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CONSTITUTIVE MODELING FOR POLYCRYSTALLINE
ALUMINUM ALLOY EXTRUSIONS AND APPLICATION
TO HYDROFORMING OF THIN-WALLED TUBES

presented by
Yabo Guan

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PhD. degree in Mechanical Emeen'ng

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CONSTITUTIVE MODELING FOR POLYCRYSTALLINE ALUMINUM ALLOY
EXTRUSIONS AND APPLICATION TO HYDROFORMING OF
THIN-WALLED TUBES
By

Yabo Guan

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Mechanical Engineering

2003

ABSTRACT
CONSTITUTIVE MODELING FOR POLYCRYSTALLINE ALUMINUM ALLOY
EXTRUSIONS AND APPLICATION TO HYDROFORMING OF THIN-WALLED
TUBES
By
Yabo Guan

Hydroforming of thin-walled hollow extrusions has become popular with automotive and
materials industries for making complex structural components. Because most of the
hydroformed parts are made with steel, very little research has been done on the
hydroforming of extruded aluminum tubes, which are lighter and environmentally
friendly. As for any forming operation, computer modeling, using finite element analysis
(FEA), is today a common tool used at the design stage of the process. However, in order
to obtain realistic predictions, accurate material descriptions (constitutive models) will be
needed. In this work, several material models have been developed for accurate
simulation of metal forming processes, specifically hydroforming of extruded aluminum
tubes.

A Taylor-type polycrystalline model, based on a rate-independent single crystal yield
surface and rigid plasticity, has been developed and implemented into ABAQUS/Explicit
finite element (FE) code using VUMAT. It has served directly as a constitutive law in the
FE to calculate the crystallographic texture evolution during the hydroforming of an
extruded aluminum tube. Initial crystallographic texture measured with OIM and uniaxial
tensile test results are input to this F EA model. Although very accurate in representing

the material anisotropy, finite element analysis based on direct use of a polycrystal model

is extremely CPU-intensive, making it unfeasible for design purposes.

To develop the most accurate phenomenological model for forming of extruded
aluminum tubes, this polycrystal model was used to predict the anisotropy coefficients of
Barlat’s Yld96 yield fianction. This phenomenological anisotropic yield function was also
implemented into ABAQUS/Implicit finite element code, using UMAT, for simulation of
bulging and hydroforming of a 6061-T4 extruded aluminum tubes. It was shown that
compared with von Mises and Hill’s 1948 yield function, the Yld96 material model’s
predictions are in better agreement with experimental results.

In order to take into account the anisotropic hardening of aluminum tubes a new flow
potential, as a function of the anisotropy coefficients and deviatoric stresses, was
proposed. The evolution of the anisotropy coefficients was proposed to be a linear
function of the strain path, with its proportionality constant determined from
experimental measurements. This model was also implemented into ABAQUS/Implicit
code and shown to be capable of predicting deformation strains very accurately. It is
concluded that in the absence of experimental results or reliable data, the anisotropic
model could be used at early design stage to evaluate the accuracy of the

phenomenological models’ predictions.

ACKNOWLEDGMENTS

I would like to express gratitude to major professor, Dr. Farhang Pourboghrat, for his
instruction, guidance, patience and encouragement during the Ph.D. Study.

Special thanks are extended to Dr. Fre’déric Barlat for his excellent support and valuable
assistance through this work.

I would like to thank my committee members, Drs. R. Averill, T. Pence and M.
Miklavcic for viewing this dissertation and for their valuable comments.

Thanks are due to the research members in our group, including Dr. W-R Yu, Dr. M.
Zampaloni and Mr. N. Abedrabbo.

I would like to thank Mr. John Brem for providing valuable assistance to perform
mechanical testing of 611 1-T4 tubes, and Mr. Edward Llewellyn for conducting and
analysis of Orientation Imaging Microscopy (OIM) of the aluminum tubes.

A final note of appreciate must go to my parents and my wife Haiyan for their support
and encouragement through all these years.

Funding of this research by National Science Foundations is most gratefully

acknowledged.

iv

TABLE OF CONTENTS

LIST OF TABLES
LIST OF FIGURES

CHAPTER 1
INTRODUCTION
1.1 What’s Tube Hydroforming
1.2 Constitutive Models
1.3 Methodology
1.4 Literature Reviews

CHAPTER 2

PRELIMINARY OF CRYSTAL PLASTICITY
2.1 Slip in Single Crystal
2.2 Deformation of Polycrystals
2.3 Self-hardening and Latent-hardening

CHAPTER 3

vii

viii

LAWN—‘fl

11
ll
13
16

POLYCRYSTAL MODELING BASED ON GAMBIN’S SINGLE CRYSTAL MODEL

AND ITS APPLICATION TO FINITE ELEMENT ANALYSIS
3.1 Grain Orientation
3.2 Gambin’s Single Crystal Model
3.3 Boundary Conditions
3.4 Strain Hardening
3.5 Spin and Rotation
3.6 Implementation into Finite Element Code
3.7 Verification of the Polycrystal Model

CHPATER 4
PHENOMENOLOGICAL MODELING
4.1 Phenomenological Yield Function
4.2 Yld96 Yield Function
4.3 Material Characterization of 6061-T4 Extruded Aluminum Tube
4.4 Implementation of Barlat’s Yld96 into ABAQUS

CHAPTER 5
PHENOMENOLOGICAL MODELING WITH ANISOTROPIC STRAIN
HARDENING

5.1 Strain Hardening in Single Crystal and Polycrystal

5.2 Isotropic and Anisotropic Strain Hardening

5.3 Formulation for Prediction of Anisotropic Coefficients

5.4 Evaluation of the Anisotropic Strain Hardening Model

18
19
20
21
25
27
29
33

49
49
50
52
55

64

65
66
7O

CHAPTER 6

EXPERIMENTAL SETUP AND PROCESS 89
6.1 Experiment Setup 89
6.2 Tube Hydroforming Process 90
6.3 Sequence of Operation in Experiment ~ 92

Appendix A

INTRODUCTION TO SPIN AND ROTATION 94

Appendix B

CALCULATION OF ANISOTROPIC COEFFICIENTS FOR PLANE STRESS CASE

OF YLD96 98

Appendix C

PREDICTED YIELD SURFACES BASED ON THE PROPORTIONAL LOADING 104

REFERENCE 1 14

vi

LIST OF TABLES

Table 2-1
FCC single crystal vertices

Table 3-1
Dimensions and mechanical properties of the tube

Table 3-2
Sensitivity with respect to total simulation times (1 grain orientation, 10 elements)

Table 3-3
Sensitivity with respect to number of grain orientations (1.0ms, 10 elements)

Table 3-4
Sensitivity with respect to number of grain orientations (1 Oms, 10 elements)

Table 3-5
Sensitivity with respect to number of elements (lgrain orientation, lms)

Table 4-1
Yld96 yield function coefficients for 6061-T4 extruded aluminum tube

Table 5-1
Hoop strain at element 1421 & 1429 for different yield functions

vii

16

37

39

4O

41

42

54

79

LIST OF FIGURES

Figure 1-1

The orthogonal anisotropy axes of tube 10
Figure 2-1

Slip elements in uniaxial tension 12
Figure 3-1

Slip systems rotation when a crystal with only one slip system is subjected to an
extension 29
Figure 3-2

Rotation axis and angle corresponding to a spin W - {2 29
Figure 3-3

Pole figure (11]) with 1005 grain orientations representing the undeformed tube 32
Figure 3-4a

OIM pole figure representing the undeformed 6061-T4 aluminum tube 33
Figure 3—4b

CODF for the undeformed 606l-T4 aluminum tube . 33
Figure 3-5

Uniaxial tension simulation (one single element) 34
Figure 3-6

Comparison of experimental uniaxial tension strain/stress curve with those obtained by
ABAQUS using polycrystal model (VUMAT) 35
Figure 3-7

Strain/stress curves predicted for uniaxial tension using polycrystal model outside of
ABAQUS (100 of 100,000 increments from VUMAT were used) 36
Figure 3-8

Finite element model used for simulating the uniaxial tension test with ABAQUS/implicit
code 36
Figure 3-9

Predicted strain/stress curves for uniaxial tension using polycrystal model outside of
ABAQUS (all 508 increments from ABAQUS/implicit code were used) 37

viii

Figure 3-10
Kinetic and internal energy histories for 10ms, 10 grain orientations, 10 elements

simulation 42
Figure 3-11

Undeformed and deformed shape of bulged tube 43
Figure 3-12

Strain and stress distribution of the bulged tube (a) hoop strain, (b) axial strain, (c) hoop
stress, ((1) axial stress 43
Figure 3-13

(a) strain path from explicit simulation (1 Oms, 10 grain orientations)

(b) strain path from implicit simulation (Yld96) 45
Figure 3-14

(111) pole figure after bulging (strain path obtained from ABAQUS/explicit) 47
Figure 3-15

(111) pole figure after bulging (strain path obtained from ABAQUS/implicit) 47
Figure 3-16

(111) pole figure after bulging obtained from OIM 47
Figure 3-17

(a) OIM pole figure representing the bulged 6061 -T4 aluminum tube 48
(b) CODF representing the bulged 6061-T4 aluminum tube

Figure 4-1

Experimental stress-strain curve measured in the extrusion direction (ED) 53
Figure 4-2

Stress vs. plastic work in various directions 53
Figure 4-3

Yield surface (Yld96) for 6061-T4 (plane stress case and shear stress is equal to zero) 54
Figure 4-4

Specimen used for uniaxial tensile test 55
Figure 4-5

Stress/strain curve comparison between experiment and Yld96 56
Figure 4-6

Hoop strain comparison for different yield functions 57

ix

Figure 4-7

Deformation and strain/stress distribution for bulged tube 59
Figure 4-8

Comer region of the tube deformed in a square die (circle grids were used to measure the
strains) 60
Figure 4-9

Major strain distribution comparison between numerical simulation and experiment 62
Figure 4-10

Deformed tube, square die and strain distribution

(a) deformed tube (b)square die (c) axial strain ((1) hoop strain 63
Figure 5-1

Predicted initial yield surface based on a 0.5% plastic strain offset for the 6061-T4
extruded aluminum tube 71
Figure 5-2

Yield surface based on von Mises model 74
Yield surface based on isotropic hardening Yld96 model, or equal to zero 75
Yield surface based on anisotropic hardening Yld96 model, or equal to -0.00001 76
Figure 5-3

Different strain paths predicted by different material models 77
Figure 5-4

(a) Strain paths for element #1429

(b) Strain paths for element #1421 78
Figure 5-5a—d

Yield surfaces comparison based on different strain paths for element 1429 80-83
Figure 5-6a—d

Yield surfaces comparison based on different strain paths for element 1421 84-87
Figure 5-7

Hoop strain distribution with different a-values 88
Figure 6-1

Experimental setup in MSU laboratory 90
Figure 6-2

The die set used for experiment 91

Figure 6-3
The principle of tube hydroforming

(a) original tube shape and (b) final tube shape (before unloading) 91
Figure 6-4

Working diagram for the hydroforming process 92
Figure 6-5

Failure modes occurring in tube hydroforming 92
Figure 6-6

Deformed tubes without failure 93
Figure A1

(a) Axes and unit base vectors; (b) two-dimensional rotation; (c) small three-dimensional
rotation 97
Figure A2

Rotation of a vector r viewed as a rotation about a fixed axis 0=6b according to Euler’
theorem; on the right a top view along the axis 0 is shown 97
Figure B]

Stress-strain curve (extrusion direction) from polycrystal simulation and experimental
result ‘ 101
Figure 82

Boundary condition in different reference frames 102
Figure Cla-d

Yield surface comparison for uniaxial tension (ED) 104-105
Figure C2a-d

Yield surface comparison for uniaxial tension (TD) 106-107
Figure C3a-d

Yield surface comparison for plane strain (822 = 0) 108-109
Figure C4a-d

Yield surface comparison for plane strain (8,, = 0) 110-111
Figure C4a-d

Yield surface comparison for balanced biaxial tension 112-113

xi

CHAPTER 1
INTRODUCTION

1.1What’s Tube Hydroforming

Tube hydroforming has shown itself to be a competitive alternative to conventional
stamping when producing tube-like components by reducing process steps and the overall
production cost. Tube hydroforming also tends to improve the structural strength and
stiffness of a component compared to those consisted of several stamped sheets. This
allows weight reduction without losing component performance.

During a hydroforming process, a tube is first placed in the closed cavity of a forming
die. Afier the ends of the tubes are sealed, the hydraulic fluid is injected into the inner of
the tube and the tube is formed to conform the shape of the die cavity [Dohmann and

Hartl I997].

Hydroforming of thin-walled extrusion tubes is receiving the greatest attention, especially
in the auto industry [Vari-Form Inc 1998], because existing multi-piece, stamped/welded
assemblies in auto body and frame structures potentially can be replaced with less
expensive hydroformed parts that are lighter, stronger and more precise. If components
are manufactured from aluminum alloy tubes, this can translate into significant weight
saving for the structure as well. Because most of the hydroformed parts are made with
steel, very little research has been performed on hydroforming of aluminum alloy tubes.
As for any forming operation, computer modeling (using finite element analysis) is today
a major tool used at the design stage of the process. However, in order to obtain realistic
predictions, accurate material descriptions (constitutive models) are needed to simulate

hydroforming processes specifically for aluminum alloy tubes.

1.2 Constitutive Models

A constitutive model is necessary for the description of the material behavior in
numerical simulations of any forming operations. Large plastic deformation taking place
in the process requires using a plasticity model. The plasticity of materials can be
described using two types of models: physical (based on crystallographic texture,

dislocation, precipitate, solute, etc) and phenomenological.
1.2.1 Polycglstal Constitutive Model

Most man-made materials are an aggregate of crystals or ‘polycrystals’. Their
crystallographic orientation is generally non-random. All non-random orientation
distributions are called preferred orientations or ‘texture’. In textured polycrystals, many
macroscopic physical (mechanical) properties are anisotropic, i.e. they depend on
direction [Kocks UF, et al. 2000]. Plasticity is one of the most anisotropic of material
behaviors. Plastic anisotropy of polycrystals arises from two causes: anisotropy of the
single crystal’s properties and texture. It is important to study the single crystal properties
and the way in which grains interact in a polycrystal since it leads to a formulation of the

constitutive relations to be used in the simulations of polycrystal plasticity.

Plastic deformation is one of the main causes of texture formation and change. Glide of
many dislocations results in slip, which is the most common manifestation of plastic
deformation in crystalline solids [Hull and Bacon, 1984]. Polycrystal plasticity embodies
the slip systems and the crystallographic orientation (texture) in a constitutive framework
that can be incorporated in numerical formulations for the analysis of the deformation

process.

1.2.2 Phenomenological Constitutive Model

The basic of a phenomenological constitutive model comprises of a yield criterion
representing a surface which separates the elastic and plastic regions of the stress space, a
flow potential (yield function if it is associated) which its gradient represents the

direction of the plastic strain, and a strain hardening rule.

1.2.3 Combination of Polycrystal Plasticity and FEM

Incorporation of polycrystal plasticity with finite element formulation can be grouped
into two categories. One category includes those simulations that have a single
orientation within each finite element. Elements represent crystals or portions of crystals.
The second category includes simulations in which every element contains an
approximation to the crystal orientation. The orientations in each element may number as
few as only tens of grains, or may be in the thousands [Dawson and Marin 1998]. In this
work we focus on the second category of the formulations.

1.3 Methodology

In this work, polycrystalline plasticity serves as a constitutive theory in essentially the
same way as a continuum elasto-plasticity model. A rate-independent polycrystal model
was used to calculate the texture evolution. This is based on a single crystal yield surface
with round-off corners, first developed independently by Gambin (1991) and Arminjon
(1991). Polycrystal behavior was based on the full constraint homogenization procedure
(Taylor assumption). The crystallographic texture of aluminum that was used as input
data in the polycrystalline was characterized experimentally using Orientation Imaging

Microscopy (OIM).

The advantage of polycrystalline constitutive models is that they can predict anisotropic
strain hardening effects due to the crystallographic texture evolution during the
hydroforming process of tubes. However, in practice, ‘ numerical simulations using
polycrystal models are difficult to perform and time-consuming; and, therefore, they
can’t be used at a design stage by process engineers. So a phenomenological constitutive
model that can be implemented in the commercial code needs to be developed.

In the phenomenological constitutive theory, the plastic behavior is assumed to be well
described with a yield surface that evolves during plastic deformation. For an aluminum
alloy tube, only yield stress in extrusion direction (ED) (see Figure 1-1) can be measured
and it is difficult to measure the yield stresses and r values (width to thickness strain
ratio) in the tangential direction '(TD), radial direction (RD), and 45° between ED and
TD, which must be used for the prediction of the yield surface. Since the crystallographic
texture of the aluminum can be characterized using OIM, we can use this crystallographic
texture as input data for polycrystalline model to numerically determine the yield stresses
and r-values in other three directions. Once these values are determined, the coefficients
of yield function (which describe the yield surface) can be computed by solving a system
of nonlinear equations, as described in the appendix B.

In order to take into account the anisotrpic hardening of aluminum tubes a new flow
potential, as a function of the anisotropy coefficients and deviatoric stresses, was
proposed. The evolution of the anisotropy coefficients was proposed to be a linear
firnction of the strain path, with its proportionality constant determined from
experimental measurements. This model was also implemented into ABAQUS/Implicit

code and shown to be capable of predicting deformation strains very accurately. It is

concluded that in the absence of experimental results or reliable data, the polycrystal
model could be used at early design stage to evaluate the accuracy of the
phenomenological models’ predictions.

