effect ofpre-drawing on formability during cold...

Effect ofPre-Drawing on Formability During Cold Heading

by

Lianzhong Ma

Department of Mechanical Engineering McGiII University Montreal, Canada

A thesis submitted to Mc Gill University in partial fulfillment of the requirements of the degree of

Master of Engineering

Un der the supervision of: Professor J.A. Nemes

McGill University

© Lianzhong Ma August, 2005

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ABSTRACT

One of the most common indus trial cold forging processes is cold heading of steel wire

or rod to produce screws, bolts, nuts and rivets. The process is limited by a complicated

interplay of many factors. The cold work (pre-drawing) is one of them. Although several

investigations into the effects of pre-drawing ~m the formability of metals during cold

heading processes have been conducted, so far no attention has been given to the

numerical simulations of this phenomenon. The CUITent work aims at examining effects of

pre-drawing on formability during cold heading through numerical simulations.

Physieal tests in the literature investigating the effects of pre-drawing on the formability

of three metals are simulated using ABAQUS 6.4, with three successive FE models: the

drawing model, the cutting model and the upsetting model. A new combined linear

kinematic/nonlinear isotropic hardening constitutive model is proposed and derived to

aecount for the Bausehinger effect existing in reverse plastic deformation. The new model

is implemented into ABAQUS/Explicit v6.4 by a user subroutine VUMAT, which is

verified by one-element numerical tests under tension, compression and reverse loading

conditions. In addition, for the purpose of comparison, the Johnson-Cook isotropie

hardening model is also applied for the materials. The Cockroft and Latham criterion is

employed to predict surface fracture.

Although considerable discrepancies between the experimental and simulation results are

observed, the proposed combined hardening model is more accurate in predicting material

behavior in the reverse loading than the Johnson-Cook isotropie hardening model. In

addition, the simulation results show that the proposed combined hardening material

mode! has the potential to correctly predict the material behavior in the reverse loading

process.

RÉSUMÉ

Un des processus industriels les plus communs de forge à froid est la formation à froid du

bout du fil d'acier ou de la tige pour produire des vis, des boulons, des écrous et des

rivets. Le processus est limité par des effets compliqués de beaucoup de facteurs. Le

travail à froid (pré-drawing) est l'un d'entre eux. Bien que plusieurs recherches sur les

effets du pré-drawing sur la formabilité des métaux pendant des processus de la formation

à froid du bout aient été conduites, aucune attention n'a été donnée aux simulations

numériques de ce phénomène. Le travail présent vise à examiner des effets du pré

drawing sur la formabilité pendant la formation à froid du bout par des simulations

numériques.

Des essais physiques dans la littérature étudiant les effets du pré-drawing sur la

formabilité de trois métaux sont simulés en utilisant ABAQUS 6.4, avec trois modèles

successifs de FE : le modèle de drawing, le modèle de découpage et le modèle de

dérangement. Un nouveau modèle constitutif du durcissement combiné de kinematique

linéaire/isotrope nonlinéaire est proposé et formulé pour expliquer l'effet de Bauschinger

existant dans la déformation plastique inverse. Le nouveau modèle est mis en application

dans ABAQUSlExplicit v6.4 par un sous-programme VUMAT d'utilisateur, qui est

vérifié par les essais numériques d'un élément sous une tension, une compression et les

conditions de chargement inverse. En outre, pour la comparaison, le modèle du

durrcissement isotrope de Johnson-cook est également appliquée pour les matériaux. Le

critère de Cockroft et de Latham est utilisé pour prévoir la rupture superficielle.

Bien qu'on observe des différences considérables entre les résultats expérimentaux et de

simulation, le modèle combiné du durcissement proposé est plus précis en prévoyant le

comportement matériel sous le chargement inverse que le modèle du durcissement

isotrope de Johnson-cook. En outre, les résultats des simulations montrent que le modèle

matériel combiné du durcissement proposé a le potentiel de prévoir correctement le

comportement matériel dans le procéssus de chargement inverse.

11

ACKNOWLEDGEMENTS

l would like to first thank my supervisor, Professor James A. Nemes, for his guidance,

encouragement, patience and support.

l gratefully acknowledge the financial support of the Natural Sciences and Engineering

Research Council of Canada and Ivaco Rolling Mills through the Strategie Grants

Program.

l would also like to thank aIl the group members under the supervision of Prof. Nemes,

specifically Christine EI-Lahham for her initial help with the simulation modeling, Amar

Sabih for his help with the documentation, Wael Dabboussi for his proofreading of the

thesis, and Desheng Deng for his translation of the abstract.

