges of the two polyethylene sheets with a heated
wire. This encapsulated coupon then required only a drop of light
oil on either side to give excellent protection and lubrication to
the coupon.

A series of nine cups were drawn using the standardized
conditions discussed, and using the ram settings specified by the

computer program.
6.5 Procedures Used to Compute Strain from Experimental Data

Radial and tangential strain computations, based on experi-
mental data, were needed for elements in the flange of partially-
drawn cups. Strains at 0° to the roll direction and at 45° to the
roll direction were determined for comparison with the results of
the theoretical study.

The tangential strains measured the change in length of a
circumferential arc subtending a 2° central angle. For this small
central angle, the chordal distances and the arc distances were

equal for a desired accuracy of 0.0001 inch.

124

Radial and chordal distances for each material element in
the flange (of the partially-drawn cup) at 0° and at 45° to the
direction of rolling had to be experimentally determined. The
corresponding distances in the blank (before drawing) then permitted
logarithmic radial and tangential strains to be computed for each
element. Each radial distance was measured five times and each
chordal distance was measured four times to permit statistical
evaluation.

Computer programs for both radial strain and tangential
strain were devised using the FORTRAN language. The data were
placed in separate data programs which could be compiled with the
associated main program as desired. Computations were performed on
the G. E. 265 Time-Sharing Computer System. The two main programs,
RADIAL and CHORD, and their associated flow charts are shown in
Figures 6.5-l to 6.5—4. The flow charts were constructed following
the suggestions of Moursund [98].

The main programs, such as RADIAL, each required at least
two data programs. The first data program had to list radial grid
dimensions for the blank along a particular radius, and the second
must list corresponding grid dimensions after the cupping Operation.
Four typical data programs associated with draw number 1, and used
with the main program RADIAL, are shown in Table 6.5-1. RADlA
program gives radial grid dimensions on the blank for draw number 1
at 0° to the direction of rolling, while RADlB gives the corre-
sponding dimensions after drawing. RADlC program lists the radial

grid dimensions on the blank for draw number 1 at 45° to the

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

AVECHORD(J) + [CHORDW(J)+CHORDX(J)+CHORDY(J)+CHORDZ(J)]/4
RANGE(J) + [Maximum Value of Chord]-[Minimum Value of Chord]
RANGE(J) I f[CHORDW(J),CHORDX(J),CHORDY(J),CHORDZ(J)]

 

 

 

 

AVECHORD2(J) + AVECHORD(J) 1

I

PRINT CHORDW(J),CHORDX(J),CHORDY(J),CHORDZ(J)
J RINT AVECHORD(J),RANGE(J),STRAIN(J)

W

IAVECHORD3(J) + AVECHORD(J) l

1 l

STRAIN(J) + LOG[AVECHORD3(J)/AVECHORD2(J)]

(:7 STOP ::) K + K + 1

FIGURE 6.5-1 FLOW CHART FOR CHORD PROGRAM

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

CH?

10
20
30
40
50
60
62
63
64
70
80
90
g 100
110
120
130
140
150
160
I65
170
180
190
210
220
230
250
26?)
270
275
310
320
325
326
327
328
329
330
340
350

126

40

0:2
DIMENSION CHOROXCIQ): CHOQDYCIQ): CHJRDZCIQ)
DIMENSION CHORDW(12): AVECHUROCIQ): RANG€(12)
SIGQANGE=O
DIMENSION AVECHOQOZCIQ): AVECHOROS‘IQ): STRAINCIQ)
2 OO 9 M=102
3 OO 4 L=IaN
STRAIN‘L)=0.0000
4 CONTINUE
PRINT" CHORDW CHOROX CHOROY CHOROZ AVECURO RANGE STRAIN"
5 DO 70 K=102
10 DO 20: I=IpN
READ:'CHORDW(I)
READ: CHOROX(I): CHOROYCI): CHOROZCI)
20 CONTINUE
PRINT
DO 60: J=laN
RANGECJ)= MAXIFCCHORDXLI): CHORDYCJ): CHORDZCJ):CHORDW(J))-
+MINIF(CHOROX(J): CHORDYCJ): CHORDZ(J): CHORDWCJ))
SIGRANGE-‘SIGRANGE + RANGECJ) '
AVECHORDCJ)=(CHORDXCJ)+CHOQDY(J)+CHORDZ(J)+CHOROWCJ))/4
IF (K-I) 35:35:45
35 AVECHORDZ(J)=AVECHORD(J)
GO TO 30
45 AVECHORDS(J)=AVECHORDCJ)
STRAIN(J)=LOG(AVECHOQDH(J)IAVECHORDQ(J))
3O PRINT 5%: CHOROW(J): CHOROX(J): CHORDY(J): CHUROZ(J)
+3 AVECHOQD(J): RANGECJ): STRAINCJ)
50 FORMATC8F7od)
60 CONTINUE
PRINT
7% CONTINUE
PRINT"SIGRANGE"
PRINT: SIGQANGE
PRINT
PRINT
PRINT
9 CONTINUE
END
SDATA CORD7A: CORO79: CORO7C: COQD7O

FIGURE 6.5-2 CHORD PROGRAM

127

 

 

 

 

 

 

<: START A:>———*:\ READ N M + N - 1 h———.i K + 1

 

 

 

 

 

 

 

 

 

 

\ READ V(I),.W(I), x(T), Y(I), 2(1) I + 1
l . YES 7
I + I + 1 I

 

 

 

 

N0

 

J + 1 HL—7

l

 

 

 

 

DELV(J) + ABS[V(J + 1) - V(J)]
DELW(J) + ABS[W(J + 1) — W(J)]
DELX(J) + ABS[X(J + 1) - X(J)]
DELY(J) + ABS[Y(J + 1) - Y(J)] A...
DELZ(J) + ABS[Z(J + 1) — Z(J)]

RANGE(J) + [Largest minus smallest value of Delta (J)]
AVERAGE(J) + [DELV(J)+DELW(J)+DELX(J)+DELY(J)+DELZ(J)1/5

 

 

 

 

 
 

 

 

 

AVERAGE2(J) + AVERAGE(J) ‘ '

PRINT DELV(J),DELW(J),DELX(J),DELY(J),DELZ(J)
PRINT RANGE(J),AVERAGE(J),STRAIN(J)

A

 

 

 

 

 

AVERAGE3(J) + AVERAGE(J)

 

 

 

J + J + l J < M

 

 

 

 

 

STRAIN(J) + LOG[AVERAGE3(J)/AVERAGE2(J)] NO

 

 

YES

 

 

K + K + l

< STOP NO

FIGURE 6.5—3 FLOW CHART FOR RADIAL PROGRAM

 

 

 

 

 

RAD

10
12
15
20
25
3O
35
M
41
43
45
46
47
48
49
50
55
6O
65
75
80
85
90
95
100
105
110
115
120
125
140
145
147
155
160
161
163
164
165
166
167
170
175
180
185
187
190
193
195
200
205