1.4 Literature Reviews

1. 4. [Polycrystal Plasticity

Quantitative descriptions of plastic flow by crystallographic slip have been developed
during the past 100 years. One of the significant descriptions of them can be attributed to
the work of Taylor and co-workers (1934,1938). Constitutive equation for elasto-plastic
behavior of ductile single crystal from the standpoint of modern mechanics were first
formulated by Mandel (1965) and Hill (1966), and extend to finite deformation by Rice
(1971), Hill and Rice (1972), Asaro and Rice (1977), and Asaro (1983a,b). In these
models, intended for low homologous temperatures, the plastic deformation of single
crystals by crystallographic slip is idealized to be rate independent.

There have been three long-standing problems in the rate-independent theory of crystal
plasticity. The first is to determine which slip systems are active and the second is to
determine the increments of shear on the active slip systems. Third, because of the typical
multiplicity of slip system in ductile crystals, the selection of slip systems required to
produce an arbitrary deformation increment is not necessarily unique [Anand et al. 1996].
In order to overcome the slip system ambiguity problem in crystal plasticity, rate
sensitive polycrystal models were developed and applied to deformation texture
simulations [Bishop and Hill 1951a,1951b, Pan and Rice 1983, Asaro and Rice 1977,

Asaro and Needleman I985, Molinari et al. 1987, Becker 1991, Kalidindi et al. 1992,

Lebensohn and Tome 1993, Beaudoin et al. 1993, Tome and Canova 1998, Dawson and
Beaudoin 1998, Dao and Li 2001].

However, the rate sensitivity is more introduced for numerical convergence issues than
for accuracy in the physical description of aluminum alloys, which are usually not rate
sensitive. In the context of rate independent plasticity, another approach using single
crystal yield surface with rounded corners was proposed by Gambin to remove the
ambiguity problem [Gambin 1991, 1992; Gambin and Barlat 1997]. Gambin was able to
correlate a parameter of the single crystal yield surface to the stacking fault energy (SFE)
of the material. The model was able to predict a different percentage of <100> fiber
texture in axisymmetric extrusion for materials with different SF E. A polycrystal model
based on Gambin’s single crystal yield function has been used to predict the texture
evolution during the hydroforming process of circular tube [Pourboghrat and Barlat
2002]. It also has been implemented into a finite element solver (ABAQUS) and served
directly as constitutive law in the finite element analysis for calculating the evolution of
crystallographic texture [Guan, Pourboghrat, and Barlat 2003].

In order to describe the relations between stress and motion at the microscopic scale and
the corresponding quantities at the macroscopic scale, a number of micro-macro linking
assumptions have been made. Most widely used and perhaps simplest one is the uniform
strain assumption [Taylor 193 8] that assumes all crystals experience identical strain equal
to the macroscopic strain. This assumption ensures compatibility and provides an upper
bound to the stress. The Sachs model [Sachs 1928], on the other hand, assumes
proportional (not uniform) loading in the grains, and satisfies neither compatibility nor

stress equilibrium. It is different from other so-called uniform stress assumption that

requires the stress to be identical from crystal to crystal [Ahzi et al. 1993]. This model
gives the low bound to the stress. Self-consistent model [Molinari et al. 1987,1997;
Lebensohn and Tome 1993; Tome and Canova 1998; Tome 1999] treats each grain as an
inhomogeneity embedded in and interacting with a homogeneous effective medium
(HEM). Self-consistent theory can give a solution that satisfies both compatibility and
equilibrium.

1.4.2 Phenomenological Plasticity

In the phenomenological approach, the plastic behavior of metals is assumed to be well
described by an analytical yield function. To account for the initial anisotropy, several
anisotropic yield functions associated with isotropic strain hardening have been proposed.
Hill’s 1948 planar anisotropic quadratic yield function is certainly the most popular for
describing the plastic behavior of orthotropic materials such as rolled steel sheets [Hill
1948]. It is an extension of the von Mises criterion for isotropic plasticity. Several non-
quadratic yield functions were developed by Hill (1979,1990), Hershey (1954), Hosford
(1972), Bassani (1977), Gotoh (1977), Logon and Hosford (1980), Barlat and Lian
(1989), Barlat et al. (1991) and Karafillis and Boyce (1993).

These yield functions can be used to improve the yielding description for aluminum
alloys. Hosford (1972) introduced an isotropic yield function based on the results of
polycrystal calculations. Barlat and Lian (1989) proposed a non-quadratic yield function
that introduces a coupling between normal and shear stress components, which is in
agreement with findings based on polycrystalline plasticity theory. The criteria suggested
by Barlat et al. (1991) account for six stress components that can be implemented to

general three dimensional elasto-plastic continuum codes. The most accurate yield

function for aluminum alloy sheet was proposed by Barlat et al. (1997). This so-called
Yld96 yield fimction simultaneously accounts for the anisotropy of uniaxial yield stresses
and r-values. This yield function takes seven parameters into account in the plane stress
condition. Lademo et al. (1999) evaluated three criteria proposed by Hill (1948), Barlat
and Lian (1989) and Karafillis and Boyce (1993) based on the accurate measurements of
the uniaxial behavior of aluminum alloys. Wu et al. (2003) predicted the anisotropic
behavior of aluminum alloys using the 3-D yield criteria including Barlat et al.
(1991,1997) and Karafillis and Boyce (1993). The evaluations of the yield functions were
extended to the comparisons of the computed and experimental forming limit diagrams.
Most recently, a new plane stress yield function Yld2000-2d [Barlat et al. 2003] that well
describes the anisotropic behavior of sheet metal, in particular, aluminum alloy sheets,
was proposed. Since it is based on the formulation using two linear transformations on
the Cauchy stress tensor, convexity of yield surface could be proven which is an
important requirement in numerical simulation to ensure the uniqueness of a solution.
1.4.3 Finite Element Analysis of Tube Hydroforming

Built on the successful application of the finite element method on sheet metal forming, a
number of finite element formulation had been proposed to simulate the hydroforming
process. In Hu et al. (1997), a rigid viscoplastic finite element formulation was developed
to analyze the gas-pressure constrained bulging processes of superplastic circular sheets
in cone disk shape dies. A computational method based on the membrane theory for the
analysis of axisymmetn'c sheet metal forming processes such as punch stretching, deep
drawing and hydroforming was presented in Hsu and Chu (1985). In Noh and Yang

(1998) work, a general formulation for hydroforming of arbitrarily shaped boxes was

expressed. A finite element program based on rigid-plastic approach and the backward
tracing scheme was developed for design in hydroforming [Kim et al. 2002].

In recent years, due to the advent of the high-speed computers, the use of 3D commercial
finite element codes (LS-DYNA3D, ABAQUS etc) to model the complex hydroforming
became possible. Simulations of forming different automotive parts by hydroforming
process were carried out using LS-DYNA3D [Ni 1994; Wu et al. 1996; MacDonald et al.
2001; Trana 2002].

Formability analysis has become a subject of increasing interest in the hydroforming
process. Davies et al. (2000) investigated the forming limits of aluminum extrusion
AA6051 in both the T4 and T6 tempers under laboratory conditions. In Asnafi et al.
(2000), forming limit curve (FLC) was determined by the hydroforming. Lei et al. (2002)
predicted the bursting in tube hydroforming process by using rigid-plastic FEM
combined with ductile fracture criterion.

Successful tube hydroforming depends on the reasonable combination of internal
pressure and the axial feed load at the tube ends. Yang et al. (2001) dealt with the optimal
process design of the hydroforming using numerical simulation by the explicit finite
element code combined with an optimization tool. In Gelin et al. (2002), different
approaches were proposed to find by numerical simulation and optimization the loading
curve versus process parameters that minimize the thickness variations. Manabe et al.
(2002) studied the effects of process parameter and material properties on deformation
process in tube hydroforming.

Two-dimensional model can provide insight into the detailed defamation and

stress/strain development otherwise lost in a more complex three-dimensional model.

Chen (2000) studied the tube expansion (also known as corner filling) based on two-
dimensional plane stress model. Most recently Guan et al. (2003) and Sharma et al.
(2003) proposed a “cross-section” analysis method for calculating tube stretching and
bending followed by hydroforming and expansion to fill the die in the hydroforming

process.

RD (Radial Direction)

TD (Tangential Direction)

 

> ED (Extrusion Direction)

 

Figure 1-1 The orthogonal anisotropy axes of tube

10

CHAPTER 2
PRELIMINARY OF CRYSTAL PLASTICITY

In this chapter, basic concepts about crystal plasticity, such as slip system, Schmid law,
Taylor assumption, self-hardening, latent hardening are introduced.

2.] Slip in Single Crystal

It is necessary to introduce the slip in a single crystal because it is the primary
deformation mechanism in most crystals. It involves the shearing on crystallographic
planes in crystallographic directions. When a crystal is subjected to stress, slip begins

when the resolved shear stress rm, on some slip system reaches a critical value, Tc (called

the critical resolved shear. stress). This simple yield criterion for crystallographic slip is
called Schmid’s Law that can be expressed as the following:

rm, = 7, (2-1)
where subscripts n and d refer to the normal of slip plane and the slip direction

respectively. If a crystal Is subjected to the stress state of 0' the shear stress on any slip

”'9

system can be written as
55221 1 a (2-2)

I

”,1,” are the direction cosines between the axes of the stress coordinate system and slip

system. In a special case, Figure 2-1 shows a crystal with normal cross-section area A. A

tensile load acting on it generates a uniaxial stress P/ A = on. The resolved shear stress
cm, in the slip plane n and along the slip direction p can be found from Eq.2-2.

Ind = [with an (2-3)

11

Defining 7L as the angle between the slip direction and the tensile axis, and (p as the angle

between the tensile axis and the slip plane normal, the Schmid’s Law for this special case

may be expressed as

1c: 0, cos]. cosgo (2-4)
or

a, =rc/ m, (2-5)
where the Schmid factor mx= cos A cos (a.
The combination of slip plane and its direction of slip is known as a slip system. There
are 12 slip systems (<110>/{111}) in the face-centered cube crystal (e.g. aluminum).

<110> denotes the slip directions on the slip plane and {111} denotes the normal of the

slip plane.

tensile axis

slip plane normal, It
slip direction, (I

 

 

 

Figure 2-1 slip elements in uniaxial tension

It can be proven that, for all FCC crystals, all orientations within the so-called basic

stereographic triangle (bounded by [100],[110], and [111] corners), the Schmid factor for

12

slip system <110>/{111}is higher than for other system. This slip system is known as
primary slip system.

2.2 Deformation of Polycrystals

The mechanical properties of polycrystal are very different from single crystals. Slip on a
single slip system cannot occur in a polycrystal because deformation in the various grains

has to be compatible.
2. 2.] Independent Slip Systems in Polycrystals

For any FCC (face centered cubic) crystal with tensile axis inside the basic stereographic
triangle, the deformation should occur at the primary slip system. However, if this crystal
is surrounded by other crystals with different crystallographic orientations (in the
polycrystalline aggregate), the strain taking place in the first crystal would have to be
compatible with the strain in the other crystals (grains). Von Mises (1928) stated that five
independent slip systems are required to produce a general homogenOus strain in a crystal
by slip. It can be presented as the following: since the plastic flow generally occurs
without voltune change (incompressibility of plastic deformation), therefore,

911+ szz+s33 = O. This relationship reduces the components of strain from six
(811,822,833,6‘12,823,£31) to five. Therefore, five independent slip systems are required for
the deformation of one grain in a polycrystalline aggregate [Meyers et al. 1984].

2.2.2 Taylor Analysis

The deformation of polycrystal is much more complicated than that of single crystals, and
great efforts have been devoted to bridge the gap between mechanical behaviors of single

crystal and polycrystals.

13

Sachs (1928) proposed that the tensile yield strength of a polycrystal should be the
average of the tensile yield strength of the crystal. It is also called lower-bound analysis.
This idea assumes that only one slip system is active in each grain. In fact, when a
polycrystal deforms, the shape change in each grain must be compatible with that of the
neighboring grains. Such shape compatibility cannot be maintained by single slip.

Consequently, Sachs analysis underestimates the polycrystalline stress-strain curve.

In order to satisfy this compatibility, Taylor (1938) proposed that all the grains undergo
the same shape change as the entire polycrystal, namely, all crystals experience identical

strain equal to the macroscopic strain. It is also called upper-bound analysis.

He assumed that this strain was imparted to each grain by the operation of five
independent slip systems. Since the FCC structure has 12 slip systems, there are 792

different ways of choosing 5 slip systems from a group of 12 using the combination
12! . . . ' .
theory (5777). Many of the 792 3le system combrnatrons are not possrble because they

can’t produce 5 independent strains. Taylor (1938) searched all the possible combinations
and calculated the set of five Schmid factors (M) for the various slip systems based on

the assumption that the critical stress for slips is the same for all active slip systems. He

obtained the average 1171 (for FCC) over all orientations of the tensile axis and found:
a = Mr = 3.06r
6:7/A7I=7/3.06 (2-6)

where r is the critical stress for slip, 7 is the accumulated shear strain for all active slip

systems. 0' ands are the macroscopic average stress and strain respectively.

14

This relationship is very important for the later strain hardening analysis in the
polycrystal modeling.

2.2.3 Bishop and Hill Analysis

Taylor’s analysis can find the slip systems that geometrically produce a specified shape
change. However, it cannot be guaranteed if there is a stress state that could physically
activate the required slip systems. Bishop and Hill (1951a, 1951b) approached the
problem by considering the stress states capable of simultaneously activating five or more
slip systems. They found that, for FCC crystals, there are only 28 such stress states. They
used the principle of maximum virtual work to determine which of the 28 stress states (or
their negatives) would be appropriate for a given shape change. These so-called Bishop
I and Hill vertices lie on the single crystal yield surface obeying the Schmid’s Law. Since
these 28 stress states are used as the initial stress guess in the polycrystal modeling
calculation that will be discussed in the Chapter 3, it is essential to discuss Bishop and
Hill vertices that are listed by Bishop (1953).

These 28 stresses are also reproduced below in terms of A, B, C, F, G and H (see

definition below). The corresponding stress states are:
S]1=(C-B)/3 822=(A-C)/3 S33=(B-A)/3

S23 = F 331 = G S12 = H (2-7)

15

 

 

 

 

 

 

 

 

 

# A=322—S33 B=S33-SII C=SII-322 F=$23 G=S31 H=512
I 1 -1 o o o o
2 o 1 -1 o o o
3 -1 o 1 o o o
4 o o o 1 o o
5 o o o o 1 o
6 o o o o o 1
7 '/2 -1 '/z 0 V2 0
8 '/2 -1 '/2 o -'/2 o
9 -1 ‘/2 '/2 '/2 0 o
10 -1 '/2 '/2 -A o o
11 V2 '/2 -1 o o '/2
12 V2 '/2 -1 o o -‘/2
13 '/2 o -'A 'A o '/2
14 '/2 o -'/2 -'/2 o '/2
15 '/2 o -‘/2 '/2 o -'/2
16 V2 0 -'/2 -‘/2 o -'/2
17 o -'/2 '/2 o '/z '/2
18 o A '/z o A 'A
19 o -‘/2 '/z o '/2 -‘/2
20 o '/2 '/2 o -'/2 V2
21 -'/z '/2 o '/2 V2 0
22 -'/2 '/2 o -'/2 '/2‘ o
23 -'A '/2 o '/2 A o
24 A V2 0 A -A o
25 o o o '/z ‘/z -'/2
26 o o o '/2 -'/2 '/2
27 o o o -'/2 ‘/2 '/2
28 o o o '/2 '/2 '/2

 

Table 2-1 FCC single crystal vertices (from Bishop, 1953)

2.3 Self-hardening and Latent-hardening (Asaro RJ, 1983c)

As mentioned before, grains (crystals) deform inhomogeneously, i.e. only certain planes
in the family of possible slip planes undergo slip. Moreover, the slip would disrupt the
imperfections (dislocations), which in turn would make it more difficult to activate

further slip and thus cause strain hardening [Ewing and Rosenhain 1900].

16

Strain hardening that occurs on the active slip system is called “self-hardening”. The
individual slip systems harden with strain and that slip on one slip system hardens other
slip systems, even if the latter are not active. This is known as “ latent hardening”.

Two very important aspects of crystal strain hardening were stated in Taylor and Elam
(1925): (1) Slip systems are hardened by slip on other systems (whether they themselves
are active or not) and (2) this latent hardening is at least comparable in magnitude to self-
hardening.

A rate-independent hardening rule can be written as

drf” = Z hapdy‘m (2-8)
p

where 2:“) is the current yield strength on the a slip system, and hat, the hardening rates.

The off-diagonal terms in the matrix h represent latent hardening.

If the latent-hardening rates is assumed to be equal to self-hardening rate i.e. hafl = h for

all a and ,6, isotropic hardening is assumed. However, experiments actually indicate that

the latent-hardening rate is slightly larger than the self-hardening rate.

17

CHAPTER 3

POLYCRYSTAL MODELING BASED ON GAMBIN’S SINGLE CRYSTAL
MODEL AND ITS APPLICATION TO FINITE ELEMENT ANALYSIS

In the polycrystalline approach, the material is assumed to be a polycrystal consisting of
many single crystals (grains). Plastic flow is assumed to occur only by crystalline slip on
given slip systems within each crystal. A homogenization scheme is necessary in order to
calculate the plastic response of a polycrystalline aggregate. Taylor scheme in which the
strain rate is considered to be identical in each single crystal and equal to the macroscopic
strain rate was used. The yield stress is derived from polycrystal calculations as an
averaged behavior over the total number of single crystals. This approach can be used to
take crystallographic texture evolution into account during forming operations. Some
specifications about the treatment of polycrystals are considered [Besdo et al., 2001]:

(l) The polycrystal consist of N crystallites (grains) with equal volume, though grain

size is known to have a consistent effect on the yield stress and work hardening.

