Biaxial friction model for sheet metal forming

Material Information

Biaxial friction model for sheet metal forming
El-Jaam, Bassam A
Publication Date:
Physical Description:
viii, 91 leaves : illustrations ; 29 cm

Thesis/Dissertation Information

Master's ( Master of Science)
Degree Grantor:
University of Colorado Denver
Degree Divisions:
Department of Mechanical Engineering. CU Denver
Degree Disciplines:
Mechanical engineering


Subjects / Keywords:
Sheet-metal -- Formability ( lcsh )
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )


Includes bibliographical references.
General Note:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Department of Mechanical Engineering.
Statement of Responsibility:
Bassam A. El-Jaam.

Record Information

Source Institution:
|University of Colorado Denver
Holding Location:
Auraria Library
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
24526427 ( OCLC )

Full Text
Bassam A. El-Jaam
B.S., University of Colorado, 1989
A thesis submitted to the
Faculty of the Graduate School of
University of Colorado in partial fulfillment
of the requirement for the degree of
Master of Science
Departement of Mechanical Engineering

This thesis for the Master of Science
degree by
Bassam A. El-Jaam
has been approved for the
Departement of
Mechanical Engineering
S' 7- 9/
John A. Trapp

El-Jaam, Bassam A. (M.S., Mechanical Engineering)
Biaxial Friction Model for Sheet Metal Forming
Thesis directed by Professor James C. Gerdeen
This thesis deals with the application of plasticity
theory to sheet metal forming problems. In particular,
a new approach called a 'Biaxial Friction Model' using the
Tresca yield condition is developed. The primary objective
of this model is to provide an accurate thickness
distribution and to solve for forming strains. Different
factors affecting the limiting strains in a biaxially-
loaded sheet are also studied. In addition, the analysis
examines different possible modes of deformations. These
modes require appropriate stress boundary conditions and
yield criteria.
The theoretical development of this model is based on
the membrane theory of shells of revolution of arbitrary
shape, Coulomb friction, the Tresca yield criteria the
potential flow rule, the 'Free Equilibrium Shape,' and
finally the plastic work concept. Also, an empirical
expression relating generalized stress and strain, which is
known as the power law is adopted.
Since it is possible to accurately solve for thickness
variation by the new approach, the reliability of the

strain results has been consequently improved. A
hemispherical punch stretched and drawn sheet for a range
of friction values and boundary conditions is studied and
compared with experiment and other results.
For constant thickness strain, it can be shown when
assuming that the meridional and hoop stresses are equal in
order to satisfy the symmetry condition at the center of
the punch, that the surface has to be frictionless. It is
also shown that for higher friction values, instability
occurs at a smaller angle and the hoop strain distribution
reached a higher peak. Furthermore, boundary conditions at
the center of the punch (namely thickness strain) altered
the angle where necking takes place. The lower the
thickness strain is at the center, the further the necking
point moves from the center. Finally, it is found that
materials with higher strain hardening index gave a higher
overall strain distributions and a higher depth of draw.
The form and content of this abstract are approved. I
recommend its publication.
James C. Gerdeen

Acknowledgements ................................ viii
1. INTRODUCTION............................. . 1
General Overview ............................. 3
Thesis Format ................................ 7
2. LITERATURE REVIEW ............................ 8
3. THEORY........................................12
Geometrical Mapping Technique ............... 12
Friction .................................... 15
Membrane Theory of Shells ................... 17
Geometric Relations ............. 19
Equilibrium of a Shell Element ......... 21
Axisymmetric Loads ..................... 23
Plastic Work ................................ 24
Tresca Criterion............................. 25
Flow Rule.....................................27
Problem Formulation ......................... 30
4. RESULTS.......................................55
5. CONCLUSIONS...................................82
A. Geometrical Methods Formulation..............84
B. Numerical Error Evaluation...................85
C. Uniform Thickness........................... 87
BIBLIOGRAPHY ......................................... 89

1.1a Schematic view of the stretching of a sheet metal
with a hemispherical punch ....................... 4
1.1b Schematic view of the drawing of a sheet metal
with a hemispherical punch ....................... 4
3. la Displacement u and w and coordinates rQ and r . 14
3.1b Change in elemental length ...................... .14
3.2a External pressures in the punch contact region 18
3.2b External pressures in the die contact region . 18
3.3 A surface of revolution and curvilinear
coordinate and 9..............................20
3.4 Meridian of a shell of revolution ................22
3.5 Modal representation of Tresca yield condition . 26
3.6 Yield stress analysis of Tresca yield condition . 28
3.7 Variation of strain hardening in different
3.8a Geometry of a sheet metal before deformation . 33
3.8b Geometry of a sheet metal after deformation . 33
3.9 Strain vector representation for different modes 35
3.10 Strain vector representation for region 1
(sphere mode 1) 37
3.11 Strain vector representation for region 2
(unsupported mode 1)..............................43
3.12 Strain vector representation for region 2
(cone mode 2) 46
3.13 Strain vector representation for region 3
(toroid mode 2) 49
3.14 Strain vector representation for region 3
(toroid Mode 3) 51

Strain vector representation for region 4
(plate mode 2)..................................54
4.1 Strain distribution for a lubricated case
(mu=. 05) .....................................56
4.2 Major strain comparison (mu=.05) 57
4.3 Minor strain comparison (mu=.05) 58
4.4 Geometry for a lubricated case (mu=.05) .... 61
4.5 Strain distribution for a dry clamped case
(mu-. 2) 62
4.6 Major strain comparison (clamped mu=.2) .... 64
4.7 Minor strain comparison (mu=.2) . .......65
4.8 Geometry for a dry clamped case (mu=.2) .... 66
4.9 Major strain distribution for different n ... 68
4.10 Minor strain distribution for different n ... 69
4.11 Geometry for different n....................70
4.12 Major strain distribution for different friction. 71
4.13 Minor strain distribution for different friction. 73
4.14 Geometry for different friction .................. 74
4.15 Major strain distribution for different depth
of draw ........................................ 76
4.16 Minor strain distribution for different depth
of draw ........................................ 77
4.17 Geometry for different depth of draw........78
4.18 Strain distribution for a draw-in case......80
4.19 Geometry for a draw-in case........................81

I would like to thank Dr. J. C. Gerdeen for his knowledge,
patience and support as I am very grateful for his
suggestions and wisdom. I also would like to extend my
thanks to Dr. Gerdeen for the great numerous works he has
done over the years which were very helpful in getting a
good understanding of sheet metals in forming. I would
like to thank the Budd Company Stamping & Frame group for
supporting this project.
I would like to thank Ron Lewton for his assistance in the
I would like to thank my uncle Jamal Delati for his
encouragement and understanding.
I would like to thank my parents and my family for their
love and prayers and a special thanks to my aunt Jacque
Delati for her unmeasurable love and support throughout my
And I thank the creator who makes all things possible.