128
IAL

N=9
DIMENSION STRAINCIZ)
DIMENSION VIIQ): WC12)
DIMENSION DELVCIQ): DELW(12)
DIMENSION X(16): Y(16): Z(16)
DIMENSION DELXIIS): DELYCIS): DELZ(15)
DIMENSION RANGE(20)-
DIMENSION AVERAGE(QO)
DIMENSION AVERAGEZCED): AVERAGE3(29)
SIGRANGE=O
M=N'1
2 DO 9 L=1:9
PRINT"- V W X Y Z RANGE AVE STRAIN"
3 DO 4 I=1:M
4 STRAIN‘I)=O:OOOD
5 D0 70 K=102
19 D0 20 I=I:N
READ: VII): N(I)
20 READ: XCI): YCI): Z(I)
PRINT
DO SO J=1:M
DELVCJ1=AOSCVCJ+1)-V(J))
DELWCJ)=ARS(W(J+1)-H(J))
DELX€J1=ARS(X(J+1)-XCJ))
DELY(J)=ARS(Y(J+1)-Y(J))
DELZCJ1=AOS(Z(J+1)-Z(J))
RANGECJ)=MAXIF(DELV(J): DELW(J): DELXCJ): DELY(J): DELZ(J))'
+ MIN1F(DELV(J): DELW(J): DELX(J): DELY(J): DELZCJ)’
AVERAGE(J)=(DELVCJ)+DELW(J)+DELX(J)+DELYCJ)+DELZ(J))l5
SIGRANGE=SIGRANGE + RANGECJ)
IF (K-I) 35:35:45
35 AVERAGE?(J)=AVERAGECJ)
GO TO 55
6M FORMATC8F7o4)
45 AVERAGE3CJ)=AVERAGE(J)
STRAINCJ)=LOGCAVERAGE3(J)/AVERAGE2(J))

55 PRINT 60: DELV(J): DELW(J): DELX(J): DELY(J): DELZ(J):
+RANGE(J): AVERAGE(J): STRAIN(J)

SO CONTINUE ‘

PRINT

PRINT

70 CONTINUE

PRINT"SIGRANGE"

PRINT: SIGRANGE

PRINT

PRINT

PRINT

PRINT

9 CONTINUE

END

TDATA RADIA: R4014. RADIO. :4110

FIGURE 6.5-4 RADIAL PROGRAM

129

TABLE 6.5-1 TYPICAL COMPUTER PROGRAMS

RADIA

560
570
580
590
6001
610
, 620
630
640

.1570:
.2564:
.3558:
045503
0554fl3
065483
.7544:
085343
.9561:

RADIB

560
570
580
590
600
610
620
630
640

0G7443
018193
0287G3
.3918:
049483
.6006:
.7035:
.8049:
091083

RADIC

560
570
580
590
600
610
620
630
640

.1572:
025773
035663
045653
.5557:
065643
.7562:
085593
095253

RADIO

560
570
580
590
600
610
620
630
640

.0000:
016453
.2G77’
03105:
041323
.5170:
061573
072g23
031963

.1568:
025643
.3557:
045513
055453
.6549:
.7538:
085343
095623

057413
.1812:
028733
039233
049553
.6007:
070343
080533
091123

.1572:
.2574:
.3565:
045623
055563
065653
075583
.8558:
095233

-.0005:

.1044:
029743
031@83
041323
051693
061843
.7201:
081973

FORIMDIALIMTA

.1557:
.2550:
035473
045383
055253
.6537:
075293
.8522:
.9543:

097453
.1815:
.2870:
039193
049543
.6008:
07g313
089493
091123

.1583:
025833
035723
045663
055623
065733
075673
085633
095363

000523

.1098:
021323
031623
.4183:
.5229:
0624Q3
072573
.3254:

01G7®3
020593
030553
043473
050323
069463
079493
.8034:
09QSS3

0Q7483
018083
028663
039173
049513
066063
07Q323
080513
.9196:

.0058:
0IQ6M3
.2056:
03QSQ3
04fl423
.5057:
.6048:
.7045:
080183

009953

011463
021763
032073
.4232:
052713
062853
073G33
.8300:

.1067
.2057
.3055
.4046
.5031
.6044
.7035
.8031
.9058

.0751

.1813
.2870
.3919
.4950
.6008
.7032
.8052
.9106

.0054
.1054
.2052
.3052
.4043
.5053
.6047
.7043
.8023

.0080
.1128
.2161
.3192
.4215
.5252
.6269
.7284
.8284

130

TABLE 6.5-2 COMPUTER PRINT-OUT FOR RADIAL
STRAIN - DRAW NUMBER 1

RADIAL
v w x Y z RANGE AVE STRAIN
.0994 .0996 .0993 .0989 .0990 .0097 .0992 0.0000
.0994 .0993 .0997 .0996 .0998 .0005 .0996 0.0000
.m992 .0994 .0991 .9992 .9991 .0003 .0992 9.0090
.0990 .0994 .0987 .0985 .0985 .0309 .0988 0.0300
.1008 .1004 .1012 .1914 .1013 .0010 .1010 0.0090
.0996 .0989 .0992 .0994 .m991 .0007 .0992 z.@000
.9990 .0996 .0993 .0994 .0996 .0006 .9994 @.@0@@
.1027 .1028 .1021 .1921 .1027 .0007 .1025 0.0000
.1066 .1971 .1070 .1060 .1062 .0011 .1066 .0714
.1060 .1061 .1055 .1058 .1057 .0006 .1058 .0610
.1043 .1047 .1049 .1051 .1049 .0034 .1049 .0557
.1030 .1035 .1035 .1034 .1031 .0005 .1033 .0443
.1058 .1052 .1054 .1055 .1058 .0006 .1955 .0438
.1029 .1027 .1023 .1026 .1024. .0006 .1026 .0331
.1914' .1019 .1017 .1019 .1020 .0006 .1018 '.4239
.1959 .1059 .1064 .1055 .1054 .0010 .1058 .0321
v w x Y z RANGE AVE STRAIN
.1005 '.1@02 .1090 .1002 .1300 .9905 .1092 0.0000
.0989 .0991 .0989 .0996 .0998 .0909 .0993 0.9900
.0999 .0997 .0994. .0994 .1000 .0006 .0997 0.0000
.0992 .0994 .0996 .0992 .0991 .mmns .0993 0.9090
.1007 .1009 .1011‘ .1015 .1010 .0008 .1019 0.0000
.0998 .0993 .0994 .0991 .0994 .4097 .w994 0.0000
.0997 .1000 .0996 .0998 .0996 .0404 .9997 0.0000
.0966 .0965 .0973 .0972 .0980 .9215 .9971 0.0000
.1045 .1049 .1046 .1051 .1048 .0006 .1048 .0449
.1032 .1039 .1934 .1030 .1933 .0004 .1032 .0387
.1028 .1034 .1030 .1031 .1031 .0006 .1031 .0335
.1027 .1024 .1021 .1925 .1923 .0006 .1024 .0307
.1038 .1037 .1046 .1m39 .1037 .mnm9 .1039 .0283
.1017 .1015 .1m11 .1014 .1017 .0006 .1015 .0297
.1015 .1017 .1917 .1018 .1015 .0003 .1016 .0189
.0994 .0996 .9997 .0997 .1000 .0006 .0997 .0260

131

direction of rolling, while RADlD lists the corresponding grid
dimensions after cupping.

A typical computer print-out page is shown in Table 6.5-2
for the RADIAL program and the radial data programs for draw num-
ber l. The top eight rows are for the eight radial grid dimensions
on the blank at 0° to the roll direction. Since each radial grid
dimension was measured five times, these are listed in the first
five columns. The sixth column lists the range for the five dim-
ensions while the seventh column lists the arithmetic average for
the five radial grid dimensions. The second group of eight rows
represents the corresponding radial grid dimensions at 0° to the
roll direction on the partially-drawn cup; the logarithmic radial
strain for each element is listed.