Finally, l thank my family, my wife, and my daughter for their love and support.

111

TABLE OF CONTENTS

Abstract ........................................................................................................................................... i

Résumé ........................................................................................................................................... ii

Acknowledgments ........................................................................................................................ iii

Table of Contents .......................................................................................................................... iv

List of Figures .............................................................................................................................. vii

List of Tables ................................................................................................................................. xi

Glossary ...................................................................................................................................... xiii

1 Introduction ......................................................................................................................... 1

1.1 Motivation ................................................................................................................ 1

1.2 Objective .................................................................................................................. 2

1.3 Organization ............................................................................................................. 3

2 Literature Review .............................................................................................................. 4

2.1 Cold Reading ........................................................................................................... 4

2.1.1 Properties and Manufacturing Procedures for Cold Reading Quality (CRQ) Steel Wire ........................................................................................ 5

2.1.2 Pararneters for Cold Reading ...................................................................... 6

2.2 Pre-Draw .................................................................................................................. 7

2.2.1 Pre-Drawing Process ................................................................................... 7

2.2.2 Effect ofPre-Drawing on Forrnability during Cold Reading .................. 10

2.2.3 The Bauschinger Effect ............................................................................. 17

2.3 Cold Readability and Ductile Fracture Criteria .................................................... 22

2.3.1 The Macromechanical Approach to Ductile Fracture .............................. 23

2.3.2 The Micromechanical Approach to Ductile Fracture ............................... 25

2.4 Constitutive Relations ............................................................................................ 26

2.4.1 Isotropie Hardening Material Models ....................................................... 27

2.4.2 Kinematic Rardening Material Models .................................................... 30

2.4.3 Combined Kinematic/Isotropic Rardening Material Models .................. 32

2.4.4 Flow Rules ................................................................................................. 34

2.5 Numerical Simulations of Metal Forrning Processes ........................................... 35

IV

2.5.1 Numerical Simulations of Drawing Processes ......................................... 35

2.5.2 Numerical Simulations of Cold Heading Processes ................................. 37

2.5.3 Numerical Simulations of the Fastener Manufacturing Process .............. 38

3 Model Development ........................................................................................................ 39

3.1 Failure Criterion Determination ............................................................................ 39

3.2 Identification of the Corresponding Material Constants for the Johnson-Cook Hardening Model ................................................................................................... 40

3.2.1 Typical Procedures to Determine the Corresponding Material Constants for the Johnson-Cook Hardening Model ................................ .40

3.2.2 Determination of the Corresponding Material Constants for the Johnson-Cook Hardening Model in this Work ....................................... .42

3.3 A Proposed New Combined Linear KinematiclNonlinear Isotropie Hardening Model ...................................................................................................................... 47

3.4 Implementation of the Proposed Combined Linear KinematiclNonlinear Isotropie Hardening into ABAQUS ...................................................................... 52

3.4.1 Overview of User Subroutine ................................................................... 52

3.4.2 The Goveming Equations ......................................................................... 53

3.4.3 Integration of the Goveming Equations ................................................... 54

3.4.4 Derivation of Temperature Increment for an Adiabatic Analysis ........... 57

3.4.5 Flow Chart and Code ofVUMAT and UMAT ........................................ 58

3.5 Verification of the User Subroutine VUMAT ...................................................... 59

3.5.1 One-Element Tests under Uniaxial Loading Conditions ......................... 60

3.5.2 One-Element Tests under Reverse Loading Conditions .......................... 62

4 Numerical Simulations ................................................................................................... 64

4.1 Numerical Simulations of Tozawa and Kojima's Tests ....................................... 64

4.1.1 Experimental Procedure ............................................................................ 64

4.1.2 Description of Simulation ......................................................................... 65

4.1.2.1 Description of the Drawing Model .......................................... 65

4.1.2.2

4.1.2.3

Description of the Cutting Model ............................................ 72

Description of the Upsetting Model ........................................ 74

4.2 Numerical Simulations of Gill and Baldwin' s Tests ............................................ 76

4.2.1 Experimental Procedure ............................................................................ 76

4.2.2 Description of Simulations ........................................................................ 77

5 Numerical Results and Discussion ............................................................................ 79

v

5.1 Determination of the Kinematic Hardening Modulus, H ................................... 79

5.1.1 Determination of the Kinematic Hardening Modulus, H, for S45C ...... 79

5.1.2 Determination of the Kinematic Hardening Modulus, H , for Mn steel. 85

5.1.3 Comments .................................................................................................. 89

5.2 Results of Simulations of Tests in Tozawa and Kojima' s Paper ......................... 90

5.2.1 Results of Simulations for S45C ............................................................... 90

5.2.1.1 Contour Plot lllustration .......................................................... 90

5.2.1.2 Results for the Material Point with the Highest Principal Stress on the Exterior Surface of the Upset Rod ..................... 93