The behavior of metal in forming is complex. Factors
such as tool-to-workpiece interaction, properties of
product, production processes, and material deformation
make it difficult to get a complete strain solution in
sheet metal problems. So it is challenging to relate,
individually these various well-known factors governing the
material behavior in press forming with press formability
or deformation behavior of sheet metals. Although, it may
be hard to describe such involved relations among many
factors, a number of approximate methods have been
suggested with varying degrees of approximation and
realization [2].
In order to present a fundamental understanding of
sheet metals in forming and justify why the thesis shall be
written from an engineering point of view, here is an
excerpt from H.W. Swift's lecture title "On the Foot-Hills
of Plastic Range" [3].
In this broad field of development there is work for
many hands; for the pure scientist and the applied
scientist, for the mathematician, the metallurgist, and the
engineer....for his own purpose the engineer needs to be
able to assess the possibilities of various processes of
formation and fabrication in relation to materials which
are economically his own field...the
metallurgist has to accept a role complementary to the
The mathematical theory of sheet metal forming is

phenomenological in nature and attempts to formalize and
put into useful form the results of macroscopic
experiments. The ultimate goal is to arrive at one unified
theory of plasticity which will both explain the material
behavior and provide the engineer and scientist the
necessary tools for practical applications [4].
From an engineering standpoint, the most worthwhile
analyses would tend to be those which predict trends in
deformation behavior and provide needed information for
proper design and control of sheet metal processes. There
would, however, seem to be definite advantages to reduce
the number of assumptions that need to be made in solving
for forming strains. In this thesis, there was a great
concern to avoid the need for any prior assumptions that
are necessary for our model.
The attractiveness of the model formed in this paper
was that it does not require any assumed quantity, in
particular thickness distribution, therefore it can be
started at a step closer to the final solution than
'Geometric Mapping'[G.M. ] technique, developed by J.C.
Gerdeen [5]. Another feature in the thesis is that the
model uses the Tresca yield condition which has not been
used much previously, and which is easier to formulate than
the Von Mises because of its linearity. The linear form of

the Tresca condition enables determination of inability
General Overview
A new approach called the 'Biaxial Friction
Model' has been developed. The objective of this model is
to predict strains in biaxially loaded sheets. This
technique is designed to accurately solve for the thickness
distribution in a pressed part knowing only the friction
and boundary conditions. The method is simply a total
approach which employs the G.M. formulation, theory of
shells of revolutions, the power law, plastic work,
friction, and the free equilibrium shape method
simultaneously. It is a combination of these different
laws to obtain a comprehensive solution that involves all
aspects of forming strains. Finally, two basic sheet-
metal forming processes, namely, the stretching of a sheet
with hemispherical head punch and deep drawing of a sheet
with a hemispherical-head punch, are solved, as shown in
The current theoretical work in the thesis assumes
that the material is strain hardening. To apply the
theory of strain hardening to sheet metal forming, an
empirical expression is adopted, which is experimentally
determined and known as the power law, relating the
generalized stress and strain.

Pig. 10. (a0 Schematic view of the stretching of a
sheet metal with a hemispherical punch
Pig. ll (b) Schematic view of the drawing of a
sheet metal with a hemispnerical punch

The G.M. method is a method that assumes a known
thickness distribution or uniform thinning or strain
distribution to solve for the forming strains. The
thickness strain can be iterated on to match friction and
boundary conditions [5].Since an assumption is needed to
start the method, its success is dependent on the closeness
of this assumption which requires some user expertise.
The G.M. method is extended in the presently formulated
Biaxial Friction model by making use of its general
differential form.
In addition, the analysis of the model used in the
thesis generates results in terms of the effective
plastic strain. Strains due to bending and twisting, in
a thin shell, are thought to be rather small and
therefore are neglected. The theoretical work in the
model assumed that the material is isotropic and rigid-
plastic; i.e, the elastic strains are neglected.
A computer program has been developed to solve for
the forming strains. The program consists of several
subroutines. The subroutines are simply numerical
integration codes that solve the various systems of
differential equations. It accounts for different modes
of deformation that may be encountered in implementing

the Tresca criteria. Consequently, appropriate matching
boundary conditions and yield criteria are required.
Also, the geometry of the computer model requires more
than one shell part, actually a combination of three or
more shells. In summary, the program solves for all
possible strains under different modes of deformation
adopted to the right shell part and yield condition.
At last, when the calculated strains are compared
with the formability limits of the material, the
technique is able to evaluate the design of the sheet
metal part. Various factors affecting stability,
friction and distance traveled by the punch will be

Thesis Format
The following is a chapter by chapter overview of
the thesis: "Biaxial Friction Model"
Chapter 2 presents a general literature review of some of
the other technique used to model sheet metal forming.
Chapter 3 addresses the theories, models, and assumptions
involved in the model.
Chapter 4 gives a discussion and comparison of the
results of the model to the experimental data and results
from other models.
Chapter 5 presents conclusions, comments and
recommendations for further research.

The behavior of sheet metals in press forming has
long been studied from various points of view and the
topic has already grown in the wider sense. It was first
experimentally examined for explaining a difference in
the forming limits and finding the material factors
affecting the pres formability of sheet metals. Then,
some fundamental tests were investigated with the actual
press performance used as a means of predicting the
As a result of past studies in sheet metal forming,
the understanding between metal properties and the
geometry of a pressed part has greatly advanced. This
makes it allowable to draw some one-to-one correlations
between the two.
Among earlier works on deep drawing are those by
Hill [6] and by Chung and Swift [7] using the incremental
theory of plasticity. More refined analysis are the
finite-difference solutions by Chiang and Kobayashi [8],
by Budiansky and Wang [9], and by Chakrabarty and Mellor
[10]. Such a refinement improves the understanding of
the deep drawing process, and the stated works are
extensive and in good agreement with experiments. These

previous works used the von Mises yield condition, and
there seems to be a lack of work using the Tresca
Adding to these investigations on deep drawing with
a flat-bottomed punch, some works are reported on metal
forming of a sheet with a hemispherical head punch. Woo
[11] analyzes this problem by breaking the deep drawing
over the flange and the punch stretching over the
hemispherical punch head. He first obtains solutions for
pure radial drawing in the flange and then used this
solution at a point initially situated near the die lip
as the boundary condition for the stretching problem, and
thereby essentially matched the punch stretching
component with the pure radial drawing component at a
particular point in the die profile region. Instead of
this matching boundary process, finite element methods
that treat the problem in a unified manner were developed
[12] , [13]. Again the referenced work employed Von Mises
yield condition and its associated flow rule and hardly
no work is reported on sheet metal forming using the
Tresca criteria.
The deformation of a metal in stretch-forming
operation has conceivably attracted more attention from
researchers than any other problems in the plastic
deformation of' sheet metals, but it seems correct to

state that there is yet only a few reliable theoretical
guides to the instability of strains in this process
[14] and [15].
An early solution by Gleyzal[16] for small strains
was based on the total strain theory of plasticity. Hill
[17] developed a more general solution based on the Mises
theory but using a method of successive approximations.
Among recent works, J.C.Gerdeen [18] developed a computer
model for simulating the forming strains. The solution
was found to be in good agreement with experimental
results. Here a similar reasoning is followed in our
computer modeling in a sense of matching conditions and
shell parts but under a different yield criterion.
There are some current systematic methods in
analyzing sheet metal forming in operations. However,
the following are not a complete list of all working
methods to model sheet metal forming. Descriptions of
these methods will be summarized. For further
information, one should refer to the referenced work.
1) Finite Elements:
The elastic-plastic theory is adopted in the "Finite
element analysis." It assumes a non-linear deformation
and uses infinitesimal increments of strain stresses.
Furthermore, the friction can be incorporated at
the die and work-piece interfaces along with the actual

properties of the material. The efficiency of the method
depends on the size of the strain increment used. For an
infinitesimal increment, results are accurate but the
computational time and hardware costs are high [19].
2) Slip Line Theory:
In this method, the deforming body is assumed to be
rigid, perfectly plastic and isotropic. It is applicable
to the plain strain problems. If the problem is not
statically determinant or the stress boundary conditions
are not sufficient, the construction of just one slip-
line can be a lengthy task and the method might turn to
a very laborious process [20].
3) Shell Theory
For axisymmetric shapes, the surface is usually
thought of as a number of shell elements such as
spherical, cylindrical or toroidal shapes. The modeling
technique normally adopted is applied in several ways;
for example either force or thickness can be used as an
initial boundary conditions and iterated on to obtain a
reasonable solution [21], [22].
Then, there is the Geometrical Mapping technique
which shall be discussed in details in the theory and