The two bottom groups of eight rows each are for the radial
dimensions at 45° to the direction of rolling. The first of these
two refer to the dimensions on the blank, while the last one refers

to the corresponding dimensions on the flange of the cup.
6.6 Results

The results of this experimental phase of the investigation
are summarized in graphical form as Figures 6.6-l to 6.6-4. On
each graph, dash lines were used to show the distribution of strain
across the flange for a particular partially—drawn cup. Any solid
lines follow the strain history of a particular element during the
process of drawing a flat circular blank into a cup. Some of the
rim element points, shown by small squares, have been identified

with a particular partially-drawn cup to facilitate comparisons.

132

The abscissa values for each graph were computed as di—
mensionless values of the current position of the element by
dividing the current radial position of the element by the rim
radius of the blank. Each partially-drawn cup was measured on the
Optical comparator to determine the rim diameter in the rolling
direction and at 45° to this direction. Then the current radius
for each element was determined from the current rim radius and
the current radial grid dimensions.

For example, the rim diameter for draw number 1 at 0° to
the roll direction was measured to be 4.6451 inches, which gave a
current rim radius of 2.3225 inches. At this rim radius, Table
6.6—1 lists the circumferential logarithmic strain as -.0525, which
was plotted on Figure 6.6-2 against a current position of the
element of r/bo = 2.3225/2.4 = 0.97.

The outermost element of this partially—drawn cup has a
radial logarithmic strain of 0.0321 as listed in Table 6.5-2.

This strain was computed from the change in length of a radial
line approximately 0.1 inch long extending in from the rim. The
current position of this element was computed by subtracting half
of the current radial length of this line from the current rim
radius, i.e. r = 2.3225 - 0.5(.1058) = 2.2696 inches.

Figure 6.6—5 gives the depth of draw vs. the current rim
position b/bo based on measurements taken from the nine partially-
drawn cups. This graph shows that, at any particular depth of
draw, the rim element at a = 45° has been displaced (radially

inward) a greater distance than the corresponding rim element

133

at a = 0°. This figure also gives an indication of the earing
which occurred during the cupping eXperiments.

Figure 6.6-6 is a plot of the strain ratio, er/ee, as a
function of the draw number for the element r0 = 2.35 inches, which
corresponds to the rim element (see Figure 6.6-5 to relate draw
number to either depth of draw or to b/bo). This figure was drawn
to determine if proportional straining occurred during the cup-
drawing process. Although the experimental evidence is limited,
it does give support to the hypothesis of proportional straining

during the draw.

CHORD

CHORDW

.0567
.0595
.0624
.0659
.0700

.0738,

.0775
.0804
.0841

.0514
.0542
.0584
.0617
.0661
.0699
.0725
.0761
.0789

CHORDN

.0563
.0588
.0629
.0665
.0697
.0732
.0773
.0800
.0839

.0536
.0577
.0607
.0645
.0676
.0707
.0736
.0770
.0797

134

TABLE 6.6-l COMPUTER PRINT-OUT FOR
TANGENTIAL STRAIN - DRAW NUMBER 1

CHOQDX

.0562
.0589
.0628
.0654
.0687
.0726
.0767
.0803
.0837

.0503

.0539

.0587
.0622
.0662
.0691
.0731
.0763
.0800

CHORDX

.0562
.0594
.0639
.0669
.0702
.0729
.0766
.0804
.0837

.0535
.0572
.0609
.0639
.0671
.0709
.0740
.0783
.0811

CHORDY

.0566
.0601
.0629
.0661
.0704
.0735
.0774
.0811
.0847

.0504
.0540
.0586
.0626
.0657
.0693
.0728
.0762
.0802

CHORDY

.0556
.0589
.0628
.0665
.0698
.0737
.0765
.0800
.0838

.0530
.0575
.0604
.0644
.0676
.0708
.0779
.0809

CHORDZ

.0557
.0595
.0627
.0656
.0698
.0732
.0773
.0805
.0839

.0509
.0541
.0586
.0624
.0659
.0689
.0733
.0760
.0801

CHORDZ

.0567
.0595
.0628
.0663
.0706
.0731
.0767
.0806
.0845

.0533
.0571
.0607
.0642
.0672
.0708
.0734
.0782
.0813

AVECDRD

.0563
.0595
.0627
.0657
.0697
.0733
.0772
.0806
.0841

.0507
.0540
.0586
.0622
.0660
.0693
.0729
.0761
.0798

AVECORD

.0562
.0591
.0631
.0665
.0701
.0732
.0768
.0802
.0840

.0533
.0574
.0607
.0642
.0674
.0708
.0739
.0778
.0807

RANGE

.0010
.0012
.0005
.0007
.0017
.0008
.0010

.0011
.0003
.0003
.0009
.0005
.0010
.0008
.0003
.0013

RANGE

.0011
.0007
.0011
.0006
.0009
.0008
.0008
.0006
.0008

.0006
.0006
.0005
.0006
.0005
.0002
.0012
.0013
.0016

STRAIV

0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

'01638
-o@961
-.0681
“OWSSI
-.0553
‘0“558
‘0“573
-.0565
-.0525

STRAIV

0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000

-.052@
-.0305
“00392
-.0352
-.0393
-.0337
“00322
-.0304
-.0392

Radial Logarithmic Strain

135

“1'

 

 

 

 

V

   

.9 1.0

Current Element Position r/bo

Figure 6.6-1 Experimental Radial Strain for

Flange Elements at a = 0°

-o 30
-.20

-010

Circumferential Logarithmic Strain

1

.. . . .-A, .1. . , ... . _ .
. J ... , . ' ; . , .
l 1 .- +| - i .. . u I- ‘1 - ~
- . ‘~.— . 1. —. ~O.-C.Y."HI-. .. -g-o o-l... .
...l ..7 *1 . v.51... .
r ‘ ‘ ‘ I ‘ .. . . ' I
‘ >1‘ , - .

136

2.4
2.3
2.1
1.9
1.7

#40013

e o
.. .....— .’ '-~ ...

‘ .

 

1 ...
1 I
4 .. . _ _
.4-. 1.- .-,. -_-<"
.. F ‘ v ; 4 “‘ '
__.r. . .
§.,I - , .

 

 

 

 

  

.‘ .‘w L
I U ‘1 9 LA - l L anaa-wyn—l . ' n
. r.',.“-¢ ; .‘ . ... 1"“.".+'.~ . , my .
a .'I:. .3 i J ...... ~ .3 A.» ‘f—‘j fl
:17.‘1-.. 7* ; . .7 ‘ A ‘
. , . .1. H14 . . ._ - - 4‘ .H ..‘..A..: . .7- -.
wan-1.4-. *0.‘ -f. 41...“-.. B-
..a.117...‘11.. ..--.JJ.. .. ..1 L1. ! ‘7
. .. J .. .. .‘ .,i... '1. ._.,,_ 11 .I . .,
“4.. I 1 4. if. .' f.
‘ -:1. 1‘ . O .4 ..-_
I L
I 1 .. r

 

 

1.0

Current Element Position r/b0

Figure 6.6-2 Experimental Circumferential Strain for Flange Elements

at a = 0°

Radial Logarithmic Strain

137

.30

.20

    

      
 

'v 'f
‘

1.;L_. 7 ..1.. _‘ .
.45039015k§,41@ u-,

.10

4*? Draw Na. 2

 
 

.6 .7 .8 .9 1.0
Current Element Position r/b0

Figure 6.6-3 Experimental Radial Strain for Flange Elgnents
at a = 45°

138

n
_- .. -"
__._,_, ..