5.2.1.3 Calculations of Reduction in Height from Simulation Results .................................................................................... 1 03

5.2.2 Results of Simulations for Mn Steel ....................................................... 110

5.2.2.1 Results for the Material Point with the Highest Principal Stress on the Exterior Surface of the Upset Rod ................... 110

5.2.2.2 Calculations of Reductions in Height from Simulation Results .................................................................................... 114

5.3 Results of Simulations of Tests in Gill and Baldwin's Paper ............................ 117

5.3.1 Contour Plot lllustration .......................................................................... 117

5.3.2 Calculations of Cold Heading Limit from Simulation Results .............. 119

6 Conclusions and Future Work ................................................................................. 123

6.1 Conclusions and Summary ................................................................................. 123

6.2 Future Work ........................................................................................................ 125

References .............................................................................................................................. 126

Appendix A 2-D Subroutine ..................................................................................................... 130

vi

Number

Figure 2-1

Figure 2-2

Figure 2-3

Figure 2-4

Figure 2-5

Figure 2-6

Figure 2-7

Figure 2-8

Figure 2-9

Figure 2-10

Figure 2-11

Figure 2-12

Figure 2-13

Figure 2-14

Figure 2-15

Figure 2-16

Figure 2-17

Figure 2-18

Page

4

6

8

8

12

LIST OF FIGURES

Tille

Schematics of the cold heading on an unsupported bar in a horizontal machine. (a) Head formed between punch and die. (b) Head formed in punch. (c) Head formed in die. (d) Head formed in punch and die. (adapted from Davis, 1988)

Conventional procedure for the manufacturing of CHQ steel wire (adapted from Sarruf, 2000)

Drawing ofrod or wire (adapted from Davis, 1988)

Cross section of a typical wire die for drawing 5.5mm (0.218 in.) diameter rod to 4.6mm (0.18 in.) diameter wire (adapted from Davis, 1988)

he ad diameter Cold heading limit ( .. ) versus percentage reduction of area by

Wlre dzameter

drawing by 7°,15° and 30° dies (adapted from Gill and Baldwin, 1964)

13 Plot of fracture true axial strain versus pre-strain by drawing (adapted from Luntz, 1969/1970)

14

15

16

16

17

18

19

19

20

20

21

21

A -A o Ir, on the upsetting limit,

Ao Effect of the reduction of area in drawing,

ho - hlr -----""- (adapted from Tozawa and Kojima, 1971) h

Effect of approach die angle at constant reduction on the reduction in height for two steels (adapted from Tozawa and Kojima, 1971)

S45C. Average axial stress versus average axial strain curves for upsetting with different reductions of area (adapted from Tozawa and Kojima, 1971)

Mn steel. Average axial stress versus average axial strain curves for upsetting with 40% reduction of area (adapted from Tozawa and Kojima, 1971)

Schematic Bauschinger effect curve

The effect of pre-drawing on strength in compression. Material K1020. Only the homogeneous drawing strain is shown (adapted from Havranek, 1984)

The effect of pre-drawing on strength in compression. Material K1020. (adapted from Havranek, 1984)

The effect of 29% pre-drawing on strength in compression. Material K1020, spheroidised 700°C/24h (adapted from Havranek, 1984)

The effect of pre-drawing on strength in compression. Material KI040 (adapted from J. Havranek, 1984)

The effect of 29% pre-drawing on strength in compression. Material K1040, spheroidised 700°C/24h (adapted from Havranek, 1984)

Fracture limits in K1020 and K1040 determined in the support upset tests (adapted from Havranek, 1984)

Fracture limits in spheroidised K1020 and K1040 determined in the support upset tests (adapted from Havranek, 1984)