In this chapter, there are several detailed
sections addressing various concepts used in the thesis.
It is intended to discuss the origin of each concept and
how it is involved in the theory. The final section will
utilize all derived formulas to formalize the solution
and develop a working technique.
Geometric Mapping Technique
The Geometric Mapping approach is an approximate
method of strain analysis that does not consume much
computer time, yet gives accurate prediction of forming
strains. This approach was first developed, by J. C.
Gerdeen, et al. [1974,1984,1985], and later by Sowerby,
et al. [1982,1985] and used in a number of applications.
Fundamentally, strain is defined by the difference
in geometry before and after deformation. Since the
final geometry of a sheet metal part is generally
specified by a mechanical drawing of the part, it should
be theoretically possible to map the flat blank and
calculate the strain distribution without knowing much
else [23].
In this thesis, the technique would be used by
applying the general differential form of the fundamental

equation of the Geometrical Mapping[21]. It is
essentially the incompressibility relation of plasticity
rewritten in terms of the deformation of a shell element,
see Fig.3.1(a,b) :
(r-u) (l--^) = [l+(-^)2]1/2 (l+ec)r (3.1)
u=radial displacement r-r0
w=the normal(transverse) displacement
r0=original radial coordinate in the flat blank, and
r=radial coordinate in the formed part.
Eq.(3.1) is the general differential equation in
terms of the original radius r0( or the displacement u)
in terms of the known shape w(r). Previously, when
Eq. (3.1) was used as a working technique to solve for the
strains, et was left explicit because it was assumed it
will be specified as a trial solution and iterated upon.
But here et is eliminated and solved for separately by
using the incompressibility and the strain hardening
The strains, for axisymmetric deformation, are
defined with reference to Fig.3.1(a,b). The
conventional definition of engineering strains are :
ds-dr Ar At
r dr
> ee et~m f.
0 ^fl


and the true strains are obtained by the following
general conversion formula:
e-ln(l+e) (3.3)
For a certain shape, the coordinates (r,w) are
known. The detailed analysis of the adopted shell parts
shall be covered in the final analysis.
Since the plastic strains are large compared to the
elastic strains during fabrication, the elastic strains
can then be neglected. The resulting deformation state
is governed by the incompressibility relationship of
plasticity :
(l + ec) (l+e0) (l + er)-1. (3.4)
It is to be noted that the above equation is
independent of the material properties and applies for
isotropic as well as the anisotropic materials, as long
as the material can be plastically deformed and the
incompressibility applies. Also, Eq.(3.4) applies for
asymmetric and axisymmetric deformation. Thus it is
applicable in general for arbitrary fabricated shapes.
Friction and lubrication are of vital importance in

most metal forming operations. Effective lubrication
results in low friction levels which reduce the
loads imposed on tooling and work pieces and which
allows more unified strain distributions. This can
eliminate problems with tooling or work-piece failures or
permit a reduction in the number of steps required to
form a part. Lower force levels also reduce tooling
deflection and can improve the dimensional accuracy of
the product. Furthermore, thermal problems are reduced
by effective lubrication.
Because of the importance of lubrication in metal
forming, there is relatively significant interaction
between researchers in the two fields. But sometimes,
it is possible to find sophisticated treatments of
plasticity combined with naive assumptions about
frictional forces.
In most metal forming processes, friction between
the work piece and tooling has an important effect on the
mode of deformation and the required forces. In
analyzing such processes, it is important to have some
method of characterizing the friction stress at the
The most commonly used method of characterizing
friction in any contact, lubricated or not, is the use of
the constant coefficient of friction jli[24]. In the

present context the tangential pressure P^ is assumed
to be given by :
P*=MPn (3.5)
P0=-^Pn (3.6)
Where Pn is the local normal pressure component. The
coefficient of friction is usually assumed to be a
property of the work piece, the material and the lubri-
cant if applied. It is usually assumed to be independent
of the process geometry and sliding speed at the inter-
face. Fig.3.2(b) shows that jk is positive when Pn is in
the negative normal direction as the case with die
contact region. In the contrary, fi is negative if Pn is
in the positive normal direction, as shown in Fig.3.2 (a) .
Qualitatively, in the contact region with the punch, P^
is induced by the external force in -0 direction, and in
the die contact region, P^ is induced by the die and
alsoin the -0 direction.
Finally, the effect of the frictional factor in the
final strain distribution is studied thoroughly in the
final results. As we will see later in the thesis, the
coefficient of friction plays a determining factor in
where the necking angle takes place.
Membrane Theory of Shells
It is assumed that the shell is sufficiently thin

r.3.2(&) External pressures in the die contact
{' 3.2(b) External pressures .in the punch contact

and that the stretching is sufficiently large that the
bending and twisting moment of a thin shell are
relatively small and, hence, of little influence on the
overall stress picture. The membrane theory is based on
the assumption that they are small enough to be neglect-
ed. This simplification reduces the number of variables
to be determined.
In the development of general thin shell theories,
including bending, simplification is accomplished by
reducing the shell problem to the study of the deforma-
tions of the middle (or reference) surface of the shell.
One approach is to begin with the governing equations in
the three dimensional theory of elasticity[25] and
attempts to reduce the system of equations, involving
three independent space variables, to a new system
involving two space variables. Another approach, the
mechanics approach is to assume a two dimensional plane
stress state to start with, [21]. These two variables are
more conveniently taken as coordinates on the middle
surface of the shell.
Geometric Relations
Fig.3.3 shows a surface of revolution and curvili-
near coordinate 0 and 6 The following notation is defined:

Fig.3.3 A surface of revolution and curvilinear


0=constant, the meridians;
r=the radius of the parallel circle;
rl=the radius of curvature of the meridian;
r2=r/sin ( surface and the axis of revolution.
For the line element ds of the meridian, see Fig.
3.4, we have:
ds=rjd0 (3.7)
and since:
dr=cos(0)d0, dz=sin(0)d0 (3.8(a,b))
we have the following relations:
-^=rxcos () -^=r1sin( Equilibrium of a Shell Element
The shell element is cut by two meridians and two
parallel circles, each pair close together. The compo-
nents of equilibrium will furnish three equations, just
enough to determine the three unknown stresses resul-
tants; the meridional stress force N^,the hoop force Nfl,
and the shear N^. The equilibrium of the shell element
yields the following equations [25]:

Fig34 Meridian of a shell of revolution

^ +r13i\rw-r1^ecos () +P dr
>} +r^ at
H-r^^cos (<|)) +Perr1=0.
Where Pr,Pfl and P^ are the loads in the radial hoop and
and meridional directions respectively.
Axisymmetric Loads
In many practical problems, the external forces have
the same symmetry as the shell itself which is the case
in the metal forming problems considered in the thesis.
Then the stresses are independent of 6, and all
derivatives with respect to this coordinate disappear.
From Eq.(3.10 (b)) this requires that Nd>0 and
Eq.(3.10(a,c)) then read:
-r^cos () =-P^rr1