'7 ,... r-.

u'.
m.
_
u...
fl.
a.

p.

‘_.

. .
‘1
‘8‘..-

I .. fDMW 10.6

‘ l
I

l,»

 

-050
-.40

0
3
_

-.20
-.10

nwmuum Uganuwumwoq amauaoummaboufiu

0

Current Element Position r/b

Figure 6.6—4 Experimental Circumferential Strain for Flange

Elements at a = 45°

Depth of Draw - Inches

139

1.50

1.00

.50

 

.6 .7 .8 7.9 1.0
Current Rim Position b/bO

Figure 6.6—5 Depth of Draw vs. Rim Position

Strain Ratio er/e6

 

140

 

 

 

 

 

 

 

 

 

 

 

 

 

"g .. .~ , >J—L5153. >. ..I

.40 e ' mi-m—a-w-T-L‘m -~ .-

 

 

 

 

'- . ‘7 -A f . AI
' . I . .. t. n '. ;.. . ,
u , ‘ I t
. 4‘ . . z. . . ‘
- r .—< A o ‘ I ‘ 1 I I ‘ ‘. ,
. 20 ‘4 ‘— .. - a... -,.. ..A J 4' Max-.4-.— -3..~-..4 ..-.r_-- ”To. --—-
a . . . -- o p' .L. I . . - ’ '
Y ..' . , L . l . . . , .. . . t , . g
" n ,. . . A a q I
10 'u , I. .1 . ,. .»
O p. ~01- _ ‘-. . .‘-~‘ . _. - ' - . --_ . .1
: o - I. - q o - s 4 .
§ A g 8 . -t s... o l r. . '1.
b i
. A A A A i ' A
v

 

1 2 3 4 5 6 7 8 9

Draw Number

Figure 6.6—6 Experimental Strain Ratios vs. Draw Number for
the Element r0 = 2.35

VII. SUMMARY AND CONCLUSIONS

7.1 Preliminary Remarks

Chapter 5 reports on the theoretical strain analysis of the
cup-drawing process, while Chapter 6 reports the strain results from
the experimental cupping tests. Section 7.2 of this chapter compares
. the theoretical results from Chapter 5 to the experimental results
from Chapter 6. No comparison of the results of this investigation
with results reported by other investigators was made, since it was
believed that a comparison of the theoretical to experimental results
from this investigation would be more direct and meaningful. The
specific conclusions resulting from this study and recommendations

relative to future possible studies are.reported in section 7.3.

7.2 Comparison of the Theoretical Strain Field with the Experi-
mentally-Determined Strain Field

As was stated in the resultsof section 5.7, the computed
strains from the theoretical investigation did show a marked difference
between the rolling direction where a = 0° and the direction a = 45°,
and the relative magnitudes did correlate qualitatively with the ex—
perimental evidence of ears at a = 0° and a = 90°. Similarly, the
graphical display of the experimentally-determined strain field,
Figures 6.5-1 through 6.5-4, indicated strain differences with a change
in orientation. The theoretical analysis does not give any indication
of the relationship of the computed strains to the depth of the draw,
since it does not provide a relationship between depth of draw and rim

radius. In order to compare quantitatively the results of Chapters 5

141

142

and 6, the theoretical and the experimental strain histories for the
rim elements at a - 0° and a = 45° are plotted as a function of the
current rim position b/bo in Figures 7.2-1 and 7.2-2.

Figure 7.2-1 is a plot of logarithmic circumferential strain
for the rim element ro - 2.4 inches as a function of the current rim
position b/bo. The theoretical strain is plotted as one curve identi-
fied by small square boxes and a solid line. The least-squares method
was used to get the curves for the experimental data. As was pointed
out in Chapter 5, the assumption of strictly radially—inward motion
of each element implies that no difference in circumferential strain
exists for different angles a, if plotted against the current rim
position. It can be seen from Figure 7.2—1 that this assumption was
only approximately correct.

Figure 7.2-2 is a plot of logarithmic radial strain for the
"rim" element as a function of the current rim position b/bo. The
radial strain at the "rim" was experimentally determined by measuring

the length of the radial line between r = 2.3 and r0 = 2.4 inches

0
at successive partial draws; hence the experimental radial strain for
the rim was associated with the mean initial radius of the element

r0 = 2.35. The curves from experimental data are "best—fit" curves.
In this illustration, the theoretical strain histories for the rim
elements (r0 = 2.4 inches) at a = 0° and a = 45° are both identified
by solid lines, the line for a = 0 is marked with small squares to
differentiate it from the line for a = 45° which is identified with

small circles. It is clear from looking at Figure 7.2-2 that the

theoretical radial strains are smaller in magnitude than the

143

experimental curves. This is probably a result of the experimental
difficulties associated with determining Hill's anisotrOpic yield func—
tion by the direct method. Possibly some of this discrepancy might

have resulted from the arbitrary choice of u = 0.06 as the coefficient
of friction between the sheet metal and the die components. Another
possible source of error might have resulted from the plane stress
assumption near the end of the draw, when the entire blankholding force
is carried by a reduced flange area. Additional errors were very likely
introduced by the approximations in the analysis, but it is believed
that the major error was in the yield function determination by the

direct method.
7.3 Conclusions and Recommendations

The following conclusions are based on results from the theo-
retical and the experimental study:

1. The method used to include planar anisotropy in the
theoretical analysis did result in stress and strain
fields of the type associated with the 0° and 90° earing
which occurred during the experimental study.

2. The radial strain fields for the 0° and 45° directions
indicated smaller magnitudes from the theoretical analysis
than was evident from the experimental analysis.

3. The strain—ratio method of measuring anisotropy resulted
in an indication of greater anisotrOpy for the aluminum-
killed steel than the direct method.

4. Rim elements experience proportional straining during the
cup-drawing Operation.

5. The method of electro etching grid lines on the blanks
used in the cup—drawing experiments was not completely
satisfactory. The lines were not consistently distinct
and uniform; this resulted in certain inaccuracies in
the experimental data.

144

The following recommendations are suggested for use in future

cup—drawing investigations of this type:

1.

2.

The strain-ratio method should be used to determine the
anisotropic yield function.

The linearized approximation to the anisotropic yield
function should not be used. This approximation was not
essential in this study and probably saved less time than
anticipated.

A total-deformation theory might be used in future analyses,
since the assumption of prOportional straining is supported
by both theoretical and experimental results of this in-
vestigation.

Other methods of imprinting grid lines on the sheet metal
blanks should be considered including the technique de-
veloped by the printed circuit industry. This is mentioned
on page 199 of the article by Palmer [90].

Circumferential Logarithmic Strain

-.30

-.20

-.10

145

      
    
 

‘ t
.47.“- '.. ‘

‘ am me...

-...