VIl

Figure 3-1

Figure 3-2

Figure 3-3

Figure 3-4

Figure 3-5

Figure 3-6

Figure 3-7

Figure 3-8

Figure 3-9

Figure 3-10

Figure 3-11

Figure 4-1

Figure 4-2

Figure 4-3

Figure 4-4

Figure 4-5

Figure 4-6

Figure 5-1

Figure 5-2

Figure 5-3

Figure 5-4

Figure 5-5

42

43

44

45

46

50

59

60

61

62

63

64

66

69

72

73

74

80

81

83

83

84

Stress versus strain in simple tension and compression tests (adapted from Tozawa and Kojima, 1971)

Stress versus plastic strain in the simple tension and compression tests

Comparison of the stress versus plastic strain curves calculated from the Johnson-Cook hardening model with the corresponding values of the material parameters obtained from tension curve fitting and those from tension tests in the literature

Comparison of stress versus plastic strain curves calculated from the JohnsonCook hardening model with the corresponding values of the material parameters obtained from tension curve fitting and those from compression tests in the literature

Comparison of stress versus plastic strain results from compression tests in the literature and those calculated from the Johnson-Cook hardening model with the corresponding values of the material parameters obtained from compression curve fitting

Symmetric strain cycle experiment (adapted from HKS Inc., 2004)

Flow chart for VUMAT

S45C. Mises stress versus equivalent plastic strain results from uniaxial tension simulations with H = 100 (MPa) and isotropie hardening

S45C. Mises stress versus equivalent plastic strain results from uniaxial compression simulations with H = 100 (MPa) and isotropie hardening

S45C. Axial stress versus axial plastic strain results from reverse loading testing models with H = 0 (MPa) and isotropie hardening

S45C. Axial stress versus axial plastic strain results from reverse loading testing models with H = 100 (MPa)

The procedure of Tozawa and Kojima's test

Geometry and mesh for FEM drawing model

The end shape ofthe eut rod (a) with adaptive mesh and (b) without adaptive mesh

History of ratio of kinematic energy to internaI energy

(a) Configuration of the drawn rod at the final increment of the drawing simulation and (b) The initial configuration of the rod in cutting model.

Initial configuration of the upsetting model

Force versus displacement curves for the simulations of upsetting after 20% pre-drawing by a 30° die for S45C

Simulation and experimental average axial stress versus average axial strain curves for upsetting after 20% pre-drawing by a 30° die for S45C

Force versus displacement curves for the simulations of upsetting without predrawing for S45C

Force versus displacement curves for the simulations of upsetting after 40% pre-drawing by a 30° die for S45C

Simulation and experimental average axial stress versus average axial strain curves for upsetting without pre-drawing for S45C

viii

Figure 5-6

Figure 5-7

Figure 5-8

Figure 5-9

Figure 5-10

Figure 5-11

Figure 5-12

Figure 5-13

Figure 5-14

Figure 5-15

Figure 5-16

Figure 5-17

Figure 5-18

Figure 5-19

Figure 5-20

Figure 5-21

Figure 5-22

Figure 5-23

Figure 5-24

Figure 5-25

84

86

86

87

87

88

88

91

92

95

95

96

96

97

97

99

99

100

100

101

Simulation and experimental average axial stress versus average axial strain curves for upsetting after 40% pre-drawing by a 30° die for S45C

Force versus displacement curves for the simulations of upsetting after 40% pre-drawing by a 30° die for Mn steel

Simulation and experimental average axial stress versus average axial strain curves for upsetting after 40% pre-drawing by a 30° die for Mn steel





Contour of the equivalent plastic strain of the rod obtained from the simulations with H equal to 300 (MPa) for the process with 20% pre-drawing by a 30° die for S45C

Contour plot of maximum principal stress (MPa) of the rod obtained from the simulations with H equal to 300 (MPa) for the process with 20% pre-drawing by a 30° die for S45C

History of hoop (0 (J(J)' axial (0 zz), shear (0 rz) and radial stress (0 rr )

components of the element with the highest principal stress from the simulation with H equal to 300 (MPa)

History of hoop (Et(J ) and axial (E~) plastic strain components of the element

with the highest principal stress from the simulation with H equal to 300 (MPa)

Time history of hoop (0 (J(J)' axial (0 zz), shear (0 r(J ) and radial stress (0 rr)

components of the element with the highest principal stress from the simulation with the isotropie hardening

History of hoop (E:(J ) and axial (E ) plastic strain components of the element

with the highest principal stress from the simulation with isotropie hardening

History of the maximum principal stress, hoop stress and axial stress of the element with the highest principal stress from the simulations with H equal to 300 (MPa)

History of the maximum principal stress, hoop stress and axial stress of the element with the highest principal stress from the simulations with isotropie hardening

History of maximum principal stress for simulations with 10% pre-drawing



History of equivalent plastic strain for simulations with H equal to 300 (MPa)

History of maximum principal stress for simulations with H of 300 (MPa).