Eq.(3.10(b)) becomes independent of the other two and
contains only the shear. We eliminate it from further
consideration by putting P6=0.0 and N^fl=0.0
Plastic Work
An important concept that is frequently used in
plasticity theory is the concept of plastic work [26].
The work done per unit volume on an element during
straining is:
dW-ajjde,^ (3.12)
=aVi (de^. (plastic) +dc7ij. (elastic) =dWe+dWp
But dW^Ujjde^. (elastic) is a recoverable elastic energy
which is negligible, where as the plastic deformation is
an irreversible process from which the energy cannot be
recovered. The remainder of the work done is called the
plastic work per unit volume:
dWp=a i j- d e ? j=ad e (3.13)
where a and e are defined to be the effective stresses
and strain simultaneously. Since the Tresca is
piecewise linear yield criterion, it is postulated that
the strain hardening in the biaxial case also occurs in
the linear manner. Therefore, strain hardening in the direction is different from the 0 direction which result

in plastic work induced in both directions. In the corner
of the yield surface where f1 intersects f2, for illus-
trative purposes deformation is called Mode 1, shown in
Fig.3.5 . Applying the plastic work concept in 0 and 0
directions, we get the following equations:
In 0-direction dWp1=ade=a^de0 (3.14)
which means that a=a, and 6=eM
In 0 direction dWp2=cjd£=CT0de0 (3.15)
a=a0 and e=e0
The total plastic work is the sum :
dW =dW.+dW-
p pi p2
Also, in the region where f2 intersects f3, deformation
is referred to as Mode 2, see Fig.3.5. Here, N0 is zero
and the plastic work equation reduces to the following :
Finally, in the region where the yield stress equals to
f1f i.e. Mode 3 deformation, the plastic work is :
Tresca Criterion
This theory (sometime called the Coulomb or Maximum
shear theory) assumes that yielding will occur when the
maximum shear stress reaches the value of the maximum
shear tress occurring under simple tension. The maximum
shear stress is given by the equation below:

Fig.3.5 Modal representation of Tresea yield

. (3.16)
Sy/2= max=+/-(l/2) (a^-a^)
where Sy is the yield strength
For the biaxial case with a3=0.0# letting a^ax and ag=a2:
fl-2 max if a2>ox >0
f2=2 max=ffl-*2 if ^>0, a2<0 (3.17(a,b,c))
A plot in the axa2 plane for this criterion is shown in
Fig.3.6. The Tresca Criterion is in fair agreement with
experiment and is used to a considerable extent by
designers [27]. It suffers, however, from one major
difficulty-it is necessary to know at any point which
yield condition fj governs yielding.
Flow Rule
The first approach to plastic stress-strain rela-
tions was suggested by Saint-venant in 1870 [28], who
proposed that the principle axes of strain increments
coincided with the principal stress axes. The general
three dimensional equations relating the increments of
total strains to stress deviators were given
independently by'von Mises in 1913 [29]

Fig3.6 Fxg*3.6 Yield stress analysis of Tresca yield
condition I

Where Sfj. is the deviator tensor and dA. is a non-negative
constant which may vary throughout the loading history.
Again the elastic strains are being neglected.
For ideal plasticity, it is assumed that
the yield function, exists as a function of stress only
and that the plastic flow takes place without limit
f(aij)=k (3.19(a))
and the material behave elastically when(3.19(a))
f(an) where k is known function. For a plastic flow therefore:
Comparing Eg.(3.19) and Eg.(3.20), it is seen that :
Where dl is a scaler. Then:
Eg.(3.21) is known as the flow rule that is going to be
associated with the Tresca yield Criterion which assumes
that the plastic strain increment at any instant of
loading proportional to the gradient of the yield

Problem Formulation
The 'Biaxial Friction Model' of sheet metals is
intended for stretch draw operations. The purpose is to
solve for possible forming strains after deformations.
As stated earlier, the model is based upon the use of
the effective(or generalized) plastic strains. These
strains are obtained from the experimentally determined
empirical equation relating the effective stresses and
strains. It is assumed that the behavior in terms of the
generalized stress and strain is the same as in a simple
tension. The analysis in the thesis is done on an
Aluminum alloy whose constitutive behavior has been
determined experimentally to have the form of:
a=71100e 179 (3.22)
Where a is the generalized stress
e is the generalized strain
k is a material constant=71100. psi
and n is the well known strain hardening index=.179.
The value of n is, according to this theory, independent
of the applied stress system for isotropic materials.
This concept of work hardening is assumed to be
taken place biaxially,i.e. in the 6 and

Because the Tresca yield condition is a piecewise
continuous and,has a corner for the biaxial stress state
oxa2, it is assumed in this thesis that the amount of

strain hardening is different in these two directions 0
and r. Therefore, it is concluded that the yield values
of the membrane stresses N0 and Nfl are not equal after
some strain hardening, shown in Fig.3.7.
The entire sheet in forming process can be can be
divided into four major regions illustrated in
region 1 : 0C is the contact angle with the punch head
region 2 : the unsupported region
region 3 : 0d is the contact angle with the die profile
region 4 : the plate over the die
Different conditions are imposed depending upon the
regions. For example, the plate is constrained to move
only horizontally along the die surface, while the
contact region with the die profile or punch head has to
satisfy matching conditions with the unsupported region.
Various dimensions involved in the problem are given
shortly. Comparison of results on the present analysis
with previous results of experiments [30] for the same
alloy is made and also with the results of J.C. Gerdeen
[18]. The dimensions used here are the same as in the
experiment. The radius of the punch, Rp, is 2.0 inches.
The clamping radius, R0, is 3.0 inches. The die

Fig.37 Variation of strain? Hardening in different

^ig*3.8 (a) Geometry of a sheet metal "before
d c forma, I; ion

centerline radius, R,., is 2.49 inches. Finally, the die
profile radius, Rd, is 0.5 inches.
Fig.3.9 shows the various possible states of
deformation. Each state is governed by a different
yield condition. Therefore, the difficulty lies in
determining the appropriate shell part without violating
any geometry or boundary conditions.
Region I eP>0:
With the advancement of the punch head, the portion
of the sheet in contact with the punch increases and
consequently the boundary separating the contact region
from the unsupported region changes. The presence of the
moving boundary is always a source of complications in
the numerical analysis of the punch because it requires
basically a trial-error or iterative approach .
From the shell theory, Eq.(3.ll(a,b)), the
equilibrium of the forces on a shell element is :
dZV£ + _A^_ ds
dr r 3 dr
Where r!=r2=r/sin(0) =a=radius of the punch head and the
s coordinate is the same as the


]?ig-.3.9 Strain vector representation for different

Because of symmetry, and Nfl are the same at the
center of the punch. Furthermore, the principal effec-
tive strains in the

only possible stress state that satisfies this condition
is at the corner of the Tresca Yield where it is
perceived as the intersection of two yield functions i.e.
Mode 1 deformation .
Fig.3.10 shows the location of the strain vector
location whose direction is at a 45 degrees angle. This
state of stress has the following boundary conditions
which correspond to the center of the punch :
e3 =efl (3.24)
Throughout the contact region with the sphere :
f l=Syl=cre=the yield stress in 6 direction (3.25)
f2=Sy2=as=the yield stress in s direction (3.26)
where fl and f2 are the yield functions which are totally
independent of each other.
Applying the flow rule :
de= (PARTIAL fl/ PARTIAL at ) dX^dXj (3.27)
des= (PARTIAL f 2 / PARTIAL CTS) dX2=dX2 (3.28)
as seen, yielding is taken place at different rates in and 6 directions. The shape of the yield surface will
vary from point to point, see Fig.3.7. There is a family

10 Strain -sector
(sphere model)
representation for region

of yield surfaces that change continuously in the
direction i.e. the magnitude of Ns will vary
continuously. Because N is tangential to the boundaries
s=constant, it is not necessary that the magnitude of
be continuous.
In order to implement Coulomb friction law between
the sheet and the punch or die, Eq.(3.5) is recalled :
P0=MPn=MPr (3.29)
Eg.(3.23) becomes:
dNs Ns-Ne Ns
£ + 5 =-u, (- +
dr r a
So, by combining Eq.(3.7) and
jy0sin(<|>) ds
Eq.(3.8(a,b)), Eq.(3.30)
ds _ 1
dr cos(<|))
dNs Ns-Ne tx , Ns ^ N0sin () ,
dr r cos(j)) a r (3*31)
By applying the strain hardening postulate,
~=d(ktg) =kteDi1n-^+ke*-^;
From Eq.(3.3) and Eq.(3.2(c)),
et=log(l+et)=log(t/t0) or t=t0e
Differentiating the above equation :