 

-O‘-~ r..- ’—
o. .4
i
‘. .-
c O
\
.—s~ -
. L.
‘.-_4~‘.-«-. Av.-
4‘. .. .
. .._¢-
3 ..
I
.. -4-
.l
4.
‘ 6.1 v- «.4 o

 

Mini}. “~41" :....‘:‘-;._;i - .
*‘ ‘iiiftftt‘; -: ~ W ~
, .1

 

 

 

 

 

 

 

 

~v
Hi 4. A w.
4 ,_ . ,1
o; o 1 .4', . ,r
—... ->+ «a... .o—a-é-q -....
3.. §-~—7¢—--> .
.1 A- 0 ,§,
.7 .- . .3
44 l . . .-
“u—ow a Hod...~ vv—a.- .
fi>~e .
o , .
y-“ :—.u W I .M4- ‘-n .“|-—.-—- H—O A< Q a4“ ‘-
. 'n.“ V - «Q a . ‘.
. ' ,
~. .1 . r . .
' 7..-. I . I. ‘4 ‘-.
' ... . .l—A 4.. .
p—w , 1 #— OMI- ck» “Md-l- J..— M
. 1 4+4; .‘. ._ -.. .
o -. 4 '
r . ML 3 i -.,. . +1-1“.
r‘.}. .. ,. .r
- J... - ,— .. - .. — - .‘_. -5“ gar-4-- -
. L. FY 1-4 . qa‘ . J- .. .
e5. . :l-A‘ a ‘ , .. '. .
i 4. I ‘ .
A A AA A
Y,
o . ‘

1.0
Current Rim Position b/b0

Figure 7.2-1 Comparison of Theoretical and Experimentally-
Determined Circumferential Rim Strains

Radial Logarithmic Strain

146

'40 " --—'"'"," --'*.--‘-. Experimental a 0°
4— a..---.......k Experimental (1 45°

. * H... 5h 1 Theoretical a 8 0°
’ 45°

:: ' '. - ~ 0 , Theoretical a

 

 

 

 

.6 .7 .8 .9 1.0
Current Rim Position

Figure 7.2-2 Comparison of Theoretical to EXperimentally-
Determined Radial Rim Strains.

APPENDIX

ANI

100
110
120
125
126
130
140
150
160
162
164
166
168
190
200
210
215
216
220
230
240
250
260
270
280
390
400
410
510
520
530
535
540
550
560

APFHHHX

C7

LET RO=203
LET R1=RO
LET 803204

-LET L330810/27050

LET L1310027

LET TO=cO35

LET T330

LET N=-.005

FOR X1=BO T0 o499*80 STEP N

IF X1>=2o2 THEN 930

IE Kl>2o0 THEN 940

IF x1>1.3 THEN 950

IE X1<=108 THEN 960

LET R3 SQRCXI'Z ' (BO'Q-RO'2)*T0/T6)
LET D1=R1-R

GGSUB 510

LET L3=27050+60700*(L0G(RO/R))70518
LET L4=L*L3

LET T7=(14073*52‘70905*L3)/(14069*L3'70288*52)
LET T8=‘01*T7/R

LET R1=R

LET E=o0001

FGR I=204 T9 1.199 STEP -o1

IF ABSCXl-I)<E THEN 700

NEXT I

LET T3=T3+T8

NEXT X1

STGP

LET $5=-06*3980/(3o14159*X1*T5)
LET 56=L*27050*L@G(X1/R)+(1'L1)*S4
DIM ZCll)

LET M313

LET H2=(X1-R)/A

LET X2=R

FER J30 T0 M

Figure Arl Computer Program for Stress and Strain History

of Flange Element r0 = 2.3 at o = 0°

147

 

Ii lll‘ll‘l‘lll‘l

 

148

Figure A-l (cont'd.)

565
570
575
580
585
590
595
600
610
620
630
640
650
660

'y 690

700

701

702
703
705
706
707
710
715
720
725
730
740
750
760
770
780
785
790
850
930
931
932
934
940
941

942
944
950
951

952
954
960

961

‘962

964
990

LET 02=80?2-X1t2*T6/T0
LET S7=.S*LBG(T6/TO + C2/K2t2)
LET ZCJ)=L*60700*(S7'.518)/X2
LET X2=X2+H2
NEXT J
LET $8=0 '
FER J=l TE M STEP 2
LET $8=58+Z(J)
NEXT J '
LET S9=0
FER J=2 TE (M-l) STEP 2
LET S9= S9+Z(J)
NEXT J
LET S2= SS+S6+H2*(Z(0)+4*38+2*S9+Z(M))/3
RETURN
IF R1<1o1 THEN 990
LET T4=TO*EXP(T3)
LET T1=-L@G(R0/R1)
LET T2=~T1~T3
LET Sl=L1*S2-L4
LET P1=(Tl-T2)?2 + (T2-T3)t2 + (T3-T1)t2
LET T=SQRCP1*2/9)
PRINT "CURR RAD="R1
PRINT "52/R1= "52/R1
PRINT "RIM RAD="X1
PRINT "R/30="R1/2o4
PRINT "THICK="T4
PRINT "RAD STR="S2
PRINT "CIR STR="SI
PRINT "CIR STN="T1
PRINT "THICK STN="T3
PRINT "RAD STN="-T1-T3
PRINT "EFF STN="T
PRINT
60 T0 390 _
LET TS=2o91002E-3*K112-2o14851E-2*X1+6o9802SE-2
LET T6=3480502E-3*X1f2-2.S4SS6Ee2*X1+7.4176SE-2
LET S4=-179-315*K1f2+621.644#X1-392.097
GE T0 190
LET TS=.003475*X1t2-2.39585E-2*X1+7.25094E-2
LET T6=;00342tx132-.023655*X1+7o 20786E-2
LET 54:215. 3*K1?2-1138-59*X1+1570. 48
60 T0 190
LET TS=4-44002E-3*X1*2-2o78091E-2*X1+7.63506E-2
LET T6= 4 34002E- 3*X1f2- 2. 73261E- -2*X1+7.57407E-2
LET S4=39S-3*X1'2-1856o64*X1+2286 58
60 T0 190
LET TS= 8o 26488E-3*x1t2-4-0809SE-2*K1+8-73855E-2
LET T6= 7 21768E- 3*X1t2- 3 72618E- -2*X1+8o 43126E-2
LET S4= 1442.58*K1t2-S436.57*Xl+5342 59
GB T0 190

SEND

149

AREA

10 DIM F(IS). xc151

20 LET N31

30 FDR I=0 Ta N

40 READ x11). F(1)

50 NEXT 1

60 LET A=0

70 PER I=1 TD N

80 LET A=A+.5*(F(I)+F(I-l))*(K(I)-K(I-l))
90 NEXT 1

100 PRINT "RIM RAD="x<01
105 PRINT "CURR RAD="K(N)
110 PRINT "AREA ="-A

115 PRINT

120 60 T0 30 .