IX

Figure 5-26

Figure 5-27

Figure 5-28

Figure 5-29

Figure 5-30

Figure 5-31

Figure 5-32

Figure 5-33

Figure 5-34

Figure 5-35

Figure 5-36

Figure 5-37

Figure 5-38

Figure 5-39

Figure 5-40

Figure 5-41

101 Maximum principal stress versus equivalent plastic strain for simulations with 10% pre-drawing



103 Maximum principal stress versus equivalent plastic strain curve of the element with the highest maximum principal stress for the simulation with the combined hardening model with H of 300 (MPa) without pre-draw

104

105

105

111

111

112

112

Evolution of the accumulated Cockroft and Latham parameter for simulations with the combined hardening model with H equal to 300 (MPa)

Evolution of the accumulated Cockroft and Latham parameter for simulations with the combined hardening model with H equal to 600 (MPa)

Evolution of the accumulated Cockroft and Latham Parameter for simulations with isotropic hardening model

History of equivalent plastic strain for simulations with pre-drawing of 10% reduction in area with the combined hardening model with H of 200 (MPa)



Maximum principal stress versus equivalent plastic strain results for simulations with pre-drawing of 10% reduction in area with the combined hardening model with H of 200 (MPa)

113 Maximum principal stress versus equivalent plastic strain results for simulations with pre-drawing of 20% reduction in area with the combined hardening model with H of 200 (MPa)

113 Maximum principal stress versus equivalent plastic strain results for simulations with pre-drawing of 40% reduction in area with the combined hardening model with H of 200 (MPa)

118 Contour plot of equivalent plastic strain for 20% reduction

118 Contour plot of maximum principal (MPa) stress for 20% reduction in area

119 Contour plot of temperature (oC) for 20% reduction in area

x

Number Page

Table 2-1 11

Table 2-2 14

Table 3-1 43

Table 3-2 45

Table 3-3 47

Table 4-1 65

Table 4-2 65

Table 4-3 67

Table 4-4 67

Table 4-5 76

Table 4-6 76

Table 4-7 78

Table 4-8 78

Table 5-1 106

Table 5-2 106

Table 5-3 106

Table 5-4 107

Table 5-5 107

Table 5-6 107

Table 5-7 108

Table 5-8 114

Table 5-9 115

Table 5-10 115

Table 5-11 115

Table 5-12 115

Table 5-13 116

LIST OF TABLES

Tille

Wire sizes for cold heading (adapted from Gill and Baldwin, 1964)

Mechanical properties oftwo steels (adapted from Tozawa and Kojima, 1971)

Values of material parameters obtained from tension curve fitting

Values of material parameters obtained from compression curve fitting

Values of material parameters for S45C and Mn steel

Chemical composition of materials used (adapted from Tozawa and Kojima, 1971)

Pre-drawing reductions in area and die approach angle for S45C and Mn steel.

Original geometry of the rod corresponding to finallength 12 mm

Original geometry of the rod for the FEM drawing model

Heights of compressed rods to fracture (a) for S45C and (b) for Mn steel

Chemical composition of AISI 1335 (adapted from EAD Inc., 1977)

Material properties of Mn steel

Corresponding heights at fracture from Gill' s paper

Predicted heights to fracture and reductions in height for simulations with H of 300 (MPa)

Predicted heights to fracture and reductions in height for simulations with H of 600 (MPa)

Predicted heights to fracture and reductions in height for simulations with isotropic hardening model

Heights to fracture and reductions in height from the experiments in the literature

Comparisons of heights to fracture between the simulation and experimental results

Comparisons of reductions in height between the simulation and experimental results

Differences of reduction in height

Reductions in height for the experiments in the literature

Calculated Cockroft and Latham constants

Comparison of the predicted and experimental heights to fracture for the process with 15°approach angle

Comparison of the predicted and experimental heights to fracture for the process with 300 approach angle

Comparison of the predicted and experimental heights to fracture for the process with 600 approach angle

Comparisons of reductions in height between the simulation and experimental results for the process with 15° approach angle

Xl

Table 5-14 116

Table 5-15 116

Table 5-16 116

Table 5-17 117

Table 5-18 117

Table 5-19 120

Table 5-20 120

Table 5-21 120

Table 5-22 121

Table 5-23 121

Table 5-24 121

Table 5-25 121

Table 5-26 122

Table 5-27 122

Comparisons of reductions in height between the simulation and experimental results for the process with 300 approach angle