(2) The statistical orientation of each crystal is given by the Euler angles 0, ¢, ([1.

No information about the spatial arrangement of the crystals in polycrystal is
available.
(3) Elastic anisotropy is neglected, namely, Young’s moduli remain constant along
different orientation.
In this chapter, a Taylor type (full constraint homogenization scheme) polycrystalline
model, based on a rate-independent single crystal yield function will be described. This
polycrystal model was directly implemented into a finite element (FE) program as a

constitutive law to calculate the crystallographic texture development during the

18

hydroforming of extruded aluminum tubes. Crystallographic texture measured using
Orientation Imaging Microscopy (OIM) and uniaxial tensile test results were used as
input to this FEA model. In order to efficiently and practically simulate the tube
hydroforming process using the polycrystal model through the explicit finite element
code, texture discretization (grain number), loading rate (step time) and element size

sensitivities were studied.

3.1 Grain Orientations

A global reference frame, x1, x2 and x3 are given where a load or a velocity field is
applied. Each grain is described by the orientation of its crystallographic axes, chc2 and

c3 , and an orthogonal transformation matrix P = l: p17] as:

x1 01
x, =1> c, A (3-1)
X3 ca

The components of a vector v are denoted by vi in x“ It2 and x3 (global frame) and v; in
chc2 and c3 (crystal flame). The components of a tensor T are denoted by ti}. in x“ 112 and
It3 and t; in chc2 and c3. The relationships between the components in the two frames
are

[VI] = [pg-Irlvj] and [v1] = [pg-Hm

[t;]=lpa][tu][P1,-] and [ta-l=[p.-,.][ti1][pi,-] (3-2)
Among the many ways to represent an orientation, the Euler angles are well accepted.

Roe’s Euler angles define the orientation of a grain, whose crystal axes (c1 ,c2 and c3) are

19

initially superimposed on the global reference frame (Itl , x2 and x3). This grain is then
subjected to the succession of three rotations, first around 2 = c3 , then around c2 , and

finally around c3. The corresponding transformation matrix P is

coswcosficosgp—siny/sino -cosy/cost9$in(o—siny/cosqp cosy/sinfl
= sing/cosdcoso-l-cosy/singo —sinI//cos6lsin¢+cosy/cosgo sing/Sim? (3-3)
— sin 9 cos (a sin 6 sin a) cos 0

3.2 Gambin’s Single Crystal Model

This model is based on the single crystal yield surface with round-off corners, first
developed independently by Gambin (1991) and Arminjon (1991). Because of the
absence of a sharp vertex, there is no ambiguity in the selection of the slip systems. This
single. crystal model is rate-independent and suitable for aluminum that is not very

sensitive to the strain rate.

This model is time independent with a yield ftmction and an associated flow rule

= 1 (3-4)

 

 

6‘ 9’
S

where r" is called critical value of the slip system and 1:5 is the resolved shear stress

C

‘6 99
S

on a slip system , defined by the slip plane n and the slip direction b

t" = bfnjsy. (3‘5)

where is so. the deviatoric stress tensor. For convenience, a homogeneous function of

degree one can be used for the yield function

20

¢=¢§ 13-6)

The associated flow rule L; --= 2:1 gives the rate of deformation D‘ = [d;] tensor and
s”.
the plastic spin tensor {2‘ = [0); ]
d; = 51— 2£+_61fi_ (symmetric part of L3.) (3-7)
2 6s”. as},
(of = 1 pi ~61 (anti-symmetric part of L;) (3-8)
’ 2 as”. as},

The plastic spin tensor expresses the rotation rate of the crystal axes with respect to the

principal deformation axes of the crystal. The superscript c denotes the crystal frame.

Note that because I and 45 are both first degree homogeneous functions of s”. the plastic
work rate can be written as follows

.fls -ié¢i-a—:s =j§£75 (3-9)

W=ds = .. _ ..
" ” as”. " 61" as”. " 61"

Therefore, the shear rate on each system is given by

(3-10)

 

 

 

 

3.3 Boundary Conditions

The boundary conditions can be stress-imposed, strain-imposed or mixed. In any case,
the material response consists of the set of a stress tensor S, a rate of deformation tensor

D and a spin tensor W. Whatever boundary condition imposed, these three tensors are

21

known after a time increment. The upper bound (Taylor) model will be described in this

section.
3. 3.] Taylor Model

The proposed formulation of the Gambin single crystal plasticity may be directly
introduced into the upper-bound polycrystal model proposed by Taylor (1938). It is
assumed the global kinematical field is imposed on all the grains and the Cauchy stress of
polycrystal can be taken as a single number arithmetic average of Cauchy stress in the
various grains.

3.3.2 Strain Imposed Boundary Condition

A velocity gradient in a global reference frame It], XL and X3 can be decomposed into a

rate of deformation tensor and a spin tensor
L‘=[v,‘j]=D‘+W‘ . (3-11)

The superscript denotes the global frame. For one crystal, the rate of deformation tensor
D‘ is imposed and model gives crystal stress SC and 0‘, the plastic spin. The Taylor
assumption is now enforced i.e. the crystal rate of deformation, D‘ , is identical to the
global rate of deformation 0" . In the crystal frame, it can be written as

PTD‘P = D‘ (3-12)

{2‘ is different for each grain whereas W 3 does not depend on a particular grain. After an

incremental time step dt, the principal rate of deformation axes rotates by the quantity
W‘dt with respect to the global frame. At the same time, the crystal rotate by

—.Q‘dt with respect to the principal rate of deformation axes. The net result is a rotation

22

of the crystal by the quantity (PTW ‘P-Q‘)dt in the crystal frame. The polycrystalline

stress deviator, E , is the arithmetic average of the stress deviator in each grain
>2 = _1_ 2 51! (3-13)
N g 8

where S‘ = PS‘PT.
3.3.3 Non-linear System of Equations for Taylor Model

Because the yield function is highly non-linear, the stress in each grain is obtained
numerically using a non-linear solver, Newton-Raphson. There are 6 unknowns, the 5

components of the stress deviator and zl. One equation indicates that yield function is

satisfied and five equations give the rate of deformation tensor. The system to solve is

given by
P] =16¢l6s“—dn =0 - (3-14-1)
F, = iawasn — d2, = o (3-14-2)
F3=¢-¢' =0 (3-14-3)
E, = 2(a¢ / as” + a¢/as,, ) - 2d,, = o (3-14-4)
F, = Marl/633, +6¢/as,3)—2d3l = o (3.14-5)
F6 = urns/as” +a¢/as,,)—2d,, = o (3-14-6)

where the d”. are the components of D which is imposed to the crystal. Using the vector

notation for the tensor (with indices 11, 22, 33, 23, 31, 12, 32, 13, 2.), the following derivatives

are used for calculating the flow rule and the Jacobian matrix of the non-linear system

flé=l£a—¢,i=l to9 (3-15-1)
6s. a (p as,

I

23

 

2 2 _1
2L_l¢[aji’l_(“ )Q‘K—afl],i=1to9,j=1to9 (3-15-2)

 

 

 

6s 16s,- ago 6s 6s. arp as,- as]-
a—2
where WELT—L ,i=1to9 (3-16-1)
25:3 132': r:
62¢ TS 0-2
=a a- -1 , i=1 to 9, '=1 to 9 3-16-2
6s] -,-6s ( )gfl‘; {g 1':— z'c J ( )

 

 

For the first step, several types of initial estimate for the linear solver can be imposed.
The Bishop and Hill vertex (Chapter 2) corresponding to maximum plastic work rate for
the imposed rate of deformation, or the rate of deformation itself can be used. If there is a
convergence problem, the solution can be obtained for a series of exponents, starting at a
= 2 (easy convergence) and finishing at the desired exponent. In this method, the solution
obtained for an exponent ‘a’ serves as the initial estimate for iterations with the next
exponent. Finally, if all of the above estimates do not lead to a good convergence, the
solution for the highest known exponent (less than a) can be used as a reasonable
approximation.

For the next steps, the solution for the previous step can be used as the initial estimate in

addition to the initial estimates discussed above. The initial estimate for A is obtained

from the Euler theorem for homogenous functions,

5115

s..—
U as”

=¢ (3-17)

because It is homogeneous of degree one. As a consequence, the initial estimate for A is

2: s0. d /¢ (3-18)

24

3.4 Strain Hardening
The classical strain hardening law is used in this work. In the time interval (It, the critical

resolved shear stress and accumulated shear strain increases on each slip system for a

given grain, where z‘c'dt and y‘dt , are related through the following expression

 

dr.’ =22; dlrl (3-19)

5
In the absence of any information concerning hardening, it is reasonable to assume

isotropic hardening, i.e., all the partial derivatives are equal. As a result,

drc’ = drc = h(z

 

 

y" )Zdly‘l = h(r)dr (320)

where F is the accumulated slip and h the ,rate of hardening for a grain. However,
experiments on single crystals (for instance, Franciosi, 1980) report that latent hardening
can be higher than self-hardening, i.e., the hardening matrix off-diagonal components are
larger than diagonal components. This trend is also suggested by cross-loading
experiments on polycrystalline materials (for instance, Sang and Lloyd, 1979; Lloyd and
Sang, 1979). Sequences of tension followed by simple shear in a manner that new slip
systems become activated on reloading lead generally in higher flow stresses (Lopes et

al., 2003). Therefore, for anisotropic hardening, it can be assumed that

 

 

62"
‘ =h F H 3-21-1
673 ( ) ’5' ( )
with
dH(F)
h F = 3-21-2
( ) dr ( >

25

where H is the shear stress — shear strain curve and Hm are the components of a matrix

with constant coefficients. The critical resolved shear stress change is therefore given by

 

 

drc’ = h(r)Z Hmd y~‘| ' (3-22)
the shear rate on a system is
s s “—2
y" = M” T 2 ’3 (323)
(r:)

 

 

As an application, the shear stress—shear strain curve H and the rate of hardening h can be

given by the Voce law
H (F) = A' — B'exp(-C'F) (3-24)
h(F) = B'C'exp(—C'F) (3-25)
If the macroscopic stress — strain curve of a material can be described by a macroscopic

Voce law
0' = A — Bexp(—C£) (3-26)
the coefficients A’, B/ and C’ can be roughly approximated from A, B and C, assuming a
constant Taylor factor M equal to 3 (o = Mr, F = Me , see Chapter 2)
A'=A/M B'=B/M C'=C/M (3-27)
Finally, if the normalized hardening matrix H is assumed to contain only two

coefficients, p and q that characterize self hardening and latent hardening, respectively,

the critical resolved shear stress changes are given by

d7: {(1)-(1)612" +qu]h(F) (3-28)

26

3.5 Spin and Rotation

3. 5.! Plastic Spin

Figure 3-1 shows an example of a crystal with only one slip system subjected to an
extension to illustrate how the lattice rotates during the plastic deformation. More

generally, the plastic spin is given by
42132-2] (3-29)

3.5.2 Total Spin

The total spin for a grain with respect to the global frame is W -.Q. This is an anti-
symmetric tensor, which operating on a vector v, gives the rate of change of this vector
(Fig. 3-2). The new crystal axes are given by 5i = ci + dci = (I + W -.Q)ci

However, in a time step dt, the new vectors c1+dc1, c2+dc2 and c3+dc3 do not keep their
orthonormal property. Therefore, it is better to find the rotation axis and angle, and to
operate the pure rotation on the crystal axes. It is easy to show that there is only one
eigenvalue for the tensor I +W 42, u =1, which corresponds to the rotation axis, r.
Then, the eigenvectors can be calculated. Another way to calculate the rotation vector is
to write the spin matrix 0 = W -[2 as a matrix (see A8 in Appendix-A) made of three

vectors 0], 02 and 03

0 012 -031
0 = W - Q = [01 02 03] = -012 0 023 (3-30)
031 ‘023 0

It can be shown that the rotation axis is

 

r=oz Xo3/loz X03 , or r=o3Xo]/|o3xol|, or r=ol on/lol on| (3-31)

27

The rotation angle is (Fig. 3-2)

tgéi = sign (Lei x dci )Idci I / (ll —(r.ci )2 (3-32)

where dci = Oci. As for the rotation axis, there are three possible ways to calculate E,i ,

depending on the choice of ci. In order to calculate the new transformation matrix P after

rotation, it is necessary to calculate the coordinates of the vectors 5i rotated from ci by
g, (see A-13 in Appendix-A)

5i = cos §ci + (r.ci )(l — cos §)r + sin gr X ci = Qilcl + Qizcz + Qi3c3 (3-33)
The transformation matrix P becomes PQ after the rotation of the crystal, where matrix

Q is given by

- r12 (1 -cos§)+ cosE, r1r2(1—cos§)+ r3 sing r1r3 (1 -cos§)—r2 sinE,
Q= r2r1(1—cos§)-r3 sing r22 (l—cosé)+cOs§ r2r3 (l—cos§)+rlsin§ (3-34)

 

 

br3r1(1-cos§)+rzsin§ r3r2(l-cos§)—rlsin§ r32(l—cos§)+cosl§ _

This transformation matrix and the corresponding Euler angles have to be updated in

order to account for the re-orientation of the grains during the plastic straining.

28

 

 

 

 

 

Figure 3-2. Rotation axis and angle corresponding to a spin W - fl .

3.6 Implementation into Finite Element Code

The polycrystal model was implemented into ABAQUS/Explicit FE code, assuming a
full polycrystal at each integration point. The orientation and hardening of individual

grains associated with each element is updated as deformation proceeds.
3. 6. l Oua_si-static Analysis

ABAQUS/Explicit can be an effective tool for solving a wide variety of nonlinear solid
and structural mechanics problems. Particularly, it is very efficient in solving certain
class of nonlinear problems that are essentially static. Tube hydroforming simulation is
one of these highly nonlinear quasi-static problems. It involves very large membrane

deformation, buckling and complex frictional contact condition. One might encounter

29

severe convergence problems if ABAQUS/Standard (implicit solver) is used instead for

the simulation.

Applying the explicit dynamics procedure to quasi-static problems requires some special
considerations. Since a static solution is, by definition, a long-time solution, it is often
computationally impractical to analyze the simulation in its natural time scale, which
could require an excessive number of small time increments. To obtain an economical
solution, the event must be accelerated in some way. The problem is that as the event is
accelerated, the state of static equilibrium evolves into a state of dynamic equilibrium in

which inertial forces become more dominant.

The computer time involved in running a simulation using explicit time integration with a
given mesh is directly proportional to the time period of the event. This is because ’

numerical stability considerations restrict the time increment to

At s min[L" i—e—J (3-35)
A+2p

where the minimum is taken over all elements in the mesh, L" is a characteristic length
associated with an element, p is the density of the material in the element, and A and p.
are the effective Lame’s constants for the material in the element. Since this condition
effectively means that the time increment can be no larger than the time required to
propagate a stress wave across an element, the computer time involved in running a
quasi-static analysis can be very large. The cost of the simulation is directly proportional
to the number of time increment required, n = T / At , if At remains constant, where T is

time period of the event being simulated. (At will not remain constant in general, since

30

element distortion will change L" and nonlinear material response will change the

effective Lame’s constants and density. But the assumption is acceptable for the purpose

n = 7/max(L" i—p—J (3-36)
l+2p

In polycrystal model approach, using the Newton-Raphson iterative method to compute

of this discussion, thus,

the Cauchy stress for each integration point through the user material subroutine
(VUMAT) is very costly. An inexpensive alternative therefore is to obtain the strain path
for any critical location (integration point) from a FE simulation with a
phenomenological constitutive model (e.g., von Mises, Yld96, etc.) and use this

information as input for the polycrystal model to predict the evolved texture.
3.6.2 Initial Crystallographic Texture Input

In this approach, the polycrystal model discussed above was directly implemented into
the ABAQUS/Explicit code via the user material subroutine (VUMAT) option. Each
integration point in the shell element was viewed as an aggregate consisting of hundreds
of single crystals. The strain increment of each step for each integration point was

imposed to each single crystal as required by the Taylor-type model.

The material used in this study was 6061-T4 extruded aluminum tubes, which were
obtained by heat treatment (heated to 520 0C, maintained 1 hour, quenched to room
temperature in one minute) of the original 6061-T6 tubes purchased from an aluminum
vendor. Figure 3-3 shows the (111) pole figure for this material using 1005 grain

orientations. This represents in the statistical sense the texture shown in Figure 3-4 (OIM

31

pole figures and CODF) without losing the characteristic grain distribution. In the finite
element simulation, the initial crystallographic texture characterized by OIM is
incrementally updated using the spin tensor. Of course, the more grain orientations are
introduced into the calculation, the more accurate the crystallographic texture will be
represented. However, considering the expensive CPU time, if all the grain orientations
(about 90,000) were to be used to represent the initial crystallographic texture inside the
VUMAT, an extraordinary CPU time will be required to simulate the tube hydroforming
process. Based on the current computational condition we have used 10, 20 and 40 grain

orientations to represent the initial texture of the tube.