*L = t e *£-(3
+ di
where Nfl/Na=(efi/es)n from the Power Law relations.
In order to eliminate de9/dr from the above equa-
tion, the general equation of the G.M. method is used.
Appendix A provides the final form of the desired
equation as a few manipulations are required:
Cfen 1 <3 _c+c . _ .
9 x M e ) (3.38)
dr x v cos ( It is to be noted here that Eq.(3.38) is valid for any
shell part, knowing only the shape of the metal after
deformation. Eq.(3.37) and Eq.(3.38) are a system of two
differential equations that involves e9 and ea as the

dependent variables and r or 0 as the independent
variable. These equations are solved using a numerical
integration method called 4th order Runge-Kutta. The
boundary conditions at the center of the punch are the
initial conditions required to begin the integration
processes. Since numerical integration is an initial-
value problem, an initial condition is required with et
at 0=0 where e3=e=-et/2 (0=0 only) .
The above formulation is only valid for the corner
of the yield surface where efl>0, see Fig.3.10. The depth
of draw HI in this region is determined from :
Hl=Rp(1-cos (0j)) where punch.
Region 2 e^n.
This region is defined as the unsupported region
where no pressure loads are applied. In this region,
boundary matching conditions are a crucial problem, and
various solutions are possible. In the past, a flat
conical shape was assumed by some experts. What is
needed is to be able to find a possible shape whose
solution satisfies the stress and strain boundary
conditions and yet maintain continuity with and the
angle 0. In addition to the conical shape, the

foregoing treatment of the unsupported region will
provide a more general approach which is known as the
'Free Equilibrium' method developed by J.C.Gerdeen [1].
The cases considered include the conditions where
eg>=0, eg=0 and ee<0, where each requires some iteration
method to match the proper strain solution.
Region 2
The Free Equilibrium method provides at least two
differential equations that solve for shape z and dz/dr
as the dependent variables and r as the independent
These equations are:
let Yl=z ----> dYl/dr=dz/dr (3.39)
and Y2=dz/dr------>dY2/dr=d2z/dr2=( (l+ fdYl/dr)2)3'2/^
(3.40) where
rj is radius of curvature of the shell and Y1 and Y2 are
temporary variables.
In addition, one more differential equation, is
required in terms of e, as the dependent variable. The
two equilibrium equations Eq. (3.23 (a,b)) with no pressure
dN3/dr=- (N-N,) /r
Ng/rj +NflSin(0) /r=0.

As in the sphere analysis where Mode 1 deformation is
taken place, Eq.(3.30) is used here, see Eq.(3.42).
Therefore, there are a total of four differential
equations where es, e9, z and dz/dr are the dependent
variables and r is the independent one are derived.
These equations are (3.38), (3.39), (3.40) and (3.41).
An explicit equation for rt is needed in terms of the
dependent variables just mentioned. That is easily found
by using the second equilibrium equation:
The above analysis is useful for the condition where
the stress state is at the corner of the yield surface.
The integration routine of the above equations can
proceed to a some final radius r=rf where efl>0(Mode 1) or
efl=0., see Fig.3.11. It is important to consider the
strain equation at which ee=0 because in such a case, the
geometry alone will determine the strain values.
Region 2 eP-P.
Discussing the stress-strain state at e9=0, leads us
to take a closer look at the G.M. equation orEq.(3.1)
r1=-r2N3/N9=- (r/sin(0)) ((es/efl)n)

liegion 2(unsupported)
Mode 1
^ig-3.11 Strain vector representation for region 2
(unsupported model)

-^£ = [1+tan2 (0) ] 1/2eGc
The fact that the strain vector has turned the
corner on the yield surface suggests that e9 will
maintain a zero value until N9 reaches a zero value, i.e.
along the side of the yield surface.
e= (1+tan2(0))1/2----->et=log(cos (0) )=-es (3.45)
Eg. (3.40(b)) is recalled with defl/dr=0. so :

de (ns-nq)
dr 1 r
since es=-et, then des=-det,by differentiating et and using
( -1) ]
sin(0) (l-sin2 (0)) es (3.47)
Now we have completed the eguations reguired to
solve for the shape desired. In order to use the above
solution, the previous unsupported solution has to
provide an appropriate et to match with the current
solution of et=log(cos (0)) Some trial-error steps are
reguired to accomplish the objective. It was almost
impossible to satisfy Eg.(3.45) when solving for the
forming strains and satisfying the matching conditions.

The reason was that the unsupported final strain value
did not correspond to an angle that satisfied Eq.(3.45).
Region 2 e-<0 ;
On the yield surface, the solution where e9<0
corresponds to the stress state corner at which N=0 or
Mode 2. When Nfl=0, the equilibrium equations then reduce
to :
dNs/dr=-Ns/r and Ns/rx=0 >r!=infinity 3.48(a,b)
the fact that r^infinity suggests that the only possible
shell is a cone or a toroid if the radius r matches that
of the toroid. Now formalize the cone solution by
recalling Eq.(3.35) and Eq.(3.47(a)). When combining
them, one can obtain :

*e1 = Ik
dr r
with the use of the G.M. differential equation, there
are have two differential equations to solve for the
conical shell part. The integration of these equations
can be carried out until et=0 or when the r matches (Rc-
Rdsin(0o)), radius of the die and Fig.3.12 shows the state of deformation where the strain

Figi3.|2 Strain vector representation for region 2
(aone mode 2)

vector of the cone is at Mode 2 region. It is important
to point out that the unsupported region can have more
than one shell part. As will be demonstrated later, to
satisfy geometry and boundary conditions, the unsupported
region consists of a combination of cone and some other
solved shape. The depth of draw :
Hc=H (unsupported) + (r2-rt) tan (0O)
where r2 is the final radius of the toroid section
rj is the initial radius of the toroid section
Region 3 :
This region consists of the die part where it is in
contact with the sheet metal, shown in Fig.3.8. In this
analysis, the die considered has 3 toroidal shape. The
contact angle between the sheet and the die varies from
one condition to another. The friction is assumed to be
of the same order as the in the punch contact region.
Three cases of circumferential stresses are considered;
positive, zero and negative.
Region 3 N(,>0:
Starting with the analysis by classifying the yield
condition to be considered. If the final ed obtained
from the unsupported region is positive,i.e. Mode 1, then
the derivation must be continued at the corner of the

Tresca yield surface. In that case, the strain hardening
postulate is used again; N0=kt en and Ns=kte. In
addition, the equilibrium equations then become:
(Jl-1) des-( deo + (- 1+ H ) ) (ea)a( 1 |xsin(<()) ,
es dr dr r acos(<}>) e6' r rcos(4>) '
(3.5 D)
Now, the current equation along with Eq.3.38 form a
system of two differential equations.
Region 3 N=0 :
Applying the yield condition at Mode 2 and the flow,
, see Fig.3.13, the yield stress is equal to the.
meridional stress :
! f=Ny=Ns (3.51)
Then the flow rule becomes:
de= (PARTIAL f/ PARTIAL CTS) d\=d\ (3.52)
And using the Plastic Work analysis by recalling
Eq.(3.12(a)) :
a0dee=ffgd6fl=a1dei +CTflde=asdes or e=e6 and a=ae (3.53)
finally our equilibrium equation, when using P^=mP:

]?ig>3.|3 Strain-vcecrtor representation for region: 3
(toroid mode 2)

The above equation along with the G.M. equation i.e
Eq.(3.38), form a complete system of equations that solve
for the forming strains.
Region 3 NP<0 :
As stated earlier, the Tresca Criteria predicts that
yielding occurs if f=ay=as-a9 or Mode 3, see Fig.3.14.
Since Ns>0 and N9<0, applying the Flow Rule equation :
des=dX( PARTIAL f/ PARTIAL CTfl)=dX (3.55)
defl=dX( PARTIAL f/ PARTIAL CTtf)=-dX (3.56)
Recalling the Plastic Work concept :
dWp=CT3de3+cT(,dfl=dXCTs-dXCTfl= (ffa-CTtf) dX=ffcdec
Thus ee=es, es=-efl and et=0. (3.58)
Using the friction and the power law equations:
P*=MPn/ rt<0. and Ny=Ns-N=kt0e3' 3.59(a,b,c)
Now the equilibrium equation is derived :
dNa__ Ny
dr r cos ()
r 3 es dr

Fig".3.14 Strain vector representation for region 3
(toroid mode 3)

Using the Geometric Mapping equation, reduce to :
drQ/dr=r/(r0cos(phi)) (3.61)
Once r0 is found, e is known. Then the obtained value is
used in the above equation and solve for Ne. The total
depth of draw in this region is :
H=Hc+Rd (1-cos (0d)) where 0d is the die contact angle
Region 4:
This is the plate region where it lies between the
toroid and the clamped edge. It is a flat plate where
r!=r2=infinity and 0=0. Two possible cases of e that
can be used are : efl>0 or e9<0. Depending on the toroid
results, in particular the ending value of e9, the yield
condition is determined. Each strain or stress state
requires a different yield condition.
Region 4 e^>0:
In this case, deformation is taking place at Mode
1 region, i.e. at the corner of the yield surface. Using
the same argument as for the sphere, Eq.3.23(a) yields
the following equilibrium equation:
[(JL-X) *£-*].
e dr dr
-l+(i )n
Then by using the G.M. equation, Eq.(3.38), a

system of two differential equations are obtained. It is
to be noted here that the boundary condition has to be
satisfied. For the clamped condition, the final eg must
be very small at r=3.0 inches. If final boundary
condition is not satisfied, then the process should be
repeated and the pressure which represents friction
must be changed.
Region 4
Here, two possibilities of circumferential stresses
Ns=0. or Nfl<0 are present. If N0=O or Mode 2, see
Fig.3.15, then Eq.(3.62) could be used. In order to use
Eq. 3.62, since Nfl=0 then the ratio of strains in the
equation is set to zero. And if N9<0. then Mode 3
deformation is used. In this mode, et is 0 and es and
e9 equal and opposite. The strains are determined by
using Eq.(3.38) and by replacing es with -ee. Then, the
stresses are easily determined by the equilibrium
The range of possible solutions have been completed.
All applicable cases have been covered. Now it is a
matter of trying these solutions to determine what shell
part to be used and what yield condition is appropriate
to arrive at a final valid solution to satisfy conditions


The configuration used for a hemispherical punch
stretching problem is shown in Fig.3.8, and the
geometrical parameters are: R,=2.0 inches, Rd=.5 inches,
R0=3.0 inches, Rc=2.49 inches and K=71100 psi. In this
chapter, two comparison cases are considered for the
punch stretching problem In addition, a theoretical
result is provided for a typical sheet metal drawing
case. Finally, the effect of the strain hardening
coefficient n and friction /z or (mu) on instability is
Fig.4.1 shows the distribution of engineering
strains in the radial and circumferential directions for
a clamped condition. The maximum depth of the punch is
.8714 inch and the coefficient of friction for a
lubricated case is .05 and the material used is an
Aluminum alloy described earlier in the last chapter.
Note that friction causes a little hump in the radial
strain. The required punch force was 5715 lbs.
When this case is compared with previous results of
J.C. Gerdeen (von Mises criteria) and with experiments
[18], shown in Fig.4.2 (major strain) and Fig.4.3 (minor
strain)., it is clear that there is a good agreement
between experiment and theory for the maximum strain at

CLAMPED, MU.05, N-.179, H-.87H
Fig.4.| Strain distribution for a lubricated case

2 0.13
| 0.12
fci 0.11
0 0.4 0.8 1.2 1.6 2 2.4 2.8
CLAMPED, MU.05, N-.179, H-.8714

o MA. OR (V ON M

n a
(J o C
S"" <
0 o v o o sx u O

\ 8 a
\ 0
<> r 4
o $ 1 1 &
V o

Fig.4.2 Major strain-comparison (mu=.05)

CLAMPED, MU-.OS, N-.17S, H-.871 +
Pig.4.3 Minor strain comparison (mu=.05)

13%. The current calculated strains (solid
lines) are higher than von Mises's results in the first
part of the sphere section (region 1) and lower than the
experiment's. If bending strains of 1% (eb=thickness/2Rp)
were included, then the current result would have been
in a better agreement with experiment than if they were
not included. But after a radius of .6 inch the major
strain was lower than the experiment and the von Mises
result. Also, because the current analysis was started
with a little higher initial conditions, there was a
slight increase of results over von Mises in the sphere
region. As for the minor strain, the current result
matched better with experiment than von Mises at first,
then it was closer to von Mises toward the end. The
experimental strains appear to be somewhat in error
toward the outer edge, where the cicumferential strain
should be zero. The accuracy of measuring the strain
circles is estimated at 1-2% Since only the membrane
strains were calculated, the distributions of midsurface
strains from the current results is smoother than those
of von Mises's solution and experiment, which both
included bending strains.
For the most part, the Tresca result case was
within the range of the other stated results. The
increase of the major strain at a radius of 2.0 in the

other results is due to the bending strains which change
significantly with the change of curvatures. In the
plate region, the pressure that represents clamping could
be iterated upon further until the boundary conditions
are fully met, i.e. where e is zero. It is to be noted
that von Mises's result and the experiment allowed a
punch depth of .93 inch where Tresca's was .8714 inch
which leads to the conclusion that Tresca gives a higher
strain for the same depth than von Mises.
Fig.4.4 presents the deformed geometry of the
stretching problem solved in Fig.4.1. The geometry
consists of five shell parts; a spherical shell at the
punch contact region where the contact angle 0C=33.O
degrees at Mode 1, the generated unsupported shell to a
radius of 2.06 inch at Mode 1, a conical shell to a
radius of 2.32 inch at Mode 2, and finally a toroidal
shell at the die contact region where 0d= 20.0 degrees at
Mode 2 which is followed by a plate at r=2.5 to r=3.0
inch at Mode 2. The five shell parts described above
are used as an example model in constructing the geometry
of sheet metal stretching problems. Several attempts
were made to arrive at the combination of the mentioned
shell parts that would satisfy boundary and geometry

0 0.4 0.8 1.2 1.6 2 2.4 2.8
CLAMPED, MU.05, N-.179, H-.8714

/ /
/ /
/ /

EIg.4.4 Geometry for a lubricated case (mu=.05)

CLAMPED, MU-.20, N-.179, H-.9181
Eigi4.5 Strain distribution for a dry clamped case