Comparisons of reductions in height between the simulation and experimental results for the process with 60° approach angle

Differences of reduction in height between results from simulations and literature for 15° approach angle

Differences of reduction in height between resuIts from simulations and literature for 30° approach angle

Differences of reduction in height between results from simulations and literature for 60° approach angle

Radius of the rods to fracture for 7° approach angle



Comparisons of ratios of the fracture radius to the initial radius of the rod between the simulation and experimental resuIts for 7° approach angle



Differences between the predicted and experimental ratios for 7° approach angle



xii

a 2a 0.

gnew

dv aij

do.ij

Ë P

Ë f

t P

êl

1

. " é

. * ê axia/

!!.ê e

!!.ê P

!!'ËP

o al

am al

GLOSSARY

semi-angle approach angle backstress tensor

backstress tensor at the end of the time increment

backstress tensor at the beginning of the time increment

backstress rate tensor

components of a backstress tensor

deviatoric backstress tensor

components of a deviatoric backstress tensor

backstress increment components

equivalent plastic strain

equivalent plastic strain to fracture

uni axial plastic strain.

tensile plastic strain at yield point of cycle i .

compression plastic strain at yield point of cycle i .

total mechanical strain rate tensor

elastic strain rate tensor

plastic strain rate tensor

equivalent plastic strain rate

dimensionless equivalent plastic strain rate for Eo= 1.0 s-J

dimensionless axial plastic strain rate for Eo= 1.0 s-J

increment of uniaxial plastic strain.

equivalent plastic strain increment

components of a plastic strain increment

increment of strain tensor over a time increment

increment of elastic strain tensor over a time increment

increment of plastic strain tensor over a time increment

increment of equivalent plastic strain over a time increment axial stress

maximum principle tensile stress

hydrostatic stress

tensile and compression yield stresses of cycle i

compression and compression yield stresses of cycle i

equivalent stress

xm

o

d trial

(Y new

o new

o ISO

(Yiso new

do df.1 t;ij

2f.1 À

dÀ yP

1z 1]

P A Ao

AI

Ac

AI

AIr

A* B B*

C

Co

CI

C2

C3

C4

Cs Cz

Cz

components of a stress tensor

stress tensor

stress tensor rate

trial stress tensor at the end of the time increment

stress tensor at the end of the time increment

stress tensor at the beginning of the time increment

a measure of the size of the yield surface a measure of the size of the yield surface at the end of the time increment increment of uniaxial stress positive scalar

the Kronecker delta

Lames constant Lames constant a positive scalar heat flux per unit volume

a material scalar quantity the plastic heat fraction material density Johnson-Cook material constant

original cross-sectional area of the rod

Oyane fracture criterion constant

CUITent cross-sectional area of the rod

finishing cross-sectional area of the rod

cross-sectional area of the rod at fracture

Johnson-Cook material constant at the strain rate of 0.002 S-I

Johnson-Cook material constant Johnson-Cook material constant at the strain rate of 0.002 S-I

Johnson-Cook material constant

fracture criterion constant

Frudenthal fracture criterion constant

Cockroft and Latham fracture criterion constant

Brozzo et al. fracture criterion constant

Oh et al. fracture criterion constant

Oyane fracture criterion constant

a material scalar quantity

the rate of change of Cz with respect to temperature and field

variables

specific heat

XIV

c F F(oij)

J; g H

h

Q

/inew

striai _new

Tnew

Tmelt

Told

t

!!lt y

kinematic hardening material constant die reaction force

function of actual stress state

field variable

plastic potential anisotropic part of the plastic hardening modulus or kinematic hardening modulus material constant representing the isotropie part of the plastie hardening modulus

initial height of the rod

height of the rod at fracture

CUITent height of the rod

ratio of the stress at elevated temperature to that at the room temperature at the same strain rate, material constant in yie1d function

finallength of the rod

originallength of the rod

Johnson-Cook material constant Johnson-Cook material constant

normal to the Mises yield surface

original radius of the rod

finishing radius of the rod

coefficient of multiple determination fraction al drawing reduction in area deviatoric stress tensor

deviatoric stress tensor at the end of the time increment

trial deviatoric stress tensor at the end of the time increment

components of a deviatoric stress tensor

CUITent temperature increment of temperature over a time increment rate of temperature normalized temperature

reference temperature

temperature at the end of the time increment

melting temperature of the material

temperature at the beginning of the time increment

time time increment yield stress in uniaxial tension (or compression)

xv

1 Introduction

1.1 Motivation

One of the most common indus trial cold forging processes is cold heading of steel wire or

rod to produce screws, bolts, nuts and rivets (Billigmann, 1953). The major consumers of

the se products are automotive, construction, aerospace, railway, metallurgical industry

and electrical product sectors (Barret, 1997).