(111) poloflguo
(1005 grain orientations)

 

 

 

Figure 3-3. Pole Figure (1 1 1) with 1005 grain orientations,
representing the undeformed tube

32

 

 

 

 

 

 

 

Figure 3-4b. CODF for the undeformed 6061-T4 aluminum tube

3.7 Verification of the Polycrystal Model
3. 7.1 Uniaxial Tension Simulation

Uniaxial tension simulations were carried out in order to verify the capability of the
polycrystal model. Two finite element models were used for this purpose. The first model

considers the uniaxial tension of a single S4R element (see Figure 3-5). MPC (multi-

33

point-constraints) was used to constrain nodes 3 and 4 to remain on the straight line
parallel to the X-axis. Nodes 2 and 3 were pulled up to 20% of the unit length of the

element using the displacement control.

YA

.....................................

 

 

 

 

 

 

 

b
X
Figure 3-5 Uniaxial tension simulation (one single element)

For comparison, 1, 10 and 20 grain orientations were used in the simulations to represent
the initial texture of the material in the VUMAT. The total calculation time for this case
was lms. The strain/stress curves from ABAQUS and the experiment are shown in
Figure 3-6. It can be seen that the experimental stresses are underestimated when only
one grain is used, as this does not properly represent the true property (anisotropy) of the
material. When 10 grain orientations were used, although the experimental stresses were
underestimated, a significant improvement was observed. However, when 20 grain
orientations were used although the stress levels increased but a large oscillation in the
stresses up to 3% strain was also observed. It is believed that this stress oscillation was
due to dynamic effects (loading and unloading), since in this strain range the kinetic
energy is still rising, as shown in Figure 3-10. Also the nature of the rigid plastic
formulation used in the polycrystal model, which calculates the total stresses rather than
incremental stresses for each incremental deformation, contributes to this stress

oscillation.

34

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

350p---.,....
soof ' V -, — 1
3250 t. 3
PT :
£200 A J r4- 1
8 150;
h . .
(75 100:. ............
g :
I: 50E 1‘”
02
.50 - 3
-0.050 0.000 0.050 0.100 0.150 0.200 0.250
TrueStrain

Figure 3-6 Comparison of experimental uniaxial tension strain/stress curve with those
obtained by ABAQUS using polycrystal model (VUMAT)

To alleviate the extraordinary CPU cost of directly simulating with the polycrystal model,
the strain path for the uniaxial tension was extracted from the explicit code and used as
input to run the polycrystal model with large number of grains outside of the ABAQUS.
Figure 3-7 shows the strain/stress curves obtained from this approach for different
number of grain orientations. Again, some improvements in stress prediction up to 10%
strain could be observed by using more grain orientations (up to 1005), but beyond this
strain level the polycrystal model consistently underestimates the experimental stresses.
The reason for this could be that only 100 of the approximately 1000,000 increments

from the explicit code were used to generate these curves.

35

 

 

 

 

 

 

 

 

  

 

 

   
    

 

 

 

 

 

 

asoL..--,-... .fi.
/’
E 300 h f
a /
m 250 _
a) .
o .
h b
25 200
a)
a —+-1g.1ms
I" 150 +209.1ms
-—10059raln,1ms I
——Experlment j
100 ~~~ UM ~
0.000 0.050 0.100 0.150 0.200 0.250
True Strain

Figure 3-7 Strain/stress curves predicted for uniaxial tension using polycrystal model outside of ABAQUS
(100 of 1000,000 increments from VUMAT were used)

Another uniaxial tension simulation was based on the exact (dog bone) shape of the
specimen used in the experiment. Due to symmetry, only a quarter model was used for
the simulation (see Figure 3-8). Barlat’s Yld96 yield function (Chapter 4) was
implemented into the ABAQUS implicit code for this purpose. Similar to previous
method, the strain path was obtained from this implicit simulation and then was used as
input in the polycrystal model to predict the stress-strain curve. Figure 3-9 shows the
strain/stress curves from this approach for different number of grains. It could be seen
that a significant improvement has resulted regardless of the number of grains used. The
reason for this is that the exact strain path (i.e., 580 increments) as used in the implicit

code was used to simulate the uniaxial tension test with the polycrystal model.

 

Figure 3-8 Finite element model used for simulating the uniaxial tension test with ABAQUS/implicit code

36

 

 

 

 

 

 

 

  
  

 

 

 

 

 

 

350 fi-,fi..,...-.... n-
’ 1
l
E 300
g 1
m 250
m
o .
h n
25 200
g —-—20 grain
|_ 150 —-—1009raln .,
- —-—10059raln
C +Experiment

 

 

 

100’~~----~
0.000 0.050 0.100 0.150 0.200 0.250
True Strain

Figure 3-9 Predicted strain/stress curves for uniaxial tension using polycrystal model outside of ABAQUS
(all 508 increments from ABAQUS/implicit code were used)

3. 7.2 Bulge Test
Geometry, Mtgeriaj Properties and Loading in the F EA

The dimensions and material properties of the sample tube are shown in Table 3-1 below.

 

Length (inch) OD (inch) Thickness (inch) E (Gpa) V I 0' (Mp8)

 

 

 

 

 

 

8.0 2.0 0.05 71 0.3 129.0

 

 

 

Table 3-1 Dimensions and mechanical properties of the tube

The tube was modeled using SAX element type, which is a 2-node, axisymmetric shell
element. Three (3) integration points through the thickness were used in each shell
element. Due to the axial symmetry of the aluminum tube, only the meridian of the tube
was modeled. The plunger mechanism was not modeled but instead the axial feed was

directly applied to the tube’s ends by using a displacement boundary condition.

The contact between the tube and the die was modeled with the *CONTACT PAIR
option. The outside surface of the tube was defined by means of the *SURFACE

DEFINITION option. The die was modeled as analytical rigid surface with the *RIGID

37

SURFACE option. An analytical rigid surface yields a more accurate representative of
the doubly curved die and results in computational savings. The rigid die surfaces are
offset from the tube by half the thickness of the tube because the contact algorithm in
ABAQUS/Explicit takes the shell thickness into account. The mechanical interaction
between the contact surfaces was assumed to be frictional contact. The coefficient of

friction between tube and die was taken to be 0.1.

For finite-strain shell elements, such as SAX], the transverse shear stiffness is computed
by matching the shear response for the shell to that of a three-dimensional solid for the
case of bending about one axis. When VUMAT (user subroutine for ABAQUS/Explicit)
is used to define a shell element’s material response, the *TRANSVERSE SHEAR
STIFFNESS option must be used. The definition of the transverse shear stiffness of the

section of a shear flexible shell element is defined in ABAQUS (see Theory Manual).

A linearly increasing hydrostatic pressure was applied on the inside of the tube. The final
pressure was 14 Mpa (2030psi) as measured from the experiment. Simultaneously, the
ends of the tube were subjected to an axial displacement of 3mm from each side, which
was measured from the experiment and provided as input for the simulation. A viscous
pressure load was applied to the surface of the shell during the simulation to damp out
transient wave effects so that quasi-state equilibrium could be reached quickly. The

coefficient of viscous pressure was chosen to be approximately 1% of the value of pcd ,

where p is the material density of the tube and cd is the dilatational wave speed.

38

Finite element sensitivity analysis

As the material (aluminum alloy) is rate-independent, time was the only parameter used
in order to drive the loading process. However, in this case, the polycrystal model (rate-
independent itself) was implemented into the explicit code to represent material response,
and therefore the speed of the analysis, namely the loading rate, became very crucial in

the simulation. The loading rate can be increased so that the same physical event occurs

in less time (compared to the natural time in physical process), as long as the solution
remains nearly the same as the true solution and dynamic effects remain insignificant (see

ABAQUS Manual).

Smooth amplitude curves were required for accurate and efficient quasi-state analysis,
even though the exact loading curve extracted from the experimental loading profile was
applied. The number of grains was also an important parameter in the FE simulations.
The prediction of the texture evolution usually required a large aner of orientations to

be used in order to track and predict texture in the pole figures.
Sensitivity with respect to total simulation time

First, the influence of the total simulation time on the material response was investigated.
The simulations were carried out using 10 SAX elements and only 1 grain was
considered for each integration point. It should be noted that one grain does not represent
the anisotropy of the extruded aluminum tube, but provides for a CPU-efficient analysis.
The total simulation times of 0.1ms, 0.5ms, 1.0ms, 5.0ms, 10.0ms and 50.0ms were
tested. Table 3-2 shows the maximum hoop strains predicted by ABAQUS/Explicit code.

It was noted that as the total simulation time was increased, approaching the actual

39

deformation time, the rigid plastic model’s prediction of hoop strain also increased.
However, the predicted hoop strain of 14.3% for a total simulation time of 10ms was far
below the experimental maximum hoop strain of 20%. On the other hand, the predicted

hoop strain of 39% for the total simulation time of 50ms was much higher than the

experimental maximum hoop strain of 20%. To find out the optimal total simulation time
at which the maximum hoop strain no longer changes, we had to try higher simulation
times. However, due to the extremely large CPU time (approaching several weeks)

needed to simulate the case of 50ms, longer total simulation times were not attempted.

 

 

 

TotalTimc (Ins) 0.1 0.5 1.0 5.0 10.0 50.0
M“‘;::n"°°" 6.88% 9.22% 9.37% 1 1.3% 14.3% 39%

 

 

 

 

 

 

 

 

Table 3-2 Sensitivity with respect to total simulation time (1 grain orientation, 10 elements)
Sensitivity with respect to number of grain orientations
Next, the sensitivity of the numerical solution to the total number of grain orientations

used in the simulation of tube bulging was investigated. For this purpose a total

simulation time of 1.0ms and 10 SAX elements were used.

 

 

 

 

 

 

 

Number of Grain
Orentations ] 10 20 40
Maximum Hoop Strain 9.37% 10.0% 9.26% 9.24%

 

Table 3-3 Sensitivity with respect to number of grain orientatins (1 .Oms, 10 elements)

As could be seen Table 3-3, the variation in predicted maximum hoop strain, as a
function of the number of grain orientations used in the finite element simulation, is

negligible.

40

 

However, when the total simulation time was increased from 1.0ms to 10.0ms, the
predicted maximum hoop strain increased significantly, as a fimction of the number of

grain orientations used in the finite element simulation, as shown in Table 3-4.

 

 

 

 

 

 

 

Grain Orientation
Number I 10 20
Maximum Hoop Strain l4.3% 20.4% 23.0%

 

Table3-4 Sensitivity with respect to number of grain orientations (10ms, 10 elements)

The explanation for this increase in predicted hoop strain can be found in equations 3-35
and 3-36. That is, by increasing the total simulation time, the accuracy of the model
improves, since simulation time approaches the actual deformation time. Also, the
number of simulation increments proportionally increases with increases in total
simulation time. Since the polycrystal model is based on the rigid plasticity theory, which
treats total strain as plastic strain, the predicted final strain increases as the simulation
increments double. Finally, the results from Table 3-4 indicate that the simulation case
with 10 elements, 10 grains and 10ms predicts a maximum hoop strain of 20% which is

similar to the experimental strain.
Sensitivity with respect to number of elements

Another set of numerical experiments was performed to study the sensitivity of predicted
maximum hoop strain to the number of elements used in the finite element simulation.
Table 3-5 shows that by increasing the number of elements two and four folds, the
maximum hoop strain increases slightly. Unfortunately, by increasing the number of

elements by four folds the CPU time also increased by four folds. Therefore, it was not

41

feasible to try larger number of elements, in order to find the optimum number of

elements for tube bulging simulation.

 

Element Number 10 20 40

 

 

 

 

 

 

Maximum Hoop Strain 9.37% 10.2% l2.7%

 

Table 3-5 Sensitivity with the number of elements (1 grain orientation, lms)

Figure 3-10 shows a typical plot of the history of the kinetic and internal energies versus
the total simulation time (e.g., 10.0ms). It can be seen that the internal energy increases
smoothly to the final value, while the kinetic energy remains as much as 3-4 orders of
magnitude smaller than that, indicating that the dynamics effects are eventually

dampened and that the simulation is close to being quasi-static.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10'
105 r/
10‘ /
/
3 103 ' ' T/
g 2
“J 10 ///- -‘-\.
101 /
10° +lnternal Energy -—
1 -+-Kinetlc Engergy
10' .

 

0.00 2.00 4.00 5.00 8.00 10.00
Time(ms)

Figure 3-10 Kinetic and internal energy histories for

10ms, 10 grain orientations, 10 element simulation

Figure 3-11 shows the undeformed and deformed shapes of the bulged tube for the case

of 10ms, 10 grain orientations and 10 elements. Figures 3-12a-d show the corresponding

42

strain and stress distributions. The axisymmetric elements can be swept circumferentially

to produce (visually) three-dimensional tube.

Figure 3-1 1 Undeformed and deformed shape of bulged tube

 

LE1]
fi-

 

 

 

 

 

 

 

VALUE
-1.l le-01

-9.48e-02
—7.88e-02
-6.27e-02
-4.67e-02
-3.07e-02
-1 .47e-02
+1 .36e-03
+1 .74e-02
+3.34e-02

 

 

 

LE22
g In-

 

 

 

 

 

 

 

VALUE
-2.28e-04

+2.256-02
+4.52e-02
+6.79e-02
+9.06e-02
+1.13e-01
+1 .36e-01
+1 .59e-01
+1 .816-01
+2.04e-01

 

 

 

 

 

 

 

 

(b)

43

 

 

Sll

 

 

 

 

 

 

VALUE

"" +6.96e+01

+9.58e+01
+1 .22e+02
+1 .48e+02
+1 .74e+02
+2.00e+02
+2.27e+02
+2.53e+02
+2.79e+02
+3.05e+02

 

 

 

 

322

 

 

 

 

 

 

VALUE

“'1 -" +1 .36e+02

+ l .6 l e+02
+1 .86e+02
+2. 1 2e+02
+2.37e+02
+2.62e+02
+2.88e+02
+3. 13e+02
+3.38e+02
+3.63e+02

 

 

(C)

 

 

 

 

 

 

((1)

Figure 3-12 strain and stress distribution of the bulged tube

(a) hoop strain, (b) axial strain, (0) hoop stress, (d) axial stress

3.8 Observations on texture evolution

Texture information for more than 90,000 grain orientations was measured from the

undeformed tube sample by OIM at Alcoa (see Figure 3-3). From that set 1005 grain

44

orientation were extracted to represent the initial texture orientation of the tube, and to
use as input data to carry out the numerical simulation of the hydraulic bulging with
polycrystal model. The evolved texture of the hydraulically bulged extruded aluminum
tube was also measured by OIM at Alcoa. For the deformed tube sample, texture
information for more than 58,000 grain orientations was collected by OIM, from which
1011 grain orientations were again extracted to represent the deformed texture of the tube

(see Figure 3-16).

For comparison with OIM, strain paths (Figure 3-13a-b) for tube bulging were obtained
from ABAQUS explicit and implicit simulations (Yld96) and then used in the polycrystal

model (outside of ABAQUS) to predict the texture evolution.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Axial/Hoop Strain Axial/Hoop Strain
10ms,10 rain orientations '
0.20 ( , g r ) 0.20 g (Yld96)
0-15 * -°-Axia| Strain I 0.15 .. —°-Axia| Strain
0,10 +H°°P Stralnl // o o --Hoop Strain
_1
"’ 0.05 / ‘0
g / = g 0.05 . /
a °-°° = = =— a
H\ 0.00
-0.05 -.
~0.10 .. \\ -0.05
-0.15 -0.10
0.00 2.00 4.00 6.00 8.00 10.00 0.00 0.20 0.40 0.60 0.80 1 .00
Time(ms) Step
(21) (b)

Figure 3-13 (a) strain path from explicit simulation (10ms, 10 grain orientations)
(b) strain path from implicit simulation (Yld96)

45

Figures 3-14 and 3-15 show the predicted (111) pole figure for the aluminum tube after
the bugle test using the strain paths from the explicit simulation (10ms, 10 grain
orientations, 10 elements) and the implicit simulation with Barlat’s Yld96; respectively.
The details of the implicit simulation of bulge test using Barlat’s Yld96 can be found in
Chapter 4. A comparison of the two pole figure plots indicate that the two predicted
textures are comparable to each other. That is, predicted location and the intensity of
grains with (111) orientation are similar in both models. A comparison of predicted
textures with that obtained from OIM (Figure 3-16) indicates that the actual intensity is
somewhat underestimated by the models. Two possible explanations for this observed
difference in texture could be offered. First, it is shown before that the Taylor model’s
texture prediction is always different from the actual texture, since grain boundary
interactions are ignored, which results in unrealistic rotation of the grains. Second, this
could have been caused by the fact that the location on the deformed tube where texture
was measured was different from the location where it was measured on the undeformed
tube. Although it is generally assumed that the texture information represents a statistical
distribution of grains that is independent of the location where it is measured on the part,
the initial texture may have in fact been very different in the two locations where they

were measured before and after the deformation.

46

 

 

 

 

 

 

Figure 3-16 (1 1 1) pole figure afier bulging obtained from OIM

47

The complete pole figure and CODF measured from OIM for the bulged tube can be seen

in Figures 3-17a-b.