13o DATA 1.3. 931.838

131 DATA 1.16094. 9408.55
140 DATA 1.4. 835.404

141 DATA 1.26675. 7412.96
150 DATA 1.5. 754.613

151 DATA 1.37174. 5947.71
160 DATA 1.6. 686.12

161 DATA 1.47613. 4840.18
‘70 DATA 107) 6270445

171 DATA 1.5801. 3981.83
180 DATA 1.8. 576.72

181 DATA 1.68377. 3301.72
190 DATA 1.9. 532.507

191 DATA 1.78702. 2759.76
200 DATA 2.0. 493.687

201 DATA 1.89001. 2315.68
210 DATA 2.1. 459.377

211 DATA 1.99273. 1945.32
220 DATA 2.2. 428.369

221 DATA 2.09533. 1627.81
230 DATA 2.3. 401.587

231 DATA 2.19772. 1340.7
240 DATA 20493770046

241 DATA 2.3. 962.883

300 END

Figure A-2 Auxiliary Computer Program AREA

AUTHIK

10
35
37
39
40
41
55
-S6
57
59
60
61
75
76
77
79
80
81
95
96
97
99
100
101
115
116
117
119
120
121
135
136
137
139
140
141
155
156
157
159

LET N=2
PRINT "KIM RAD=204"
PRINT "A=-O35"
PRINT
LET 3:3057808E-2
LET 3=3+3057336E‘2
PRINT "RIM R003203"
LET 9=RIN
PRINT "3="B
PRINT
LET C=3o66198E'2
LET C=C+3o65611E'2
PRINT "RIM RAD=2-8"
LET C3C/N
PRINT "C="C
PRINT
LET 0:3675214E'2
LET D=D+3o74494E-2
PRINT "RIM RAD=2-1"
LET D=DIN
PRINT "D="D
PRINT
LET E=3o84925E‘2
LET E=E+3084049£-2
PRINT "RIM RQD=2-D"
LET 838/1“
PRINT "E="E
PRINT
LET F=3o95418E-2
LET F=F+3o94356E'2
PRINT "RIM RQD=169"
LET E=EIN
PRINT "E="E
PRINT
LET 6:4006799E-2
LET G=G+6040551
PRINT "RIM R4031-8"
LET G=GIN
PRINT "G="G
PRINT

Figure A—s

150

160
161
175
176
177
179
180
181
195
196
197
199
200
201
215
216
217
219
220
221
235
236
237
239
240
241
255
256
257
259
300

LET H=4~19196E‘2
LET H=H+40I7632E-2
PRINT "RIM R403167"
LET H=H/N

PRINT "H="H

PRINT

LET I=4-32765E*2
LET I=I+4630347E'2
PNINT "RIM RAD=1-6"
LET I=I/N

PRINT "I="I

PRINT

LET J=004477

LET J=J+4o45324E-2
PRINT "RIM RAD=105"
LET J=J/N

PRINT "J="J

PRINT

LET K=oO46424

LET K=K+6046127
PRINT "RIM RAD=1-4"
LET K=KIN

PRINT "K="K

PRINT

LET L=4o82692E'2
LET L:L+4078942£-2
PRINT "RI“I RAD=1-3"
LET L=L/N

PRINT "L="L

PRINT

END

Auxiliary Computer Program AVTHIK

BIBLIOGRAPHY

10.

ll.

BIBLIOGRAPHY

Willis, J., and Blade, J. C., "A Proposed Earing Test for Sheet Metal,"
Sheet Metal Industries, Volume 43, Number 468, April 1966, pp 316-320.

Atkinson, M., and'Maclean, I. M., "The Measurement of Normal‘Plastic
Anisotropy in Sheet Metal," Sheet Metal Industries, Volume 42;
Number 456, April 1965, pp 290-298.

Atkinson, M., "Assessing Normal Anisotropic Plasticity of Sheet Metals,"
Sheet Metal Industries, Volume 44, Number 479, March 1967, pp 167-178.

Roster, W., "Beobachtungen an Kupfer Zum gesetzmassigen Gefugeaufbau
nach der Rekristallisation," Zeitschrift fur Metallkunde, Volume 18,
Number 4, 1926, pp 112-116.

Coler, Frhrn V., and Sachs, G., "Walz-und Rekristallisationstextur
regular 2 flachenzentrierter Metalle," Zeitschrift fur Physik,
Parts 1 and 2: Volume 41, 1927, pp 873-906.

Parts 3, 4, and 5: Volume 56, 1929, pp 477-502.

Schwartzbart, H., Jones, M. H., and Brown, W. F., Jr., "Observations
on Bauschinger Effect in Copper and Brass," National Advisory
Committee for Aeronautics, Research Memorandum E51D13, pp l-37.

Lode, W., "Versuche fiber den Einfluss der mittleren Hauptspannung
auf das Fliessen der Metalle Eisen, Kupfer und Nickel,"
Zeitschrift ffir Physik, Volume 36, 1926, pp 913-939.

Mises, R. Von, "Mechanik der Plastischen Forménderung von Kristallen,"
Zeitschrift Fur Angewandte Mathematik und Mechanik, Volume 8,
Number 3, June 1928, pp 161-185.

Mises, R. von, "Mechanik der festen K6rper im plastisch-deformablen
Zustand," Gattingen Nachrichten, 1913, pp 582-592.

Sommer, M. H., "Versuche fiber das Ziehen von Hohlkfirpern,"
Maschinenbau, Volume 4, Number 24, December 3, 1925, pp 1171—1178.

Eksergian, C. L., "The Plastic Behavior of Metal in Drawing," The

Metal Industry, Volume 30, 1927; pp 405-408, 433-436, 459—462,
483-484.

151

12.

13.

14.

15.

16 .'

17.

18.

719.

20.

21.

22.

23.

24.

152

Geckeler, J. W., "Plastisches Knicken der Wandung von Hohlzylindern
und einige andere Faltungserscheinungen an Schalen und Blechen,"
Zeitschrift ffir Angewandte Mathematik und Mechanik, Volume 8,
Number 5, October, 1928, pp 341-352.

Linicus, W. and Sachs, 6., "Die Bedeutung des Faltenhalters beim
Tiefziehen," Werkstattstechnik, Volume 26, June 15, 1932,
pp 233-235.

Herrmann, L. and Sachs, G., "Untersuchungen fiber das Tiefziehen,
"Metallwirtschaft, Volume 13, 1934; Number 40, pp 687-692;
Number 41, pp 705-710.

Sachs, C., "New Researches on the Drawing of Cylindrical Shells,"
Proceedings of the Institution of Automobile Engineers, Volume 29,
1934-35, pp 588-600.

Swift, H. W., "Drawing Tests for Sheet Metal," Proceedings of the
Institution of Automobile Engineers, Volume 34, Session 1939-40,
pp 361-432.

Fukui, S., "Researches on the Deep—Drawing Process," Scientific
Papers of the Institute of Physical and Chemical Research,
Volume 34, 1938, Number 849, pp 1422-1527; Volume 35, 1939,
Number 885, pp 373e384.

Swift, H. W., ”Plastic Strain in an Isotropic Strain-Hardening
Material," Engineering, Volume 162, 1946, pp 381-384.

Swift, H. W., "Plastic Bending Under Tension," Engineering, Volume
166, 1948; pp 333-335; pp 357-359.

Seed, E. C., and Swift, H. W., "An Experimental Crank Press,"
Proceedings of the Institution of Mechanical Engineers, Volume 163,
1950, pp 125-132.

Chung, S. Y., and Swift, H. W., "Cup-Drawing from a Flat Blank,"

Proceedings of the Institution of Mechanical Engineers, Volume
165, 1951, pp 199-223.

Lubahn, J. D., and Sachs, C., "Bending of an Ideal Plastic Metal,"
Transactions of the American Society of Mechanical Engineers,
Volume 72, Number 2, 1950, pp 201-208.

Swift, H. W., "The Mechanism of a Simple Drawing Operation,"
Engineering, Volume 178, Number 4627, 1954, pp 431-435.