Fig.4.5 shows the major and the minor engineering
strains for a friction value of .2 (a dry case) and a
depth of draw of .9181 inch. It was found that for a
higher friction value, a larger peak in the major strain
and a larger difference between the major and the minor
strain was taking place with a punch force of 6231 lbs.
Fig.4.6 shows a comparison in the major strains for
the experiment, von Mises and Tresca (solid lines). For
the experiment and von Mises the depth of draw was .97
inch. It is to be noted in region 1 that the Tresca
result was closer to the experimental value than von
Mises's. Overall, Tresca was in a better agreement with
experiment than von Mises for the major strains. It is
worth to mention that if bending strains were added and
initial strain conditions were lower, then Tresca's
solution would have been closer yet to experiment. As
for the minor strains, see Fig.4.7, again in region 1,
Tresca's solution was closer to experiment than von
Mises. But for region 3 and 4, the minor strains were
closer to von Mises than experiment.
Fig.4.8 shows the deformed geometry of the problem
in Fig.4.5. Here, The punch contact angle 0C=34.O
degrees at Mode 1, the generated unsupported region was
carried to a radius of 2.0 inches at Mode 1, the conical

CLAMPED, MU.20, N-,179, H-.9181
Fig*.4.6 Majjor straim comparison (clamped1

CLAMPED, MU,20, N-.179, H-.9181
0 0.4 0.8 1.2 1.6 2 2.4 2.8


OT o

o 0 1 V
o \
i> a a F \ \
0 \
i ,v
c 1 o , n *-
3 sl£ a
Pig.4.7. Minor strain comparison (mu=.2)

CLAMPED, MU-.20, N-.179, H-.9161
Fig.4.8 Geometry for a dry clamped case
(mu=.2) '

shell to a radius of 2.3 inches at Mode 2, and at last
the die contact angle A is 21.86 degrees and the plate
region at Mode 2. The region with the highest major
strain is strain hardened the most and in that section
failure is likely to occur. This location of the necking
region is tangent to the punch at the start of the
unsupported region.
Fig.4.9 shows the major strain distributions for
three different materials for a strain hardening index n
of .179, .279, and .379. It was found that the material
that strain hardens the least showed a little higher peak
in major strain, but the major strain dropped faster
than the material with higher strain hardening indices.
Also, the higher the strain hardening indices, the
broader the strain distributions were. As for the minor
strain, shown in Fig.4.10, the minor strain showed a
similar behavior as the major strain but it was not as
sensitive to the change in n as the major i.e. the effect
of n is greater in the s direction than the 8 direction.
Fig.4.11 shows the geometry for the three different
materials stated above. It is to be noted that a
material with a higher strain hardening index n could be
stretched slightly further than a material with a lower
n but strains would also be little higher overall than

0.02 -
CLAMPED, MU.05, N-.179/.279/.379

179 279 379
s n=.
/ ja=.
\ \<
V sX


0 0. 4 0. 8 1. 2 1. 6 2 1 2.4 2.1
Fig-4.9 Major strain; distribution for different n

O 0.4 0.8 1.2 1.6 2 2.4 2.8 .
FigA.IO Minor strain distribution fir different n

CLAMPED, MU.05, N-.179/.279/.379
Eig.4-.ll Geometry for different n

CUMPED, MU-.0S/.075/.1, N-.179
Pig. 4./2 Major strain distribution for different
friction |

the strain for the lower n material. For n=.179, .279
and .379, the punch contact angle was the same 0C=3 3.O
degrees and the punch forces were 5715, 4509 and 3562
lbs respectively.
As for the effect of friction, Fig.4.12 shows the
major strain distributions for the same Aluminum alloy
with n=.179 and the same initial conditions. The major
strain profile for mu=.l is higher than for the mu=.075
or .05. The major strain increases with the increase of
friction. When the material deformation gets close to
the necking mode, then there is a significant increase in
the strain value. For example, if a friction of .2 is
applied in the same problem, failure would have taken
place because then the value of the major strain would
reach the strain hardening index n, see Eq.3.37. The
minor strain distribution, shown in Fig.4.13, is effected
the same way as the major strain. As noticed, the effect
of friction on the minor strain is greater than the
effect of strain hardening index. The geometry is shown
in Fig.4.14 where friction mu=.l has the largest depth of
draw for the same initial strain, but with the highest
localized strain values. The punch contact angles for
mu=.05, .075 and .1 were 33.0, 34.75 and 37.23 degrees
and the punch forces are 5715, 6365 and 7180 lbs

CLAMPED, MU-.05/.075/.1, N-.179
Pig.4.13 Minor strain distribution for different

CLAMPED, MU-.05/.075/.1. N-.179
Eig.4.|4 Geometry for different friction

respectively. So, it is found that high friction
produces high traction between the metal and die or punch
which causes more stretching, high strain profile and
punch force.
More uniform strain profiles are desirable. For
example, if one assumes that the hoop and the
circumferential stresses are to be equal, then a
thickness strain differential equation in the sphere
section could be derived to be directly proportional to
friction. And if a uniform thickness is considered, then
it is found that the surface must be frictionless see
Appendix C.
If a shallow depth of draw is desired for a clamped
case, then the sphere computer routine could be started
at low initial strain conditions. For example,
Fig.4.15 shows the major strain distribution for three
different depth of draw. For an initial thickness strain
condition of -10%, a .69 inch depth of draw was allowed.
Also when initial thickness strain was -7%, then depth of
draw h was .52 (in). Fig.4.16 shows the minor strain
distribution for the same problem. The same behavior is
observed; the higher the depth of draw, the higher the
strain profile. Fig.4.17 shows the geometry of the three
different cases of initial thickness strain -13%, -10%
and -7% where the punch contact angles are 34.0, 26.8 and

CLAMPED, MU.2, N-.179
Pig.4.)5 Major strain*distribution for different depth
of drav/

CLAMPED, MU-.2, N-.179
Fig.4.16 Minor strain distribution for different.depth
of draw j

CUMPED, MU.2, N.179
Pig,4.17 Geometry for different depth of draw

21.1 degrees respectively. The construction of these
geometrical profiles was consistent with the previous
geometries. In all clamped cases, five shell parts were
used: spherical, generated unsupported, conical, a
toroidal and a plate.
Now, a typical draw-in case is considered for the
same Aluminum alloy and a theoretical result is provided.
If the friction is .05 and allow the material to draw in
until depth of draw is 1.53 inch, then the punch contact
angle is 47.0 degrees and the strains are shown in
Fig.4.18. When drawing is allowed, the material is
pulled in smoothly without being stretched at the holder
region. This condition causes the major strain to be
minimum or zero and the minor strain to be high in
compression as shown in Fig.4.18. Fig.4.19 shows the
geometry of this case where four shell parts were used;
spherical at Mode 1, conical (unsupported) at Mode 2,
toroidal at Mode 3 and a plate at Mode 3. The punch
force required to deform the sheet was 10087 lbs..

Pig.4.|8 Strain distribution for a draw-in case

MU=.05, CTO=.25. H=1.53, N=.179
Fig.4. | 9 Geometry for a draw-in case

It is apparent from any major strain differential
equation in the thesis that Tresca predicts failure when
the ultimate strain es reaches the value of the strain
hardening index n. Since von Mises predicts the ultimate
strain for a frictionless biaxial tensile stress state to
be equal to n/2 [14], one can conclude that Tresca yields
higher values for the ultimate strain for a friction less
case. But Tresca predicts failure sooner than von Mises
when the same depth is considered as shown in Fig.4.2.
In addition, the strain hardening index is known to be
higher for the biaxial case [18]. So if an accurate n
were used, then the predicted strains would give more
accurate results yet.
Since friction on the punch and die causes sheet
metal forming to be a path dependent process, it is
important to note that results would improve if given as
a function of punch travel. For example, when using von
Mises [18], at least three punch steps of travel were
needed to be analyzed to predict forming strains.
The results in this thesis were obtained on a 286
PC, and therefore the analysis methods developed are
beneficial to many engineers using such low level work

stations with reasonable accuracy which was 05 % and
total time of 3 minutes per solution, see Appendix B. If
a main frame computing system were used, the numerical
error would be reduced and computing time would be less.
The computing time could further be reduced if a
corrector method were used to adjust the integration step
size in using the Runge-Kutta method.
Even though Tresca's yield condition is linear and
simple in form, because it is piecewise continuous,
matching boundary and geometrical conditions is a lengthy
task. So an improvement could be established if an
automatic iteration loop were coded into a master program
that would include all of the subroutines and account for
all possible solutions.