It is of great significance to assess the formability of cold heading materials (metals), a

major feature of forming processes, since failures, because of insufficient formability of

heading materials, result in expensive equipment downtime, material waste, tooling

damage, and unpredictable potential loss to end users. The forming process is limited by a

complicated interplay of several factors, namely, material microstructure, temperature,

deformation rate, tool and workpiece geometry, the friction at the interface of the

workpiece and tool (Sowerby et al., 1984), the surface quality of the workpiece (Muzaket

et al.,1996; Maheshwari et al., 1978) and the amount of cold work (pre-drawing)

performed on the workpiece prior to cold heading (Jenner and Dodd, 1981).

The ductility of a material, which can be defined as "the ability of a material to withstand

deformation without fracture" (NickoletopouIos, 2000), is strain-history dependent

(Rogers, 1962). Many researchers have shown that wire drawing after process annealing

can in sorne circumstances increase ductility in subsequent upsetting operations

(Billigman, 1951; Gill and Baldwin, 1964; Luntz, 1969/1970; Tozawa and Kojima, 1971;

Havranek, 1984). The Bauschinger effect has been observed in the upsetting of steel wire

following pre-drawing, which is regarded as the cause for increased ductility in the

subsequent upsetting (Havranek, 1984). Therefore, the quantitative evaluation of effects

of pre-drawing on fracture would be of considerable use to the cold heading industry

(Nickoletopoulos, 2000).

1

To evaluate the effects of pre-drawing on formability during cold heading for metals,

trial-and-error, by repeating the real physical process and taking measurements, is not a

feasible approach since it is time consuming, costly, difficult and sometimes even

impossible. Alternatively, estimating the effect of pre-drawing with the finite element

method (FEM) , which has proved to be a powerful tool to simulate a metal forming

operation, is far more cost-effective. However, although several investigations into effects

of pre-drawing on the formability of several metal materials during cold heading

processes have been conducted, no attention has been given to the numerical simulations

of this phenomenon so far. Most of the numerical simulations of bulk forming processes

simulated only a single process such as drawing, extrusion, or upsetting. Although one of

the simulations (Petrescu et al., 2002) involved both pre-drawing and subsequent

upsetting, the intent was not to evaluate the effects of pre-drawing, but to simulate as

accurately as possible the typical procedures employed in the fastener manufacturing

process. To address the above concerns, the present project was initiated.

1.2 Objective

The objective of this work is to examine effects of pre-drawing on formability during cold

heading through numerical simulations. In this work, physical tests from the literature

(Gill and Baldwin, 1964; Tozawa and Kojima, 1971) were simulated with finite element

software ABAQUS 6.4; a new combined linear kinematic/nonlinear isotropic hardening

constitutive model was proposed and implemented in the simulations. The Johnson-Cook

isotropic hardening constitutive model was also applied to make a comparison between

the isotropic hardening and combined linear kinematic/nonlinear isotropic hardening

models. The well-known Cockroft and Latham criterion (Cockroft and Latham, 1968)

was employed to prediet the surface fracture.

The material constitutive model is one of the cri tic al inputs required for an accurate

numerical simulation of a metal forming process. To date, most simulations of bulk metal

forming processes in literature are with isotropie plastic hardening material models as

material constitutive model inputs, whieh are reasonable for monotonic bulk forming

2

processes. To simulate reverse-loading processes, such as pre-drawing followed by cold

heading in this work, kinematic hardening material models must be used instead due to

the fact that isotropic plastic hardening material models are incapable of taking into

account the Bauschinger effect. Therefore, a combined linear kinematic/nonlinear

isotropic hardening constitutive model, which is able to account for the Bauschinger

effect, was proposed to simulate the inelastic behavior of three materials, Mn steel and

C45S, investigated by Tozawa and Kojima (1971), and AISI 1335 investigated by Gill

and Baldwin (1964). Since the chemical compositions of the Mn steel and AISI 1335 are

similar, in this work, they are treated as the same material.