Swift, H. W., "Plastic Instability Under Plane Stress," Journal of
the Mechanics and Physics of Solids, Volume 1, Number 1, 1952,
pp 1-18.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

153

Brown, W. F., Jr., and Sachs, C., "Strength and Failure Characteristics
of Thin Circular Membranes," Transactions of the American Society
of Mechanical Engineers, Volume 70, Number 3, 1948, pp 241-249.

Hill, R., "A Theory of the Plastic Bulging of a Metal Diaphragm by
Lateral Pressure," Philosophical Magazine, Volume 41, 1950,
pp 1133-1142.

Hill, R., The Mathematical Theory of Plasticity, Oxford University
Press, London, 1950.

Jackson, K. L., "Determination of Plastic Strains in Sheet Metal,"
Sheet Metal Industries, Volume 26, Numbers 267, 268, 1949,
pp 1447-1451, pp 1715-1719.

Cook, M., "Directional Properties in Rolled Brass Strip," Journal of
the Institute of Metals, Volume 60, 1937, pp 159-172.

Phillips, A., and Dunkle, H. H., "Directional Properties in Rolled
and Annealed Low Carbon Steel," Transactions of the American
Society for Metals, Volume 23, 1938, pp 398-408.

Baldwin, W. M., "Effect of Rolling and Annealing upon the Crystallography,
Metallography and Physical Properties of Copper Strip," American
Institute of Mining.and Metallurgical Engineers, Technical
Publication Number 1455, 1942, pp 591-611.

Palmer, E. W., and Smith, C. 8., "Effect of Some Mill Variables on

the Earing of Brass in Deep Drawing," Transactions of the
American Institute of Mining and Metallurgical Engineers,
Volume 147, 1942, pp 164-182.

Burghoff, H. L., and Bohlen, E. C., "Directional Properties of
68-32 Brass Strip," Transactions of the American Institute of
Mining and Metallurgical Engineers, Volume 147, 1942, pp 144-152.

Baldwin, W. M., Howald, T. S., and Ross,A. W., "Relative Triaxial
Deformation Rates," American Institute of Mining amd Metallurgical
Engineers, Technical Publications Number 1808, 1945, pp 86-109.

Wilson, F. H., and Brick, R. M., "Textures, Anisotropy and Earing
Behavior of Brass," American Institute of Mining and Metallurgical
Engineers, Technical Publication Number 1803, 1945, pp 173-202.

Klingler, L. J., and Sachs, C., "Plastic Flow Characteristics of

Aluminum-Alloy Plate," Journal of the Aeronautical Sciences,
Volume 15, Number 10, October 1948, pp 599-604.

Jackson, L. R., Smith, K. F., and Lankford, W. T., "Plastic Flow in
Anisotropic Sheet Steel," American Institute of Mining and
Metallurgical Engineers, Technical Publication Number 2440, 1948,
pp 1-15.

154

38. Lankford, W. T., Snyder, S. C., and Bauscher, J. A., "New Criteria
for Predicting the Press Performance of Deep Drawing Sheets,"
Transactions of the American Society for Metals, Volume 42, 1950,
pp 1197-1232. '

39. Hug, H., Siebel, C., and Buser, P., "Untersuchungen fiber das Wesen der
Zipfelbildung," Metall, Volume 6, 1952, pp 579-586.

40. Aust, K. T., and Morral, F. R., "Directional Properties of 28 Aluminium,"
Journal of Metals, Volume 5, Number 3, 1953, pp 431-436.

41. Siebel, G., "De la Formation de Cornes dans L'Emboutissage des Toles
D'Aluminium," Congres International de L'Aluminium (Paris), Volume 2,
1954, pp 127-137.

42. Chevigny, R., "Formation des Cornes a L'Emboutissage de L'Aluminium
et Moyens d'y Remédier," Congres International de L'Aluminium (Paris)
Volume 2, 1954, pp 109-125.

43. Trapied, G., "Anisotropie des Laminés en Aluminium Pur," Congrés
International de L'Aluminium (Paris), Volume 2, 1954, pp 139-150.

44. McEvily, A. J., and Hughes, P. J., "An Experimental and Theoretical
, Investigation of the Anisotropy of 3S Aluminum-Alloy Sheet in the
Plastic Range," National Advisory Committee for Aeronautics,
Technical Note 3248, 1954, pp 1-45. '

45. McEvily, A. J., "Analysis of Ear Formation in Deep-Drawn Cups,"
National Advisory Committee for Aeronautics, Technical Note
3439, 1955, pp 1-7. ,

46. Thorley, R. T., and Tucker, G. E. G., "The Control of Earing in
Aluminum and its Alloys," Journal of the Institute of Metals,
Volume 86, 1957-58, pp 353-361.

47. Tucker, G. E. C., "Texture and Earing in Deep Drawing of Aluminum,"
Acta Metallurgica, Volume 9, 1961, pp 275-286.

48. Blade, J. C., "The Influence of Constitution on the Earing of
Commerical-Purity Aluminum," Journal of the Institute of Metals,
Volume 90, 1961-62, pp 374-379.

49. Blade, J. C., and Pearson, W. K. J., "The Requirements for a
Standard Earing Test," Journal of the Institute of Metals,
Volume 91, 1962-63, pp 10-18.

50. Whitely, R. L., and Wise, D. E., "Relationship among Texture, Hot
Mill Practice and the Deep Drawability of Sheet Steel, "AIME
Fourth Mechanical Working Conference on Flat Rolled Products,
Chicago, 1962, pp 47-63.

51.

52.

53.

54.

55.

S6.

57.

58.

59.

60.

61.

62.

63.

64.

65.

155

Siebel, G., and Mack, K., "Untersuchungen fiber den Einfluss der
Prfifbedingungen auf das Ergebnis der Zipfelprfifung," Metall,
Volume 17, Number 10, 1963, pp 1024-1027.

Wright, J. C., "Influence of Deep-Drawing Conditions on the Earing
of Aluminum Sheet," Journal of the-Institute of Metals, Volume 93,

Wright, J. C., "The Phenomenon of Baring in Deep Drawing," Sheet
Metal Industries, Volume 42, Number 463, 1965, pp 814-831.

Bramley, A. N., and Mellor, P. 3., "Some Strain-Rate and Anisotropy
Effects in the Stretch-Forming of Steel Sheet," International

Journal of Machine Tool Design and Research, Vol. 5, 1965, pp 43-55.

Hill, R., "A Theory of the Yielding and Plastic Flow of Anisotropic
Metals," Royal Society of London, Proceedings A, Volume 193, 1948,
pp 281-297.

Bourne, L., and Hill, R., "0n the Correlation of the Directional
Properties of Rolled Sheet in Tension and Cupping Tests,"
Philosophical Magazine, Volume 41, 1950, pp 671-680.

Chiang, D. C., and Kobayashi, S., "The Effect of Anisotropy and Work-
Hardening Characteristics on the Stress and Strain Distribution
in Deep Drawing," Journal of Engineering for Industry, Volume 88,
Series B, Number 4, November 1966, pp 443-448.

Wilson, D. V., "Plastic Anisotropy in Sheet Metals," Journal of the
Institute of Metals, Volume 94, 1966, pp 84—93.

Lubahn, J. D., and Felgar, R. P., Plasticity and Creep of Metals,
John Wiley and Sons, Inc., New York, 1961.

Hoffman, 0., and Sachs, 6., Introduction to the Theory of Plasticity
for Engineers, McGraw-Hill Book Company, Inc., New York, 1953.