 

 

 

 

 

 

Figure 3-l7b CODF representing the bulged 6061-T4 aluminum tube

48

CHAPTER 4

PHENOMENOLOGICAL MODELING
As shown in Chapter 3, the main advantage in using polycrystalline constitutive model is
that it can predict anisotropic strain hardening effects due to the crystallographic texture
evolution during the hydroforming process. However, in practice, numerical simulation
with polycrystal models is difficult, time-consuming and therefore not feasible for
iterative design process. So, developing accurate phenomenological constitutive models
that could be implemented into commercial FE codes would be desirable.
4.1 Phenomenological Yield Function
In the phenomenological approach, the plastic behavior of the metal is well described by
an analytical yield function. The yield surface defines the critical stress levels for plastic .
yielding, and it often serves as a plastic potential for the plastic strain increment
according to the associated flow rule. For finite element analysis of forming processes,
the yield surface and the stress-strain curve (strain hardening) are important material
property data.
Several anisotropic yield functions associated with isotropic work hardening have been
proposed. Hill’s 1948 yield function is certainly the most popular for describing the
plastic behavior of anisotropic materials such as rolled steel sheets [Hill 1948]. Hosford
(1972) introduced an isotropic yield function based on the results of polycrystal
calculations. Hill (1990) proposed a nonquadratic yield function that can be used to
improve the yielding description for aluminum alloys. However, Barlat et al. (1997)
proposed the most accurate yield function for aluminum sheet alloys. This so-called

Yld96 yield function simultaneously accounts for the anisotropy of the uniaxial yield

49

stresses as well as r-values (i.e., width to thickness strain ratio). In order to describe the
anisotropic behavior of extruded aluminum tubes one must calculate the anisotropic
coefficients of the Yld96 yield function using known yield stresses and r-values.
4.2 Yld96 yield function
The yield function proposed by Barlat et al. (1997) is given by

¢ =a, |s,—s,|"‘ +a, |s,-s,|'“ +a, |s,—s,|"‘ = 26'“ (4-1)
where 6 is the effective stress and m is a material parameter which is typically set to 8

for FCC and 6 for BCC materials. Parameters SM,3 correspond to the principal values of

a symmetric matrix Sufi defined with respect to the components of the Cauchy stress as

 

 

 

C (an —a )—C (an—0'“)
3 W 3 2 C6 0,, C50“
C — -c - 4-2
S: C6 0,“ 1(ayy an) 3 3(0):); ayy) C4 02y ( )
C2‘“n“’u’f""w“’u’

 

 

In the yield function, ah“3 are defined as

a. = 0'.pr + 61.12221 + 0.1932. (4-3)
where p”, represents the components of the transformation matrix p between the
orthogonal anisotropy axes of the material and the principal axes of S. Here, a, , ay are

constant coefficients. For plane stress condition in which 02 = 0 and the z-direction is the
third principal direction,

a: = ago cos2 2,6 + 61:] sin2 2,6 (4-4)

50

where B is the angle between the major principal direction of the stress tensor S and the

rolling direction of the sheet or the extrusion direction of the tube. Also S can be

 

 

 

 

simplified to
C — +C 2
3 (axx 03y) 2 axx C6 Ox), 0
s,“ s,‘y 0 C C ( )
a - a -a
[Sun]: sxy s” 0 = C60,” ' W 33 x" W 0 (4'5)
0 0 Sn
0 —S,m—Syy
1. .l

The quantities Cl, C2 , C3, C6 , a, , ay, azo and or,l are the material coefficients which
describe the anisotropic properties of the material.
Utilizing the normality rule, the associated plastic strain increment A65” is obtained from

the yield function It as

A55, = 7 59¢ ‘ (4-6)

60",”

 

where y is a scalar function. Using plastic incompressibility, we can express the r-value

(width to thickness strain ratio) for any angle 6 as [Barlat and Lian, 1989]

£22 _ £11 _1 = all _1= 0' _1 (4_7)

r =— _
9 e e o e e e _ _
511+£22 £11+322 gn+sw [ 60' 60' ]
a

 

 

 

—+ —— a
60',lur 60'”

For plane stress (0'z = 0 ), when the shear stress sxy is equal to O, the yield function is

 

. +a,o sx—syl = 25" (4-8)

 

 

a
“hay

¢(S) = a.

 

 

sy—sz sz—sx

where

=_'—_0' _.._.

51

sy =— 3 03. + 3 y
52 =-%Zo'x -E3'—0'y . (4-9-1-3)

Assume that v is a constant, then

a ,v “:0
{V—z,7,—‘;a—,wl,vcz,vc3,ax,ay,a,] = ¢(ax,ay,a,o,c,,c2,c3,0',,0'y,0',) (4-10)

Therefore, among the C,, C2 , C3, (1 a, and azo material coefficients that describe

anisotropy, five of them are independent. To that end, it is suggested to set ago =1. When

the shear stress is not equal to zero, two other coefficients are also used to characterize

anisotropy, C 6 and at,I . In this case, the total number of anisotropy coefficients increases

to seven. Therefore, it is necessary to use seven experimental tests or polycrystal model
results to calculate these coefficients (see Appendix B for details).
4.3 Material characterization of 6061-T4 extruded aluminum tube
The 6061-T4 extruded aluminum tube samples were characterized using polycrystalline
model (Chapter 3), while working at the Alcoa Technical Center during the summer of
2002. The strain-stress curve along the tube axis (Extrusion Direction, ED) was measured
using a tensile test and the following Voce model was fitted to the experimental data (see
Figure 4-1):

a = 338.56 - 206.29 exp(—-9.822§) (4-1 1)
Using the method described in Appendix B, the stress-strain curves in other three
directions (tangential direction (TD), radial direction (RD) and 45 degree between ED
and TD) can be calculated. Figure 4-2 shows the stress vs. plastic work curves. The flow

stresses (alimaknm'npau) used to calculate the anisotropy coefficients were

52

determined at the plastic work amount of 50 Mpa (mm/mm). Flow stresses, r-values and

yield function coefficients for 6061-T4 are shown in Table 4-1. Figure 4-3 shows the

corresponding Yld96 yield surface for the 6061-T4 extruded aluminum tube.

N

Uniaxial Tensile ResuIt of 6061-T4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.5 10 , 1

A - Experiment l

g 3.0 102 .. —Fltthg CUNL/f’”

g 2.5 102 /

tn /

U)

2 2.0 102 /

5 o = A - B OXp(-Ce)

3 1 5 102 MA = 338.56 Mpa ......

F: ' ’ e . 205.29 Mpa

c=9.822
1.0 102 1 1
0.0 0.050 0.10 0.15 0.20 0.25

True Strain (s)

Figure 4-1 Experimental stress-strain curve measured in the extrusion direction (ED)

50

6061-T4 Uniaxial Tension

 

 

 

w: «on!
ED tension

 

 

 

 

 

a= 309.0(1.WW) "‘
TD tension

 

a=32&4(1.0527) "'4
RD tension

 

 

 

as 311.3 (1.0073)

 

-

 

 

45° tension "-
a=320.6(1.0325)

 

 

 

 

Polyemtalfmodelislmuaftlom

 

o i L 1 i i 1
0.0 10 20 30 40 50 50 70 80

Plastic Work

Figure 4-2 Stress vs. plastic work in various directions

53

 

 

 

 

 

 

 

 

 

Coefficients of
Input Data
Yld96

0'51) 1.0000 C1 0.9068
0'45 1.0375 C2 1.0313
cm 1.0627 C3 0.9474
0RD 1.0073 C6 0.9265
rED 0.6667 or,” 1 .265
1335 0.8519 Otyl 1.15
rm 0.9011 all 1.3
I'RD 0.6667 (120 1.0

 

 

 

 

 

 

Table 4-1 Yld96 yield function coefficients for 6061—T4 extruded aluminum tube

Yield Surface for 6061-T4

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

1.5
1.0 t,
0.50 1’ i
1b .
‘; 00
b I
-0.50 3'
I /
'1'0 -------- (von Mises)
010961
15 o (Gambin-Barlat)
'-1.5 -1.0 0.50 0.0 0.50 1.0 1.5
0' I

Figure 4-3 Yield surface (Yld96) for 6061-T4 (plane stress case and shear stress is equal to zero)

54

4.4 Implementation of Barlat’s Yld96 into ABAQUS

Barlat’s criterion (Yld96) was implemented into ABAQUS/ Standard via the user material
subroutine (UMAT) (see ABAQUS manual). With UMAT, users can implement their
own material properties into ABAQUS by providing the expression of a stress increment
with respect to a strain increment. Isotropic (von Mises) and anisotropic (Hill) material
models were also used to compare with the Yld96 yield function.

4. 4. 1 Finite element simulation of the uniaxial tensile test

The specimen used in the uniaxial tensile test was out directly from the extruded

aluminum tube sample, as shown in Figure 4-4.

’ fl. -. .‘u ‘
if. I .
"-méw , ‘
k .

Figure 4-4 Specimen used for uniaxial tensile test

Due to the shape of the tube, this tensile specimen had a slight curvature. For simplicity
and without the loss of accuracy, a flat dog bone shape model was used to simulate the
tension test (see Figure 3-8). Due to symmetry, only a quarter model was set up and 250
84R elements were used. The predicted stress-strain curve was then compared with the
experimental result from Alcoa, as shown in Figure 4-5. The predicted stress-strain curve

matched the experimental one very accurately.

55

Uniaxial tension stress vs strain
comparison between experiment and Yld96
350 ..-. .-.T

 

30° : WATT :
250 45/

/

 

 

200 : I

150:/

(

 

 

100

 

 

True Stress (M pa)

 

50 - Experiment .1
. —Yld96 ‘

 

 

 

 

 

 

 

 

0‘---~--
0.000 0.050 0.100 0.150 0.200
True Strain

Figure 4-5 Stress/strain curve comparison between experiment and Yld96

The material properties and tube geometry shown in Table 3-1 are used for the bulge test
and square die simulations. In this implicit analysis, two different dies; hydraulic bulge
die and square section hydroforming die were used for the numerical simulation. Both
dies were modeled as rigid body in the simulations. The *ORIENTATION option was
used with the S4R elements to define a local orientation that is initially coincident with
the global direction. The reason for using this option was to obtain the stress and strain
output given in a co-rotational framework (i.e. material axes).

4.4.2 Hydraulic Bulge Test

Hydraulic bulge test was conducted to study the formability of the 6061-T4 extruded
aluminum tube. A maximum internal pressure of 2030 psi (14 Mpa) was applied just
before the tube burst in the experiment. Due to symmetry, again only a quarter model was
used to simulate the tube bulging process.

Figure 4-6 shows a comparison of the measured and predicted hoop strain distributions.

In the numerical simulations, von Mises and Hill’s 1948 material models were also used

56

with ABAQUS’s S4R shell element and compared with Yld96 result. As expected, the
isotropic material model (von Mises) had the worst results and significantly
underestimated the magnitude of the maximum hoop strain (16.7%). Hill’s model, which
accounts for transverse anisotropy of the tube showed some improvement, but still
underestimated the measured hoop strain (18.1%). Although not perfect, the best strain
predictions were obtained with Yld96 yield function (19.18%), accounting for planner
anisotropy but assuming isotropic hardening. The maximum hoop strain measured

experimentally was about 20.2%.

Hoop Strain Distribution

. L L —o—m
I —-—st5
. -—E eriment
0.15 \ 'p
0.10 / ‘
0.05 ' i/ \\

r34 "”‘N

AAA; LAAA

 

 

 

 

 

 

Hoop Strain

 

 

 

 

 

 

o.00~ -- ---. --
-100.00 -50.00 0.00 50.00 100.00

Location (mm)

 

Figure 4-6 Hoop strain comparison for different yield functions

Figure 4-7-a-e show the bulged tube and the stress/strain contour in the ABAQUS

simulation.

 

(a)

57

 

 

 

VALUE
-4. 14e-02
-3.71e-02
-3 .28e-02
-2. 866-02
-2.43 e-02
-2 .0 1 e-02
-1 .5 8e-02
- l . l 6e-02
-7.30e-02
-4. 14e-03

 

 

 

 

 

VALUE
+2.73e-03
+2.41e-02
+4.55e-02
+6.69e-02
+8.83e-02
+1 . 10e-01
+1 3113-01
+1 .53e-01
+1 .74e-01
+1 .956-01

 

 

 

(b)

 

(C)

58

 

Sll

VALUE
-1 .94e+02

-l .46e+02

 

 

-9.98e+01
-5.1 1e+01
-3.48e+00
+4.41 e+01
+9.17e+01
+1 .39e+02
+1 .87e+02
+2.35e+02

 

 

 

 

 

 

 

 

(d)

 

 

$22 VALUE
""'"'" "' -9.23e+00

+3.09e+01
+7.1 1e+01
~---- 1- +l.l 1e+02
+1 .51e+02
+1 .92e+02
+2.32e+02
+2.72e+02
+3.12e+02
+3.52e+02

 

 

 

 

 

 

 

 

0‘ .
H ".1". ,' 3 ~ o...“
a -ALWAg-fi:

 

(6)

Figure 4-7 Deformation and strain/stress distribution for bulged tube
(a) deformation, (b) hoop strain, (0) axial strain, (d) hoop stress, (e) axial stress

59

4.4.3 Hydrotorming a round tube into a square section

The simulation of hydroforming a round tube into a square section was also performed.
Due to the complex geometry of the square die, the geometry of the square die was
created with the UNTGRAPHICS software and then imported into HYPERMESH to
create the mesh used in the ABAQUS. The major strains at the comer region of the

hydroformed tube were measured for comparison, as shown in Figure 4-8.

 

Figure 4-8 Corner region of the tube deformed in a square die
(circle grids were used to measure the strains)

Figures 4-9a-c show the comparison between the numerically predicted major strains at
the comer of the tube with the experimentally measured ones obtained from three
different tube samples (designated as H3, H4 and H5). These figures show that strain
predictions with Yld96 are higher than those predicted by the von Mises material model.
The comparison with the experimental data is more difficult, since there is quite a bit of
variation in the experimental data depending upon which sample is used. One consistent
observation that could be made is that the magnitude of the measured strains in the
middle of the tube are lower than those predicted by either model. However the
distribution of the measured strains along the length of tube are similar to those predicted

by both models. That is maximum at the ends and dropping in the middle of the tube.

60

Due to the difficulties that were experienced in the strain measurement process, the
experimental results were not very consistent, which made verification of the numerical

results more difficult. More accurate strain measurement, will be done on the deformed

 

 

tube in the near future.
Major Strain d the corner reglon
(sample H3)
14.00
12.00
5‘; 10.00
.E
2 8 00
a .
-% 5.00
2
"00 —o—von III“
—~—mss
—o— Experiment
2.00
-30.00 -20.00 -10.00 0.00 10.00 20.00 30.00
Location(mm)
(a)
Major Straln at the corner region
[sample H4]
14.00
12.00
is 10.00
.E
m
8 8.00
i
'10 6.00
2
4'00 —+——von ”I“!
-—--Yl096
—~—Eaperiment
2.00
-30.00 -20.00 -10.00 0.00 10.00 20.00 30.00
Location(mm)
(b)

61

Major Strain (%)

14.00

12.00

10.00

8.00

6.00

4.00

2.00

Major Strain at the corner region
(sample H5)

 

1
4
d
1
4
1
1
q
1
I
‘
a
4
1

0.. coo-ou-r

YrY UV

 

 

 

'1
f
. x
E"

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

—a A
-30.00 -20.00 -10.00 0.00 10.00 20.00 30.00

Location(mm)

(C)

 

Figure 4-9a-c Major strain distribution comparison
between numerical simulation and experiment

Figure 4-10a—d show the shape of deformed tube, mesh of the square die and the strain

distributions of the deformed tube.

 

a

, .
. \\s A I I
‘\ ‘
I‘tOI.“:“‘ ‘ ‘u

t \\
.
a . ‘ .u- ‘ \qu::
‘A u“

n
”flu:
“HA
4 .u '.

 

 

LEll VALUE
""1" -4.93e-02
--~- -4.28e-02
-3.63e-02
-2.98e-02
-2.33e-02
-l.68e-02
40162-07.
-3.84e—03
+2.66e-03
+9.16e-03

 

 

 

 

 

 

 

       

 

 

 

 

LE22 VALUE
"""1 "" +4.95e-02

“-1"- +1.94e-02
+3.38e-02
+4.82e-02
+6.26e-02
+7.70e-02
+9 Ida-0’)
+1.06e-01
+1.20e-01
+1.35e-01

 

 

 

 

 

   

 

((0

Figure 4-10 Deformed tube, square die and strain distributions
(a) deformed tube, (b)square die,(c) axial strain (d) hoop strain

63

CHAPTER 5
PHENOMENOLOGICAL MODELING WITH

ANISOTROPIC STRAIN HARDENING

5.1 Strain Hardening in Single Crystal and Polycrystal

When metals are deformed plastically, the strength and hardness usually increase. This
phenomenon is known as work hardening or strain hardening. It should be clarified that
this kind of hardening is caused by cold working, not due to precipitation hardening. As
mentioned in Chapter 2, slip in single crystal (or dislocation, more accurately), is the
primary mechanism for the plastic deformation. In microscopic scale, the strain
hardening also can be viewed as the required increase in the stress causing additional slip

[Dieter 1976].

In a single crystal, strain hardening is caused by dislocations interacting with each other
and with barriers that impede their motion through the crystal lattice. These interactions
include dislocation pile up, cross slip etc. In polycrystals, this basic idea remains valid.
However, due to the mutual interference of neighboring grains and the problem of
compatible deformation among the adjacent grains, multiple slip occurs rather easily and,
consequently, there occurs an appreciable work hardening right in the beginning of
straining. In a manner similar to that in the single crystals, primary dislocations interact
with secondary dislocations, giving rise to dislocation dipoles and loops, which result in
local dislocation tangles and eventually a three-dimensional network of sub-boundaries.
Generally, the size of these cells decrease with increasing strain. In high stacking fault

energy (SFE) FCC metals like Al, the dislocation tangles rearrange into a well-defined

cell structure while in the low-SFE metals (brasses, bronzes, etc) where the cross-slip is

rather difficult and dislocations are extended, the sharp sub-boundaries do not form even

at very large strains.