The conventional definition of the engineering
strain ee is expressed in Eq.3.2(b) :
e=(r-r0)/r0 or efl+l=r/r0
The true e9 strain then becomes:
efl=log(l+efl)=log(r/r0) (I)
Where r can be written as follows : r^e", (II)
Differentiating the above equation with respect to r:

Recalling Eq.3.1 and substituting (r-u) by r0, (1-du/dr)
by drQ/dr and (l+et) by e/t :
dr0 re*t
dr racos ((j>)
In above equation, r/r can be replaced by eg. Now by
equating the Eq.(Ill) with equation Eq.(IV) :
d*e=l { l-e~e*ea)
dr r v cos () '


As the step-by-step numerical integration of a
differential equation progresses, the solution being
obtained may tend to depart more and more from the true
answer, owing to cumulative per-step error. In the
thesis, the solution had to be obtained by using a
relatively large number of steps, it was very important
to assume that the per-step error be kept small. The
per-step error using Runge-kutta's method of the 4th
order is of order h5, where h is the step size.
A 286-CPU IBM compatible personal computer with a
287 math-coprocessor was used with double precision. Six
significant digits were retained in the solution. Since
the exact solution to the problem being solved in the
thesis is not feasible, a test program has been developed
to approximate the numerical error. The test program
solves numerically for a differential equation of a known
exponential function, ex, and compute the percent error
of the difference between the actual and numerical value
of that known function.
When a step size of .001 was used, the cumulative %
error was .004 after 3000 steps, and for a step size of
.0001, the % error was .05. It is clear there is a trade

off between the two step sizes. In solving for the punch
contact region, a step size of .001 did not yield an
increase in the hoop strain which is the case with the
experimental solution. But when a step size of .0001 was
tried, a more reasonable strain variation was obtained.
Therefore, the smaller step size was used in the thesis.

If one assumes that the hoop and the meridional
stresses are equal i.e. pure biaxial loading, then the
equilibrium equation reduces to the following:
Applying the plastic work law:
Thus :
Using the power law: u^a^ke^Nj/t
By combining the above equations:
dt nde v
fcd ed
By differentiating thickness :
dt=t0e=tdet where t0=initial thickness
Using the above equation :
det_ 2\i _ 2\l
d (-2.+1) (JL+i)
et -e
If /i=0.f then det/d0=O. which means that t=constant. But
if Eq. (3.31) were recalled and fi is set to zero, a des/dr

term would be left in the numerator which would effect
the thickness variation and thickness would not be
constant. It is clear that for an ideal biaxial loading,
thickness could be forced to be constant by having a
frictionless surface. As for the postulate adopted in
thesis which assumes that strain hardening is different
in the s and 6 direction predicts a uniform thickness
only when jlj=0. and plane strain is considered i.e. e=0.

[1] Gerdeen, J.C., "The Prediction of Free Equilibrium
Shapes in the Forming of Sheet Materials,"Michigan
Technological University,1989.
[2] Yohida, K.,andK. Miyauchi, "Experimental Studies of
Materials Behavior Related to Sheet Metal Forming," The
Insitute of Physical and Chemical Research, Japan,1977,
GMR Symposium "Mechanics of Sheet Metal Forming."
[3] Swift, H. W., Inst, Metals, 81 (1952), 109.
[4] Mellor, P.B., "Sheet Metal Forming," International
Metal Reviews, 1981 No.l.
[5] Gerdeen, J.C., "Geometric Mapping of Computer
Modeling of Sheet Metal Forming," Submitted as a Working
Paper for the IDDRG Conference in Toronto Canada, 1988.
[6] Hill, R., "The Mathematical Theory of Plasticity,"
,Oxford Press, (1950).
[7] Chung, S.Y. and H.W. Swift, Inst, of Mech. Engs.
(London),165 (1951), 199.
[8] Chiang, D.C. and S. Kobayashi, J. Eng. Inst. (Trans.
ASME,b), 88 (1966), 357.
[9] Budiansky, B. and N.M. Wang, J. Mech. Phys. Solids,
14 (1966),357.
[10] Chakrabarty, J. and P.B. Mellor, Proc, 5th Biennial
Congress, IDDRG; Metall. Ital. (1968), 791.
[11] Woo, D.M. Int. J. Mech. Sci., 10 (1968), 83.
[12] Gerdeen, J.C. and D.J. Soper, "Computer Simulating
of Sheet Metal Forming," SAE Paper NO.851500, 1985, Int.
Off-Highway and Power Plant Congress, Milwakee, WI, Sept,
9-12, 1985.
[13] Kobayashi, S. and J. Kim, "Deformation Analysis of
Axiymmetric Sheet Metal Forming Process by the Rigid-
Plastic Finite Element Method," University of California,
Berkeley, California, 1977.

[14] Marin, J. and M.G. Sharman, "Design of a Thin-Walled
Cylindrical Pressure Vessel Based Upon the Plastic Range
and Considering Anisotropy," WRC Bulletin No. 40, May,
[15] Gerdeen, J.C., "A Critical Evaluation of Plastic
Behavior Data and A Unified Definition of Plastic Loads
for Pressure Components," WRC Bulletin No 254, Nov.,
[16] Gleyzal, A. J. Appl. Mech.(Trans. ASME, E),70
(1948), 288.
[17] Hill, R. Philos. Mag., 41 (1950), 113.
[18] Gerdeen, J.C., "Strain Analysis of Hemispherical
Punch Stretched Specimen of Aluminum," Michigan
Technological University, 1986.
[19] Kalpakjian, Serope, "Manufacturing Process for
Engineering Materials", Addison-Wesley Publishing Company
Publ., 1984.
[20] Gloeckl, H. and Lange, K., "Computer Aided Deign of
Blanks for Deep Drawn Irregular Shaped Component,"
University Stuggart, D-7000 Stuggart 1 .
[21] Gerdeen, J.C., "An Analysis of Axiymmetric of Sheet
Metal Forming," Proc. NAMRC-II (1974), 350.
[22] Flugge W., "Stresses in Shells," 2nd ed. Springer-
Verlag, Berlin, Heidelberg New York Publ., 1973, 48
[23] Gerdeen, J.C. and P. Chen, "Geometric Mapping Method
of Computer Modeling of Sheet Metal Forming," Michigan
University, NUMIFORM 89, 437.
[24] Wilson, W.R.D., "Mechanics of Sheet Metal Forming,"
Plenum Press, 1978, 174.
[25] Flugge, W.,"Stresses in Shells," 2nd ed. Springer-
Verlag, Berlin, Heidelberg New York publ., 1973, 110
[26] Mendelson A., "Plasticity: Theory and Application,"
Kreiger Publishing Company, 1968,.104
[27] Mendelson A.,"Plasticity: Theory and Application,"
Kreiger Publishing Company, 1968, 73.

[28] Saint-Venant, B., "Memoire sur l'etablissement des
equations differentielles des mouvement interieurs
operes," Compt. Rend.,70 (1870), 473-480.
[29] Levy, M., "Memoire sur les equations generales des
mouvements interieurs des corpes solides," Compt. Rend.70
(1870), 1323-1325.
[30] Hecker, S.S., "Metals Engineering Quaterly," 14
(1974), 30-36.