 

Nadai, A., Theory of Flow and Fracture of Solids, McGraw-Hill Book
Company, Inc., New York, Volume I, 1950.

 

Fukui, 8., Yuri, H., and Yoshida, K., "Analysis for Deep-Drawing of
Cylindrical Shell based on Total Strain Theory and Some Formability
Tests," Aeronautical Research Institute, University of Tokyo,
Report Number 332, June 1958, pp 43-75.

Woo, D. M., "Analysis of the Cup-Drawing Process," Journal Mechanical
Engineering Science, Volume 6, Number 2, 1964, pp 116-131.

Alexander, J. M., "An Appraisal of the Theory of Deep Drawing,"
Metallurgical Reviews, Volume 5, Number 19, 1960, pp 349-411.

Warwick, J. 0., and Alexander, J. M., "Prediction of the Limiting
Drawing Ratio from the Stress/Strain Curve," Journal of the
Institute of Metals, Volume 91, Part 1, 1962, pp 1-10.

66.

67.

68.

69.

70.-

71.

72.

73.

74.

75.

76.

77.

78.

156

Moore, G. C., and Wallace, J. F., "The Effect of Anisotropy on
Instability in Sheet-Metal Forming," Journal of the Institute
of Metals, Volume 93, Part 2, 1964, pp 33-38.

Budiansky, B., and Wang, N. M., "0n the Swift Cup Test," Journal of
the Mechanics and Physics of Solids, Volume 14, 1966, pp 357-374.

Woo, D. M., "On the Complete Solution of the Deep-Drawing Problem,"
International Journal of Mechanical Sciences, Volume 10, 1968,
pp 83-94. '

Mir, W. A., "Drawability of Sheet Metals with Reference to the
Influence of Anisotropy, Strain-Hardening, Blank Holding Methods
and Hydrostatic Pressure," Doctoral Disseration submitted to
the University of Waterloo, Waterloo, Ontario, April 1967.

Whitely, R. L., "The Importance of Directionality in Drawing Quality
Sheet Steel," Transactions of the American Society for Metals,
Volume 52, 1960, pp 154-169.

Mir, W. A., and Hillier, M. J., "Cup Drawing from an Anisotropic
Blank," Paper Number 68-WA/Prod-2, Paper presented at the Winter
Annual Meeting, New York, December 1-5, 1968, of the American
Society of Mechanical Engineers.

Fenn, R. W., and Clapper, R. B., "Evaluation of Test Variables in the
Determination of Shear Strength," Proceedings, American Society for
Testing Materials, Volume 56, 1956, pp 842-858.

Breindel, W. W., Seale, C. L., and Carlson, R. L., "Evaluation of a
Single-Shear Specimen for Sheet Material," Proceedings, American
Society for Testing Materials, Volume 58, 1958, pp 862-868.

Yen, C. 8., "Stress Distribution in Single-Shear Sheet Specimens,"
ASTM Special Technical Publication Number 289, Symposium on
Shear and Torsion Testing, Sixty-third Annual Meeting Papers,
American Society for Testing Materials, 1960, pp 15-20.

Instron Engineering Corporation, Canton, Massachusetts, Operating
Instructions for the Instron Tensile Testing Instruments,
Manual #10-29-1.

Bradley, W. A., "Discussion on Stress Distribution in Single-Shear
Sheet," ASTM Special Technical Publication Number 289, Symposium
on Shear and Torsion Testing, Sixty-Third Annual Meeting Papers,
American Society for Testing Materials, 1960, pp 21-25.

Bramley, A. N., and Mellor, P. B., "Plastic Flow in Stabilized Sheet
Steel," International Journal of Mechanical Sciences, Volume 8,
February 1966, pp 101-114.

Wylie, C. R., Jr., Advanced Engineering Mathematics, Second Edition,
McGraw—Hill Book Company, Inc., New York, 1960.

 

79.
80.

81.

82.

83.

84.

85.

86.

87.

88.

89.

90.

91.

92.

93.

94.

157

Prager, W., An Introduction to Plasticity, Addison-Wesley Publishing
Company, Inc., Reading, Massachusetts, 1959.

Johnson, W., and Mellor, P. B., Plasticity for Mechanical Engineers,
D. Van Nostrand Company Limited, London, 1962.

Thomsen, E. C., Yang, C. Y., and Kobayashi, S., Mechanics of Plastic
Deformation in Metal Processing, The Macmillan Company, New York,
1965. '

Svensson, N. L., "Anisotropy and the Bauschinger Effect in Cold
Rolled Aluminum," Journal Mechanical Engineering Science,
Volume 8, Number 2, June 1966, pp 162-172.

Mendelson, A., Plasticity: Theory and Application, The Macmillan
Company, New York, 1968.

Fung, Y. C., Foundations of Solid Mechanics, Prentice—Hall, Inc.,
Englewood Cliffs, New Jersey, 1965.

Ludwik, P., Elemente der tecknologischen Mechanik, Springer,
Berlin, 1909.

 

Brewer, G. A., and Glassco, R. B., "Determination of Strain Distribution
by the Photo-Grid Process," Journal of the Aeronautical Sciences,
Vol. 9, No. 1, November 1941, pp 1-7.

Zaat, J. H., "The Production of Line Networks on Sheet Metal," Sheet
Metal Industries, Vol. 34, No. 366, October 1957, pp 737-740.

Radchenko, A. T., "Investigation of the Plastic-Strain of Specimens
by Means of Applied Grids" Translated from Zavodskaya
Laboratoriya, Vol. 30, No. 2, February 1964, pp 218-221.

Keeler, S. P., "Use of Grid Systems for Strain Determinations,"
Paper submitted to the International Deep-Drawing Research Group -
Working Group I - Processes, London, June 1964.

Palmer, D. R., "Electro-Chemical Marking of Grids in Autobody Press
Working," Sheet Metal Industries, Vol. 46, No. 491, March 1968,
pp 198-204.

Keeler, S. P., "Determination of Forming Limits in Automotive
Stampings," S.A.E. Paper No. 650535, Mid-Year Meeting, Chicago,
Illinois, May 1965.

Pearce, R. and Drinkwater, I. C., "Some Aspects of Electrochemical
Marking," Sheet Metal Industries, Vol. 45, No. 498, October 1968,
pp 751-755. -

Moroney, M. J., Facts from Figures, Penguin Books Inc., Baltimore
Maryland, 1956.

 

Duncan, A. J., Quality Control and Industrial Statistics, Richard
D. Irwin Inc., Chicago, Illinois, 1952.

 

95.

96.

97.

98.

158

Lloyd, D. H., "Lubrication for Press Forming - 2," Sheet Metal
Industries, Vol. 43, No. 468, 1966, pp 307-315.

Wilson, D. V., "Lubrication and Formability in Sheet-Metal Working,"
Sheet Metal Industries, Vol. 43, No. 476, 1966, pp 929—944.

Rao, V. S., "Polythene as a Lubricant for Deep-Drawing," Sheet Metal
Industries, Vol. 44, No. 486, 1967, pp 673-678.

Moursund, D. G., How Computers Do It, Wadsworth Publishing Company,
Inc., Belmont, California, 1969.

 

8
m
A
R
nl.
L
Y
.1.
5
R
E
w
N 3
U
E
T
A
T"2

IIHIHMIIIU