Dislocation density and distribution will be changed (dislocation multiplication) due to
these interactions when plastic strain increases. Both dislocation density and distribution
of dislocations are very sensitive to the crystal structure, stacking fault energy,

temperature, and the rate of deformation [Meyers 1984].
5.2 Isotropic and Anisotropic Strain Hardening

Most computations in metal forming are performed using the phenomenological theory of
plasticity, which treats the metal as an isotropic material. It is a challenge to incorporate
the initial anisotropy of metal into the theory of plasticity and to describe the subsequent

anisotropic plastic deformation.

To account for the initial anisotropy, anisotropic yield criteria have been proposed by
many researchers (see literature reviews for more detail). When these yield functions are
implemented into the finite element analysis, the assumption of isotropic strain hardening
is made because it is a reasonable approximation for proportional loading. However, in
most metal forming processes, including tube hydroforming, non-proportional loading,
unloading, and springback occurs which may require the use of the anisotropic hardening.
The experimental investigations of anisotropic hardening of metals have been proposed.
Phillips et al. (1972) presented the results of experiments in which thin-walled tubes of
Commercially pure aluminum in the annealed condition were loaded in combined tension
and torsion in the plastic region at elevated temperatures. In this procedure, the yield

Surfaces were determined by probing in many different stress directions from the elastic

65

region into the plastic range. Kuwabara et al. (2000) and Kuroda et al. (1999) proposed a
method for determining the subsequent yield surface in the vicinity of a current loading
point by using an abrupt strain path change and also applied this idea to real experimental
studies. Large strains can occur in many different loading paths. In the search for a
unified description of strain hardening, it is important to study the strain hardening under
different strain paths. Kopacz et al. (1999) presented the construction of equivalent stress-
equivalent strain curves. The equivalent strain and equivalent stress are based on the
average accumulated crystallographic shear and resolved shear stress respectively. Hu
(2000) proposed several anisotropic hardening models for sheet metal based on the
combination of the experimental stress-strain curves in different directions. Lopes et al.
(2003) obtained the uniaxial and simple shear stress-strain curves for an aluminum alloy
sheet sample in different specimen orientations with respect to the material symmetry
axes. The stress-strain curves were predicted using the self-consistent polycrystal code

[Lebensohn and Tome 1993].

The macroscopic parameters, such as normalized yield stress and r-values, are used to
determine the constant coefficients in the anisotropic yield functions [Barlat et al. 1997]
using the initial texture or experimental mechanical behavior data as input. However,
sometimes these yield functions with constant coefficients are not sufficient to describe
the anisotropic strain hardening after the onset of the initial yield. Lee (2002) proposed an
optimization method to predict the change in the anisotropic coefficients of the yield
functions [Barlat et al. 89; Barlat et al. 1991] using von Mises and Tresca yield potentials

as the upper and lower bound surfaces, respectively. This method assumes that the

66

direction of the change in anisotropic coefficients varies predominantly in the direction of

the yield surface gradient with respect to the anisotropic coefficients.

In this chapter, the linear directionality of the yield potential assumption is made and the
change of anisotropic coefficients is simply determined by a single parameter. The
evolution of the yield surface (Yld96) based on different loading path were calculated

and compared with those based on the crystallographic approach developed in chapter 3.
5.3 Formulation of the Anisotropic Strain Hardening
5.3.1 Evolgtion of the Anisotropy Coefficients

In order to describe the anisotropic strain hardening after the initial yield, the coefficients
of the yield function must evolve and can no longer remain constant. Therefore, the yield
surface will be assumed to be a function of the anisotropy coefficients and deviatoric

Stresses:
¢=¢(ci’0y) (5'1)

Furthermore, it will be assumed that the evolution of these anisotropy coefficients can be

expressed as following:

n+1 n 6¢

0 =c.+a— 5-2
ac ( )

where a is a scalar that should be calibrated based on experimental observations, C,

W

represents all the coefficients involved in the yield function, and 6— represents the
C.

I

variation of the yield function as a function of the anisotropy coefficients. This former

parameter could also be regarded as representing the dependency of the yield surface on

67

the strain path. In this research, for simplicity, it will be assumed that the value of the

parameter a remains constant and independent of the strain path
5.3.2 Evolution of the Yld96 Anisotropy C oefi‘icients

For the plane stress case of the Yld96 yield function, the variation of the yield function

W

with respect to the coefficients of anisotropy, -a—— , can be determined based on the
c

r'

quantities derived in Yoon et al. (2000), as follows.

Given that the stress components are:

0'

xx

[an]: a” (5-3)
0'

1'."

And the symmetric matrix S in Eq.4-5 is:

 

 

rC3(o, -0’2)+C20', -
S 3
Z = S = C102 —C3(al _02) (5_4)
* 3
Sxy C603

 

 

The principal values of S can be found as follows:

68

 

 

 

 

 

 

 

 

 

 

Sl ~ ~ ~ ~ 2
~ 52 L,+L,_ Ll—LZ +13
[77*]: S = 2 2
3
_0d —(s,+s,)
tan" EEITZH
L3

Then, E(o)’n Eq.4-1 can b e written as

| l
_ 1 ;; 1 m "' "' ;
A.» hat—Sr wig-s) W...) i

Given that the coefficients a, , a2 and a3 are defined as (see Eq. 4-3)
al at,r cos2 6 + ay sin2 6
[0,]: a2 = a, sin9+ay c0520

a3 a“, cos2 6 + orzl sin2 6

And by defining the following new vectors:

 

 

a!
[firl= ay ’ “:0 :1
dz]
r C: -
- __ 6;
[ck] ' 6'3
-06 -
. . 65 66 . . .
Then, the quantities —a—~— and -—~— can be derived, usrng the chain rule, to be
‘11: ck
2;: = {2m5"""’}" 1?
aqk 6%
_ 3 ~
2 {ZmEm-n} lZ-aj—Qfi- (for k=l~3)

a 660 62;,

69

(5-5)

(5-6)

(5-7)

(5-3)

(5-9)

(5-10)

2i={2ma<~-w}"3€
as, 65,

ant—I :a¢67?aaL b —l~

(5-11)

 

5.3.3 Evolution of the Yield Surface Based on the Polycrystal Model

In Chapter 4, it was shown that the coefficients of Yld96 are determined by fitting the
yield function to the yield stresses and r-values obtained from uniaxial tension, plane
strain and balanced biaxial tests (see Table 4-1). Given a plastic strain path, initial
crystallographic texture (Figure 3-3), and an offset plastic strain, the polycrystal model
can be used to define the evolved shape of the yield surface. In this method, however, the
value of the offset plastic strain has a direct impact on the size of the predicted yield
surface [Kuwabara et al. 2000 and Kuroda et al. 1999], and must be carefully chosen. In
this work, the value of the offset plastic strain is determined iteratively such that the
initial yield surface obtained by the phenomenological model matches that obtained by
the polycrystal model very closely. For the 6061-T4 extruded aluminum tubes it was
determined that an offset plastic strain value of 0.5% would result in the best match, as

shown in Figure 5-1.

70

Weld Surface

 

 

 

 

 

 

 

 

 

 

 

 

 

(initial texture)
1.50firv. -..- ---.,:....
: Maura.” ; ‘
1.00 Am ”rs
. / )1
l i
.0 0.50 ’ " i
\ i
E:
b l
0.00
-o.50 ’ /
X 1
x vsw 5%) l
- wage (derived from uttest) l
-1.00 - ~ -

-1 .00 -0.50 0.00 _0.50 1 .00 1 .50
0' I 0'
xx

Figure 5-1 Predicted initial yield surface based on a 0.5% plastic strain offset
for the 6061-T4 extruded aluminum tube

5. 3. 4 Evolution of the Yield Surface Based on the Yld96 Yield Function
Based on the method described above, the scalar parameter a was introduced into the
ABAQUS implicit code through the UMAT (user material subroutine) written for

Barlat’s Yld96 yield function. During a finite element run, the anisotropic coefficients c,
are evolved based on Eq. 5-2 for a given value of the scalar parameter a, and the
deformation strain path. At the end of the FE run, the evolved anisotropic coefficients c,

are retrieved and used to plot the new yield surface. In this work, all yield surfaces are
presented in a normalized form. That is, all stresses are normalized by the effective flow
stress, before being plotted. This method allows one to see the evolution of the yield

surface shape clearly.

71

5.4 Evaluation of the Anisotropic Strain Hardening Model

5. 4. 1 Yield Surface Evolution under Proportional Load_ir_rg

The evolution of the Yld96 yield function following proportional loadings, such as
uniaxial tension, plane strain and balanced biaxial tension were investigated. For
comparison, the shape of the evolved Yld96 yield function was compared to those
predicted by the polycrystal model. Appendix C shows the results of all these yield
surface comparisons for those specific proportional loadings mentioned above. In
general, it can be concluded that for all loading cases the yield surface shape predicted by
the Yld96 yield function and the polycrystal model are in good agreement for all plastic
strains less than 25%. This is particularly true for uniaxial and balanced biaxial cases.
This comparison is not as good, however, for plane strain cases, and this deviation
becomes exacerbated with increased plastic deformation. This deviation could be
attributed to that fact that the initial anisotropy coefficients for the Yld96 yield function
were fitted to the yield stresses and r-values obtained from uniaxial and balanced biaxial
tests and not to the plane strain values.

5.4.2 Yield Surface Evolution under Non-Proportional Loading —Tube Bulging

For the bulge test, with a few iterations it was determined that a value of a = - 0.00001
would predict a maximum hoop strain of 18.4%, which although not optimal is
reasonably close to the experimentally measured value of 19%. The result from the
isotropic hardening case (a=0.0) was 19.18%. Figures 5-2a-c show the yield surfaces
predicted by several material models for the location where the maximum hoop strain
occurs. In this study, strain paths were first obtained from finite element analysis of tube

bulging using von Mises, isotropic hardening Yld96, and anisotropic hardening Yld96

72

models. Figure 5-3 shows that predicted strain paths are different for different material
models. Furthermore, Figure 5-3 shows that the von Mises material model predicts a
maximum hoop strain of 15.9%. Figures 5-2a-c show the comparison of the
phenomenological yield surfaces versus those predicted by the polycrystal model. They
also show the corresponding stress paths leading to the final deformation. Figure 5-2a
shows that l) the two yield surfaces do not match very closely, and 2) at the final
deformation point, the von Mises yield surface is above the polycrystal yield surface.
Figures 5-2b and 5-20 show that l) the two yield surfaces match closely, and 2) at the
final deformation point, both isotropic and anisotropic Yld96 yield surfaces and
polycrystal yield surfaces are very close to each other. Considering that the von Mises
material model underestimated the experimental hoop strain, while the isotropic and
anisotropic Yld96 material models did not, it can be concluded that the above comparison
method has some merits. That is, the polycrystal model’s result could be used as a mean
to evaluate the accuracy of the phenomenological models’ strain predictions, where

experimental results are either missing or not reliable.

73

Yield Surface Comparison (element #1429)
(Tube bulging with von Mises)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.50 LT fT T 1" T V T r .r f r r r r r T r f I 7 T V f
1.00
.o "-5" """"" 1*
\g 4
b 4
0.00
,
. . : : 1
-050 - i :- --------- ~
. x 11901591.) :1 .
' - von Mises ' ' ‘
’ Stress history , :
-100 . . . . r L . a . l L L . . I A . . . l . L 2 A
-1.00 -0.50 0.00 _D.50 1.00 1.50
o I o
)01
Yield Surface Comparison (element #1429)
(Tube bulging with von Mises)
1.1a -...,...- r-fi
:o 0 9 '5'. o . I
_ _.'§ 0 . .
114 b {fix E ‘x i :
.. .a E . :X 3
u- .l .. E 4
112 72-4 --------------- :1. x g
x I
1.10 E ---------------------------------- . ----------------- --------- .29 --------------- -
I 5 1x
I b _ E . :
\ 108 r- ---------------- p ----------------- 5 ----------------- :h-u-u-o-u-o-ui ------------ -1
g l- 3 b x 1
D t : E x 1
; J
103 ----- ,1 ----------------- 1m. ---------- q
r a x 1
E 0 I
1.04 b... ............... 1 ............................. ’5. "d
I ‘ 5 s 0 I
- x YS(0.5%) ; x.
1-02 f I von Mises """"""""""" """""""""" """"""" 31
_ - Stress history 5 5 5 .
1.00' ..,--.1...-i...-1--..‘
0.50 0.60 0.70 .080 0.90 1.00

ale
)0(

Figure 5-2a Yield surface based on von Mises model

74

Yield Surface Comparison (element #1429)
(Tube bulging with alpha=0.0)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.50s........,.-......,.
. ' r
A
1.00
I l
| b 0.50
x _ 4
E . .
O .
0.00
r 5 : : ‘
-0.50 . - -5 :- -------- -
. x rerun) if i ‘
’ - YldQB(alpha=0.0)
’ Stress history
-100 . . . L x . . . . 1 L . . A I . I A I I . . 4
-1.00 -0.50 0.00 _ 0.50 1.00 1.50
o I o
)0(

Yield Surface Comparison (element #1429)
(Tube bulging with alpha=0.0)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.16 L I I T T V I Tfi T V Y I 7 fl V I I I V
. x 1
L. C I . . . . . x ‘
1.14 Jug-:3 ------ x -------------- .
,. . a ' . X
. - ' x
o ' I
1.12 l _ , 3 x
_. . ' 9x
1.10 - ----------------------------------- . ----------------- . ------------------ .... ---------- -
l D x
' ' O
\ r- l
g: 1.00 - ---------------- 1 ----------------- » ----------------- g ----------------- :.. X.
0 - x j
1.05 r """"""""""""""""""""""""""""" i """"""""""" ‘ """""" 5‘"
10‘ L ..... , ................. s ................................ X:
l x YS(0.S%) l
1-07 .' a wasararpnazon) """"""" g """"""""" s """""""" ‘.
- Stress histo : E .
100 . r . I L. J . . r W. . . . 14 . . . I i L r m
0.50 0.60 0.70 _ 0.00 0.90 1.00
o I a
)or

Figure 5-2b Yield surface based on isotropic
hardening Yld96 model , or equal to zero

75

Yield Surface Comparison (element #1429)
(“the bulging with alpha-0.00001)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.50 I W I I I I I I I I I I I I I‘T I I I I I I I I
h . ' a . 1
. i
)- 4
1.00
ID 0.50 ~ ----------- —
N . 4
g . .
O .
0.00 — ------------------------------------------------- -
i ..
050 l ‘ ' l
' - x vsrosqs) é’r 4
- - YldQG(alpha=-0.00001) s s 1
: Stress history
_1'oo . . I . 1 J . . i 1 A I . . I . . r . I . . I A
-1.00 -0.50 0.00 _ 0.50 1.00 1.50
o I 0
>01
Yield Surface Comparison (element #1429)
(Tube bulging with alpha=-0.00001)
1.16 I I ' I—I I I I I I
h.I . j
t . . . 0 n . .
1 14 X 0...“......_ ..................
L b . ' 4
. - c
. - 2".
112 " - ' 1 ’5
i' I ' ' a
Z' ' "'1 1
1,10 .................................. . ................. 1 ................. ' .......... _
. x . .
\ .. 4
g 1.03 _ ................ , ................. . ................................................... ..
b : x . ‘
105 ' 1 . X .3
b x :
134 ----- . ----------------- 1 --------------------------------- u
I ' x 1
- x vswses) : 5
1-02 f e ch196(alpha=-0.00001) """""""""""""" """""""" ‘.
» - Stress histo : 5 x
1.00 L . . . 1*.4. . . 1 A A . A I I A I . I J a .
0.50 0.50 0.70 _ 0.00 0.90 1.00
o I o
)0(

Figure 5-2c Yield surface based on anisotropic
hardening Yld96 model , or equal to —0.00001

76

Strain paths

 

 

 

Bul etest
0.20 2.- .l.,g..zfi)-.,...
/ -°—von Mises
0.15 ' —-Yld96(alpha-0.0)

 

 

-°- Yld96(a|pha=-0.00001)

\
\\

 

Hoop Strain
9
8

 

0.05 '

P
b

 

i
l

0.00 -
-0.100 -0.080 -0.060 -o.040 -o.020 0.000
Axial Strain

 

 

 

 

 

 

Figure 5-3 Different strain paths predicted by different material models

5.4.3 Yield Surface EvolLion junder Non-Proportional Loaging — Square Die

Hydroforming

For the square die simulation, two different locations on the deformed tube were
investigated. Element number 1429 was located on the top of the corner region and
element number 1421 was located on the region where the tube contacted the die. The
strain paths for these two locations obtained from von Mises, isotropic and anisotropic
Yld96 material models are shown in Figures 5-4a-b. It is clear that for element 1429, the
strain paths for different material models are very different. Interestingly, the anisotropic
Yld96 material model with a value of or = - 0.0002 follows a similar path as that for the
von Mises, however, strain paths for Yld96 model with values of or = - 0.00001 and or = O
are very different. It can also be seen that the von Mises material model predicts a
maximum strain of 12.5%, while Yld96 material model predicts 11% (with or = - 0.0002),

13.8% (with or = - 0.00001), and 14% (with or = 0), respectively.

77

Strain paths Strain Paths

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(Element #1429) (Element #1421)
0.14 0.03 .
: /: : - 1
0.12 / I ~ 1
: : 0.02
c 0'10 (ll/1, // c 3
E 0.08 I / 2 E “2 .
('75 I / 1 5 t 1
§ 0’06 I -—von Mises 7 8 0'01 L ' l
11 0 » ——von Mises 1
I 0.04 : . —Yld96(alpha=0.0) .3 I : —-—arpha=o.o 1
“if —~—Yld96(a|pha=-0. 00001) 3 0.01 —-—a|pha=0. 00001 l l
°-°2 : +Yld96(alpha=-0. 0002) j ; -—alpha=-0 0002 .
l “l ....... i in ‘ ......... infui .......

 

 

0. 00 0 .00
-0. 005-0. 004-0. 003-0. 002- 0. 001 0. 000 0. 001 0 .002 -0. 07 .0. 06 -0. 05 -0. 04 .0. 03 -0. 02 -0 01 0. 00
Axial Strain Axial Strain

Figure 5-4 (a) Strain paths for element #1429 (b) strain paths for element # 1421

The corresponding yield surfaces with different strain paths for element 1429 are shown
in Figures 5-5a-d. Figure 5-5a shows that at the final deformation point the von Mises
yield surface is slightly above the polycrystal yield surface, indicating that strain
prediction with von Mises will slightly underestimate the experimental value. Figures 5-
5b and 5-5c show an almost perfect match between the two yield surfaces, indicating a
very close match between the predicted and measured strain values. Figure S-Sd
represents the worse case, where the Yld96 yield surface with or = - 0.0002 is grossly
above the polycrystal yield surface, indicating a gross underestimation of the predicted
strain. In fact, this case had the lowest strain prediction. A comparison of these strain
predictions with the experimentally measured strains in Figures 4-9a-c indicates that

probably strain measurements for sample H4 are not reliable.

Figures 5-6a—d show the yield surface comparisons for element 1421. It can be seen that
Yld96 isotropic and anisotropic material models, regardless of the value of or, obtain the

best matches. The worst match is for the case of von Mises material model (Figure 5-6a)

78

where it is expected for the strains to be underestimated. It can be concluded that since
total strain for element 1421 is very small (~ 3%), any small variation in the strain
prediction by the four models may be difficult to distinguish. Table 5-1 shows the hoop

strain of element 1421 for different yield functions.

 

 

 

 

 

 

Element #1421 Element #1429
Von Mises 0.0247 0.1231
Yld96 (a=0.0) 0.0246 0.1337
Yld96 (a=-0.00001) 0.0246 0.1323
Yld96 (a=-0.0002) 0.0246 0.1057

 

 

 

 

TableS-l Hoop strain at element 1421&1429 for different yield functions

Finally, Figure 5-7 shows the variation in the strain prediction for square die
hydroforming as a function of different a-values. It can be seen that with different or-
values, vastly different strain distribution can be predicted, which is not possible with
other material models such as von Mises, or the original Yld96 with isotropic hardening.
However, to verify which a-value is correct will require the comparison of the
phenomenological yield surfaces with those obtained from the polycrystal model. In the
case of 6061-T4 extruded aluminum tube, the value of or = - 0.00001 is the correct value

to use.

79

Yield Surface Comparison (element #1429)
(Deformation in square die with von Mises)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.50.....ss..,...---..,....
. : : E 1
1.00
i 1
lo 0.50 --------- —
\ 4
k .
b .
0.00 ------
-0.50 -- ' = ----------------- '1 -------------- --------------- A
. x YS(0.5%) 9‘ S r
L - von Mises ; .
' Stress history = 1
-1.00#~'U~4 :' .. i -+~
-1.00 -0.50 0.00 _ 050 1.00 150
o I o
)0(
Yield Surface Comparison (element #1429)
(Deformation In square die with von Mises)
1.20
1.10
1.00
'b 0.90 '
\ .
o; .
0.00
0.70 .
_ , _- x YS(0.5%) :
0.60 . . -------------- -; ----------------- I vonMises -
I . - 5 5 - Stress history 1
' e g ; 4
050 ; .' . . . . i I . . . I . + 4 I . . . I
0.00 0.20 0.40 _ 0.60 0.00 1.00
o I o
)0(
(a)

80

Yield Surface Comparison (element #1429)
(Deformation in square die with alpha=o.0)

 

 

 

 

 

 

 

 

 

 

1.50 V V I ‘r .rfi V T T I Y Y Y T I Y Y T r I T f if
P . E E . 1
- s s +
1.00
lo 0.50 ----------- -
\ .
E l
O l
0.00
“0'50 f" x YS(0.5%) """""" § """"""" 5‘ """""""" i
- Y|096(alpha=0.0) 5 5 4
Stress history 4
-1.00-*“‘i““i“"i““i‘ ‘T

-1.00 -0.50 00 0.50 1.00 1.50

0. _
0' I o
)0(
Yield Surface Comparison (element #1429)
(Deformation In square die with alpha=o.0)

1.20 V I 7— ' U I I I

 

5 coo-0:30...
; .............. "°“ ... ................. ..... /- ......

1.00 ................ ................. .. ......

 

 

 

 

 

 

 

 

 

'0 0.90 - ---------------- --------------- ,v f7 ----------------- ----------------- --------------- i
\ ' S ' i : : ‘
O; ' = E a = '
0.00 —- --------------- --------- 3..."; ----------------- ----------------- g- --------------- —
0.70 _- ---------------- §;-'-'- ------------ ----------------- ----------------- --------------- —
. x YS(O.5%)
0-60 _- ---------- _- v-"g ----------------- ---------------- e Yl096(alpha=0.0)
: _ - 5 - Stress history
050 I. 1 A4; ‘ i t . . l A A4 i 4

40 0.60 0.80 1.00

(b)

81

Yield Surface Comparison (element #1429)
(Defamation In square die with alpha=-o.oooo1)

 

 

 

 

 

 

 

 

1.50b..v.!fi1..!.v.1! v! v
r

t , , . .
r : ' : ' ‘
100 . ............... 1 ............ g ................. .......................... _
I b 050 .. ................ 3 ................................................................. —4
4
\ 4
3: r
b r
(100 .. ................ . .................................................................. d
'0'50 f’ x YS(0.5%) """"""""""""""""" T
t . Yl096(alpha=-0.00001) . ; r
L Stress History 5 ‘
.100....r....r..,.i-.t.i....‘

-1.00 -0.50 0.00 _ 0.50 1.00 1.50

olo
XX

Yield Surface Comparison (elemet #1429)
(Deformation In square die with alpha-0.00001)

1.20 ’

 

  

1.10

 

 

 

x YS(0.5%)
e YI096(alpha=-0.00001)
- Stress History

 

 

 

 

 

 

A; %l + L A l A A A

0.00 0.20 0.40 _ 0.60 0.80 1.00

 

82

Yield Surface Comparison (element #1429)
(Deformation in square die with alpha=-0.0002)

 

 

 

   

 

 

 

1.50
1.00
lo 0.50
\
5‘.
b
0.00
050 ~-- ‘ i ----- f
- x YS(0.5%) 5X
- YldQB(alpha=-0.0002) i 5
- Stress history
'100 L . . . . l a L a . A
-1.00 -D.50 0.00 _ 0.50 1.00 1.50
o l o
)0!
((0

Figure 5-5a-d Yield surfaces comparison based on different strain paths for element 1429

83

Yield Surface Comparison (element #1421)
(Deformation in square die with von Mises)

 

1.00 r I I I l I I I I I 1 I I I l I I I I I I I I I
. . . = I
I . . .

0.50

   
 

 

 

 

(b 0.00
\
E
b
-0.50
-1130 _............... . . . _
i = YS(U.5%) ‘
: von Mises ‘
r 5 : 5 - Stress history .
_1.50...JltlLli.... ________
-1.50 -1.00 -D.50 _ 0.00 0.50 1.00
6 I o
)0(
(a)

84

Yield Surface Comparison (element #1421)
(Deformation in square die with alpha=0.0)

 

1.00 I I If I ! fiI I I ! I I I I I I I I I I I

F' q
0.50 § ..... ................. ............... _

x Z
31 1

D 1
-0.50 ---------------------------------------------------- s --------------- s

 

 

-1 .00 f ‘ ------------- --------------- -
’ j i x vscnm

_ 5 - YidOOCabhFD .0)

' : : Stress iistory

  
    

 

 

 

 

-1.50 ~1.00 -0.50 0.00 0.50 1.00

85

Yield Surface Comparison (element #1421)
(Deformation in square die with alpha-0.00001)

 

 

 

 

 

 

1.00 I I I I E I I T I I I I I I ! I I I I I r I l I
0.50
I b 0.00
\
g
b
-0.50
-1.00
; . . . “momma-000001)
P - Stress history
450 LtLAJLSA.r. ..
-1.50 -1.00 -0.50 _ 0.00 0.50 1.00
o I o
)0(
(C)

86

Yield Surface Comparison (element #1421)
(Deformation in square die with aipha=-0.0002)

 

 

 

 

 

 

1.00P.r..!....!r.1t!....!.,.r‘
0.50
| b 0.00
x
k
b
-0.50
-1.00 ------------- : --------------- -
X “30.50
C . Yid00(alph38-0.0002)
_ . . Stress ”story
_150.4t.igL4tL.s.tl
4.50 -1.00 -0.50 _ 0.00 0.50 1.00
c I c
)0(
(d)

Figure 5-6a-d Yield surfaces comparison based on different strain paths for element 142]

87

Hoop Strain Distribution

 

 

 

0.25 (Different a)
0.20 —0— -0.0001
—-0— 00002

.E —o—-0.0005
E 0.15 —o—mro
5 _._ 0.0001
0. —0— 0.0002
8 0.10 —e—0.0005 ‘
I

0.05

 

0.00
-30.00 -20.00 -10.00 0.00 10.00 20.00 30.00

location (mm)

Figure 5-7 Hoop strain distributions with different a-values

88

CHAPTER 6
EXPERIMENTAL SETUP AND PROCESS
6.1 Experiment Setup
The experiments on tube hydroforming were conducted in the MSU laboratory using a
tube hydroforming press system purchased from Interlaken Technology Corporation (see
Figure 6-1). This hydroforming press system consists of the following sub-systems:
hydraulic press (ServoPress), hydroforming feed actuators, pressure intensifier, tooling

and controller.

The ServoPress is a single acting servo-controlled hydraulic press including Interlaken’s
UniPress control hardware and sofiware. The ServoPress provides clamp force and punch
force that is used for sealing of tube ends and end feed of specimen. Both punch and
clamp are controlled using either load or position as the primary feedback. High speed

data acquisition records and displays all measured variables on the computer screen.

The pressure intensifier system is a cabinet mounted system powered by shop air (80-120
psi). The system is completely plumbed and includes all manifolds and pressure control
valves. The pressure intensifier system provides fluid pressure of up to 30,000 psi and is
also controlled by the system computer.

The standard tooling is designed for tubes with 2.00” OD (outer diameter). Standard
tooling provides changeable inserts for hydroforming various shapes and various strains.
Complete die set (see Figure 6-2) for performing both constrained forming and
unconstrained burst tests of 2.00 OD tubes is available. More specifically, the following

Cavity shapes are available: Unconstrained burst; square section 2.00 inches (. 250 comer

89

radii); square section 2.50 inches (.375 comer radii); constrained hydraulic bulge (2.00
spherical radius). The system also includes docking rods and pressure porting to bring

high pressure fluid from pressure intensifier.

 

Figure 6-1 Experimental setup in MSU laboratory

6.2 Tube hydroforming process

Generally in internal high pressure forming processes a tube is sealed at both ends. Then
the tube is filled with a hydraulic fluid e.g. a mixture of oil/water. The sealed tube is
placed into an encapsulating die and the component is formed by the application of
hydrostatic pressure. In addition, an axial force is applied with horizontal rams (plunger)
to seal the tube and to introduce axial compressive stress into the tube wall and force the
material to move towards the forming zone (Figure 6-3).

Important for the successful employment of this process is the knowledge of the
necessary process characteristic. The load paths of the internal pressure and the axial
forces have decisive influence on the forming process [Asnafi 2000, Trana 2002, Manish
et al. 2003]. Too low internal pressure and too high axial forces can lead to bucking or

wrinkling in the tube. Too high internal pressure together with too low axial forces can

90

lead to bursting of the tube (Figure 6-4). Failure modes occurring in tube hydroforming

can be seen in Figure 6-5.

 

Figure 6-2 The die set used for experiment

 

 

 

 

 

 

’ 4
7 was“ F

(b)

91

Figure 6-3 The principle of tube hydroforming: (a) original tube shape and (b)
final tube shape (before unloading). The figure is extracted fromAsnafi (2000)

 

 

A
Max. axial force

on

Buckling, wrinkling E

and folding back 8
Z . 2
0 Safe zone 3“"?th g
0 o
In L-
5 s
.53 . . E
:5 Yielding .3
E

Scaling limit

 

 

 

lntemal pressure

Figure 64 Working diagram for the hydroforming process
(extracted from Trana (2002))

 

Figure 6-5 Failure modes occurring in tube hydroforming

6.3 Sequence of operation in experiment
The typical sequence of operations comprising the tube hydroforming process in MSU is
as follows:

1. The tube is placed in the lower die.

2. The upper die is brought down to closed position.

3. The seal are advanced to sealing position.

4. The tube is filled with liquid and pressurized to hydroforming pressure.

92

7.

8.

If axial feed is required, the end seals are simultaneously advanced to push the
required amount of material into the die cavity.

Pressure in the tube is reduced to atmospheric level.

The upper die goes up.

The hydroformed part is withdrawn from the lower die.

Figure 6—6 shows some deformed tubes made from both bulge and square die without

failure.

 

Figure 6-6 Deformed tubes without failure

93

APPENDIX A
INTRODUCTION TO SPIN AND ROTATION
A rotation matrix for small (infinitesimal) rotations
(From Crisfield, 1991)

In the Figure Al(b), a vector ro is rotated in the 1-2 plane through Aflto become a vector
r" . We could therefore write
r“T = r0 {cos(6i0 + AiEI),sin(6l0 + A6),O} (A1)

Alternatively, an approximation would involve

rn = r0 + Ar = rot + roAi9n (A2)

where r0 = ||r0|| , n is the unit vector orthogonal to both r0 and the z-direction(or e3) so that
tTn = egn = 0 and hence

nT = {—sin 00,c0560,0} (A3)

Hence (A2) can be rewritten as

1 O O 0 —A0 0
rn=010+A6 O Oro (A4)
0 0 1 0 0 O
or
rn = Rr0 = [I + S(Air)]ro (A5)
This relationship can be accompanied by
0n = 00 + A0 = 60e3 + AISIe3 = (60 + A6)e3 (A6)

94

because in two dimensions rotations are additive. For the two-dimensional problem of
Figure A1(b), there is no point in adopting (A4) instead of the simpler exact form of
(A1). However, (A2) can be generalized (Figure A1(c)) to

rn =ro+Ar=ro+(A0xr0) (A7)
where the symbol x denotes a cross-product so that (A0 x r0) is orthogonal to both A9

and r0 and is of magnitude Afir0 sin ,6 , where (Figure A1(c)) ,5 is the angle between A9

and r0. Equation (A7) can be recast in the form of (A5) with

0 —A9, A6,
S(A0) = A9, 0 —Ae, (A8)
—A0, A9, 0

With infinitesimal rotation, A0 represents the ‘spin’.
Large rotations ( Euler ’s theorem)

(From Belytschko, 2000)

A ftmdamental concept in large rotations is the Euler’s theorem. This theorem states that
in any rigid body rotation, there exists a line which remains fixed: the body rotates about
this line. From this theorem general formulas for the rotation matrix can be developed.

Consider the rotation of vector r by an angle 0 about the axis defined by the unit vector

e a el . The vector after the rotation is denoted by r' as shown in Figure A2. The rotation

matrix R relates r' to r by
r' = Rr (A9)
where R is to be determined. We will first derive the formula

r'=r+sin0exr+(1—cos6’)ex(exr) (A10)

95

where e is a unit vector along the axis of rotation, whose existence we know by Euler’s
theorem. The schematic on the right hand side of Figure A2 shows the body as viewed

along the e axis. It can be seen from this schematic that
r' = r+ rPQ = r+orsin0e2 +a(1—cos6’)e3 where a = rsin¢ (All)
From the definition of the cross-product it follows that:
ae2=rsin¢e2=exr, ae3=rsin¢e3=ex(exr) (A12)
Substituting (A12) into (A1 1) yields (A10).
Replacing ex (e x r) = (e . r)e — (e - e)r = (e - r)e — r into A(10)
We obtain
r'=c050r+(e-r)(l—cost9)e+sint9exr (A13)
This will be implemented to calculate the spin for each single crystal in the polycrystal

modeling in Chapter 3.

96

 

 

 

 

 

A2(y)
n
5
AC: A6 rn Atm""'t
_.___. m) 60 a
f0
/ el 9»
'3(2)
3 b
A
rn 'Al’ ..... x ..................
a” ....... ‘5... B
to '5 A9
..................... a
.......... 1“ A9

Figure A1 (a) Axes and unit base vectors; (b) two-dimensional
rotation;(c) small three-dimensional rotation.

62 Q

83 9 r P

 

 

 

 

ey

Figure A2 Rotation of a vector r viewed as a rotation about a fixed axis

9=6b according to Euler’s theorem; on the right a top view along the 6
axis is shown

97

APPENDIX B

CALCULATION OF ANISOTROPIC COEFFICIENTS
FOR PLANE STRESS CASE OF YLD96

Definition of the Input Data

The yield function coefficients can be calculated from the results of the following four
mechanical tests; uniaxial tension test in three different directions and hydraulic bulge
test), or from polycrystalline model which can numerically predict the flow stresses and
r-values for the material.

For an aluminum sheet, balanced biaxial yield stress a", , the three uniaxial yield stresses
(0'0 ,0'45 ,090) and r-values (ro,r45,r9o) are determined at O, 45 and 90 degree from the

rolling direction. For an extruded aluminum tube, the yield stress in the extrusion
direction (ED) can be measured from a uniaxial tensile test; however, it is very difficult
to measure the yield stress and r-value in the tangential direction (TD), radial direction
(RD) and the 45 degree between ED and TD. In this work, all the coefficients were
determined from the polycrystal model using the undeformed tube’s crystallographic
texture information as input. Actually, we used the flow stresses as input data instead of
yield stresses. The reasons being that: l) The yield stress is determined in the region of
the stress-strain curve where the slope is the steepest and this may provide additional
inaccuracy, 2) The yield stress is associated with a very small plastic strain and might not
reflect the anisotropy of the material over a larger strain range, 3) for aluminum alloys,
experimental observation show that after a few percent plastic strain, this flow stress does

not vary significantly [Barlat et a1. 1997].

98

Calculation (j the Yield Function C oefficients

For an aluminum alloy sheet, seven experimental input data (0",, ,0'0 ,045 , 0'90 , ro,r45,r90)
and for an aluminum extruded tube, flow stresses and r values in 4 directions are
numerically determined by polycrystal model (0'5“,0'RD,0",,0'45,rED,r.,.D,r,.D,r45) in
order to calculate the seven independent anisotropic coefficients (C,,C2 ,C3,C6 ,
a, ,ay,a,,

For uniaxial tension in the extrusion direction, the components of the tensor 3 (see Eq4-9-

1-3) becomes

 

s =-—3—o,,,, (3.1)

Combined with the yield function (Eq.4-8), it is possible to write the following equation:

”—2( 36] =0 (3.2)

El)

 

 

F, = a, c2 —c3|" +aty|2c2 +c3|a +azolc2 +203

 

Similarly, for uniaxial tension in the tangential direction and compression in the radial
direction (equivalence of balanced biaxial tension in hydraulic bulge test), it is possible to

write the following equations

 

F2 = ax C3 +2clla +ay|c3 ”clla +a:0|263 +c||a —2[_3:] = 0 (B3)
on)
.. a a 36 "
F3=ax|2c,+c2| +ay|cl+2c2| +a,0|c,—c2| —2[—] =0 (B.4)
0RD

99

It is possible to find cl ,c2, and c3 through these equations (B.2-B.4) using Newton-
Raphson method. To solve this system, it is necessary to assume some values for a, and
ay, for instance a, = ay = 1. Then a, and cry are modified in an iterative manner in

order to modify the slope of the yield surface at the uniaxial loading stress state which is
related to the corresponding r values [Yoon et al. 2000].

The above method allows the calculation of cl ,c;,_,c3,ai,r and cry. A similar iterative
procedure is used to calculate c6 and a:l by matching the flow stress 0'45 and r-value,

r45 , for the uniaxial tension at 45° between ED and TD.

Flow Stresses and r-values in 4 Different Directions A long the Tube

The predicted uniaxial tension stress-strain curves up to a strain of s=0.25 in different
specimen orientations were used to numerically determine r values and flow stresses at
equal amount of plastic work [Barlat et al. 1997]. The uniaxial directions considered were
the extrusion direction, ED; the tangential direction, TD; the radial direction, RD; and the
direction at 450 between ED and TD. In order to simulate uniaxial tension tests in
different orientation, either the test boundary condition was keep constant with respect to
the reference frame, but the texture information was transformed into different reference
frames, or the texture information was keep constant with respect to the reference fiame,
but the test boundary condition was transformed into different reference frames. The
latter approach was used in this work.

To calculate the flow stresses (a£.,,,om,,0'm,0'45) and r-values (rH,,r.,.,,,r.,.,),r45) m 4

different directions, the crystallographic textures, as measured by orientation imaging

microscopy (OIM) provided by the ATC, were used as input to the polycrystal model that

100

was introduced in Chapter 3. The crystallographic textures were expressed in the
reference frame (RD, TD, ED), the radial, tangential, and extrusion directions
respectively. Before the polycrystal computations, these textures were slightly rotated in
order to recover the expected symmetry due to the material flow.

For simulations, the voce parameters A' , B' , C ' (see Eq.3-24) were determined by a trial-
and-error method, repeating the polycrystal simulations until the experimental ED stress-

strain curve was satisfactorily predicted (see Figure B1).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Stress-Strain Curve In ED
300
A
Q 250
E
3 200 .
in
in
g 150
(D
O
a 100
l- a - A' - B' exp(-C'e)
—e—experiment ' = 118.5 Mpa
50 r'" Polycrystalmodel """"" 3' = 55.5 Mpa .....
’ C' = 1.9 _
o i i

 

 

 

 

0.00 0.05 0.10 0.15 0.20 0.25 0.30
True Strain (a)

Figure Bl Stress-strain curve (extrusion direction)

from polycrystal simulation and experimental result

As mentioned before, to predict the stress-strain curve in other three directions, we have

to keep the texture information in reference frame and transfer the boundary condition to

101

different flame. For instance, in the case of uniaxial tension simulation in TD, the strain
increment tensor was expressed in the reference flame (RD' ,TD' ,ED'). It was
transformed in the reference flame (RD, TD, ED) and the stress calculation was carried
out by polycrystal model while the input texture orientation was updated with respect to
reference flame (RD, TD, ED). The stress result was then transformed back to reference
flame (RD, TD, ED). Various possible strain increment tensors (depending on the
different r-values) were implemented as input along with texture orientation until
approximately uniaxial stress state was reached. Using this approach, the r-value in ED
could be obtained. Similarly, r-values for other directions are obtained using a similar
approach (See Figure B2).

The work hardening behavior (trends) in the different directions was consistent for all
amounts of plastic work (isotropic hardening assumption), the flow stress used to
calculate the anisotropy coefficients were determined at the plastic work amount of 50

Mpa mm/mm.

102

 

RD RD’
f TD ’

Rotate 90° , ~ ED,
> ED about RD TD

Reference (ED,TD,RD) Reference (ED’,TD’,RD’)

 

 

Uniaxial tension in ED

 

 

 

 

a” 0 O Polycrystal Model 0' 0 O
o 5,, 0 p 0 o o
O O 833 ED.TD.RD O O 0 5010.120
Strain Input in Stress Output in
Reference (ED,TD,RD) Reference (ED,TD,RD)
Uniaxial tension in TD or ED’
5” O O 822 0 0
0 5,, o > 0 a” 0
O O 833 ED'.TD'.RD' O 0 £33 5010'”
Strain 111th in _ Strain Input transformed to
Reference (ED ,TD ,RD ) Reference (ED,TD,RD)
Texture orientation fixed in
Reference (ED,TD,RD)
V
0' 0 O O O O
O O 0 ¢ 0 0' O
0 0 0 £0,103”, 0 O 0 £D,TD.RD
Stress Output transformed to Stress Ou ut in
Reference (ED ,TD ,RD) Reference (EETDRD)

Figure 32 Boundary condition in different reference flames

103

Appendix C
Predicted yield surfaces based

on the proportional loading

Case 1: Uniaxial tension in extrusion direction

Ylold Surface Comparlson
Unlaxlal Tonsionhxtruslon dlrecfloln) (1 0%)
(cw—0.00025, —o.0001 )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1:50 V V Vfi V V V V V V V V V V V V V V V V

1.00 / .

050 J ‘
ID ' .
\ 4
b: l

0.00

'0‘50 '" x a1p11a=-o.00025'

[ o alpha-0.0001
, YS(polycrystal)
-100 “UL-ubeu “A eeu
-1.00 .050 0.00 _o.so 1.00 1.50
a I 6
xx
Figure Cl-a
Ylold Surface Comparlson
Unlaxlal Tonsloruoxtruslon directloln) (25%)
( a=-0.00025, —0.0001 )
1.50 V V V V V V V V V V V V V V f V TV‘T V ‘V
t 1
L 1

1,00 1... ............ .. .

l 4/ 1

I b 0.50 ' ‘

\

br >
0.00

b 1
L 1
. . 1

'050 f" x mung-0.00025 /

_ o alpha=-0,0001 ‘ j
. Ysmolvcrvstal)
-1_oo AAAAi‘AAAl'AA-A ---- ALL.
.100 -0 so one _o.so 1 oo 1 so
a I o
X!
Figure Cl-b

104

Ylold Sulface Comparison
Unlaxlal Tonalon(axtrualon directloln) (50%)
( a.=-0.00025, —0.0001 )

150 T'rv IVVYIV'VV V'I1IT‘V—VV
D I
o

 

 

AAAA

 

 

 

 

 

V V V V

 

 

 

'0-50 "" x alphaz-OOOOZS """""
o alpnaz-O 0001 :

 

 

 

 

V V V Y

 

 

 

YSmomrvstaD
.100 “AruuiUAi.-..1.--.‘
-1.oo -a.5a 0.00 _o.so 1.00 1.50
o I a
K!
Figure Cl—c
Ylold Sulfaca Comparison

Unlaxlal Tanalon(oxtrualon dlroctloln) (15%)
(«F-0.00025, —o.0001 )
1.50 e. ...1.....r......

 

A A A A

 

 

 

 

 

A A A A

 

 

 

 

'0 5° r" x alpha=-0 00025 """"""" "
. alpha=-0 0001

 

 

A A A A

 

 

 

 

 

 

I YStpowcwstal)
-100. -.-i----1-.-. a“-
-1.00 41.50 0.00 _u.50 1.00 1.50
c I 0
xx
Figure Cl-d

Figure Cla-d Yield surface comparison for uniaxial tension (ED)

105

Case 2: Uniaxial tension in tangential direction

Yield Surface Comparison
Uniaxlal Tensionmngential directloln) (10%)
(a=—0.0002, —0.0000$ )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1,50...............w....
1.00 " ' "1
F
050 r
10 ' . 1
K " 1
b:
0.00 1
- 1
i 1
'0-50 :" x alpha-0.0002 """"""""" ' +
> o alphe=-0.00005 J
. Yaoowcmtao .
.1.oo’-...i..-xi.-x- LAAJAALA
.100 .050 0.00 __0.50 1.00 1.50
a I o
X!
Figure C2-a
Yield Surface Comparison
Uniaxial Tension(tangentlal directioln) (25%)
( a=—0.0002, —0.00005 )
1.50.... .....-.. ........
1.00 " " N
ID 050 ‘
‘ 1
i
o 0.00
r 1
’0-50 "‘ x alpha:0.0002 """""""" ' C
+ . man-0.00005 1
. Ysoommtao 1
_1_ou..--1A..-i.--- a“- .ALQ
.100 .050 0.00 _0.50 1.00 1.50
a I 0
xx
Figure C2-b

106

Yield Surface Comparison
Uniaxial Tension(tangentlal directloln) (50%)
(a=—0.0002, -0.00005 )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.50.... ....--..
r 1
t 4
1.00
L
[150 ........... .
10 T ,.
‘ . )1!
>- ,0
°>~
0.00
'0-50 7' x alpna=-0.0002 ' ‘
_ o alpha-0.00005 ,
. Ysoowcmtao .
.1.00'~~‘- ‘4 -. - U”
.100 0.50 0.00 0.50 1.00 1.50
o I o
I
Figure C2-c
Yield Surface Comparison
Uniaxial Tenslon(tangentlal direclloln) (75%)
(a=-0.0002, —o.oooos )
150 V V f V V V V I V V V V V V T V V—V
. 1
1
1
1
1.00 .
r 1
L 1
050 ' 1
lb ' T """"""
‘
o; : "
0.00 x
> Xe
'0-50 f" x alpha-0.0002 ' ‘
. alpha-0.00005 ‘
Y8(polycrystal) .
.100 .-1-...i-.-. ‘4‘
.100 .050 0.00 _0.50 1.00 1.50
o I o
K!
Figure C2-d

Figure C2a-d Yield surface comparison for uniaxial tension (TD)

107

Case 3: Plane strain (t:22 = 0)

Yield Surface Comparison
Plane Strain (522:0) (1 0%)

( m=-0.0001, —0.0000$ )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.00 . - . .
1
b 1
0.50 X"
lo h 1
\ b I
= » .
b L .
000 -— ----------
J
x alpha-00001
0 alpha-000005
_ YS(porycrystal)
-0.50 .....1
-050 0.00 _ 0.50 1.00
o I 0
xx
Figure C3-a
Yield Surface Comparison
Plane Strain (322:0) (25%)
( a=—0.0001, —0.00005 )
100 . a . . . . . . . .
L
b
050 5.
lb
\: P 1
O
0.00
X alpha=-00001
0 alpha=-000005
YS(poM:tystel)
0.50 ‘5 . ~4ge A454 -L-
050 0.00 _ 0.50 1.00
o I 0
xx
Figure C3-b

108

Yield Surface Comparison
Plane Strain (822:0) (50%)

(a=—0.0001, —o.oooos )

V '
q

1.00 r .

 

0.50

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lb
‘15
:1. .
D
0.00
’ x along-00001
r 0 alpha-0.000%
l YS(poiycrysial)
0.50 A A A A i
-0.50 0.00
o I 0
xx
Figure C3-c
Yield Surface Comparison
Piano Strain (32150) (75%)
( a=—0.0001, —0.00005 )
1.00 .f f . - . . .
0.50
lb
\
g .
O 1
0.00
x alpha-0.0001
0 elpha=-0 00005
Y8(polycrysiai) )1
fi
050 ‘4 A A#/
-0.50 0.00 0.50 1.00
o I 0
xx
Figure C3-d

Figure C3a—d Yield surface comparison for plane strain (822 = 0)

109

Case 4: Plane strain (all = 0)

Yield Surface Comparison

Plane Strain (3‘50) (1 0%)

 

 

( a=-0.0001, —0.00005 )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.00 - - . .
i ' a I I I I .
I
I
I 1
050 I,
| b x
\> ' ‘
>. I
D .. 4
0.00 :
x alphaz-O 0001
. alpha-0.00005
YS(po|ycrysiel)
0.50 A 1% A A A ' A
-0.50 0.00 0.50 1.00
a I u
xx
Figure C4-a
Yield Surface Comparison
Plane Strain (3‘50) (25%)
( a=—0.0001, -—0.00005 )
1.00 . . . . . T . . .
x I I I I
I I T l I
I I
I
I
I
0.50 I ,
| D ’ x
\> ” l
. fl 1
D :e
0.00 "
x alpha=-0 0001 ‘
0 alpha=~0 00005
Y8(poiycrystal) ¢
0.50 A A ' A A A A 1 e
050 0.00 0.50 1.00
c I 0
xx
Figure C4-b

110

Yield Surface Comparison
Plane Strain (3"=0) (50%)

( a=-0.0001, —0.00005 )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.00 . . r . . - . .
g I I i I
b I I ‘ 1
I
’ I
0.50 --------
L/
10 1
‘5:
° 1
0.00
’ x alpha=-0 0001
. o alpha=-0 00005
_ Ysmomrvstal) J
.0.50AAAA14AAAAAA%
-0.50 0.00 _ 0.50 1.00
c I 0
xx
Figure C4-c
Yield Surface Comparison
Plane Stain (5'50) (75%)
(a=-0.0001, -0.00005 )
1.00-..- .-T. ....
‘ I x I I I fl I g
f— ” ,
* 1
n 1
0.50 *
ID >'
\g ’9
w 1
D )e .
000
x alpha-0,0001
, 0 elpha=-0.00005
YS(poiycrystal)
0.50 -AAAJA.-- AAAA
.050 0.00 _ 0.50 1.00
o I o
X!
Figure C4-d

Figure C4a-d Yield surface comparison for plane strain (8ll = O)

111

Case 5: Balanced biaxial tension

Yield Surface Comparison
Balance Biaxlai Tension (1 0%)

 

1 a=—o.ooor, —o.oooos )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.00
L
0.50 f
.o 1
A: 1
b
000 —
x alpha-00001 / ‘
e Noam-000005 ‘
_ YS(poiycryslal) ‘
0.50 A A 1 A 7 A A
-0.50 0.00 _ 0.50 1.00
o I 0
xx
Figure CS-a
Yield Surface Comparison
Balance Biaxial Tension (25%)
( a=—0.0001, -0.00005 )
100 . a . . . . . .
W T
050
ID i
‘s
b L
0.00
x alpha-00001 r ‘
. o alpha-.OODOOS A
L Ysmowtwstaio
0.50 A A i A A A A
-050 0.00 _ 0.50 1.00
o I 0
xx
Figure CS-b

112

Yield Surface Comparison
Balance Biaxial Tension (50%)
(a=-0.0001, —0.00005 )
100 . . . . . . T . . .

0.00 .. .............
L
. x 51052:.00001] If}

 

 

 

 

 

0 alpha-0.00005
YS(poiycrystai)

0.50 A A A A i A A
-0.50 0.00 0.50 1.00

oIo
X!

 

A A A

 

 

 

 

 

Figure C5-c

Yield Surface Comparison
Balance Biaxial Tension (75%)
(a=—0.0001, -0.00005 )
1.00fi... .r.. r..-

b " W 1
050b J

 

 

 

 

  

 

’ x slant-0.0001
, o alpha-0.00005

 

 

 

1

Ystpoivcnstal) 1" 5 1
0.50 A- A A i A A A A i A

-o.50 0.00 _ 0.50 1.00

olo
I!

 

 

 

 

Figure CS-d

Figure CSa-d Yield surface comparison for balanced biaxial tension

113

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