1.3 Organization

This thesis is divided into the following chapters: Chapter 2 presents a literature review of

cold heading, pre-drawing processes, ductile fracture criteria, the Bauschinger effect,

constitutive relations and numerical simulations of met al forrning processes. Chapter 3

presents the determination of ductile fracture criteria, the identification of material

constants for the Johnson-Cook hardening model, the derivation of a new combined linear

kinematic/nonlinear isotropic hardening constitutive model and the implementation of the

combined constitutive model through user subroutines including the verification of the

user subroutines. Chapter 4 describes the numerical simulation models. Chapter 5

presents the simulation results and discussion. Finally, conclusions and recommendations

for future work are presented in Chapter 6.

3

2 Literature Review

2.1 Cold Heading

Cold heading is a cold-forging process in which the force developed by one or more

blows of punch( es) is used to upset the metal in a portion of a wire or rod blank contained

between the punch(es) and die(s) in order to form a section of a pre-determined contour.

(Davis, 1988). The cross-sectional area of the initial material is increased as the height of

the workpiece is decreased. Figure 2-1 illustrates the cold heading on an unsupported

bar in a horizontal machine (Davis, 1988).

(0) (b)

o (cl)

Figure 2-1: Schematics of the cold heading on an unsupported bar in a horizontal

machine. (a) Head formed between punch and die. (b) Head formed in punch. (c)

Head formed in die. (d) Head formed in punch and die. (adapted from Davis, 1988)

High production rates, low labor costs, and materials savings grant cold heading a

productive and economical process. Not only is cold heading widely used to produce

fasteners, su ch as bolts, rivets and nuts, but it can also successfully and economically

4

form a variety of other shapes. According to part size, production rates range from about

2000 to 50 000 pieces per hour. Many parts traditionally manufactured by machining

have been produced with cold heading (Janicek and Maros, 1996). Advantages of the

cold heading process over machining of the same parts from suitable bar stock include

less waste material, increased tensile strength from cold working, and controlled grain

flow (Davis, 1988).

2.1.1 Properties and Manufacturing Procedures for Cold Heading

Quality (CH Q) Steel Wire

There are two distinct sets of properties required for cold heading quality materials. One

is good cold headability required for cold forming processes; the other involves the

properties relating to product end use (Matsunaga and Shiwaku, 1980). Good cold

headability requires the materials to be adequately soft and ductile to aid the operation,

while product specification usually requires higher yield strength. There is a trade-off

between them. For exampIe, increased carbon content in the steel results in increase of

yield strength but the impact properties and toughness are adversely affected

(Maheshwari et al., 1978).

The very nature of the cold heading process demands that cold heading quality steel wire

should possess an essentially defect-free surface, an internaI soundness, a coating with

excellent lubricating properties and a ductile microstructure (Muzaket et al., 1996;

Maheshwari et al., 1978). To meet these requirements, several sophisticated processing

steps need to be carried out during the production of CHQ steel wire, as shown in Figure

2-2 (Muzaket et al.,1996; Sarruf, 2000).

5

SPHEROIDIZING & ANNEALING

CLEARING & COA TING

Figure 2-2: Convention al procedure for the manufacturing of CHQ steel wire

(adapted from Sarruf, 2000)

2.1.2 Parameters for Cold Heading

There are man y processing parameters influencing cold heading processes. The main

parameters include strain rate and temperature during deformation, friction between die

and workpiece, and pre-draw prior to cold heading (Nickoletopoulos, 2000). In this work,

these processing parameters are taken into account during simulations of the cold heading

processes.

Cold heading is a high-rate deformation process, in which the average strain rates usually

exceed 100 S-l (Yoo et al., 1997). There is a considerable difference between mechanical

behavior at high strain rates and at quasi-static or intermediate strain rates; therefore, it is

necessary to de termine the mechanical properties, such as flow stress, strength, and

ductility, at the deformation rate close to the ones observed during actual cold heading

(Kuhn et al., 2000).

6

Due to the high production rates and high speed, cold heading is essentially an adiabatic

process (Nickoletopoulos, 2000). During the forming process, approximately 90 to 95%

of the mechanicai energy required is transferred into heat (Farren et al., 1925), and as

much as 400 degrees of temperature rise in a workpiece is observed during cold heading

(Osakada, 1989). The flow stress decreases with . . mcreasmg temperature

(Nickoletopoulos, 2000). At constant temperature, increasing strain rate increases the

flow stress (Dieter, 1984A).

Friction conditions between the die and workpiece have a large influence on metai flow,

formation of surface and internaI defects, load of the die, and energy requirements

(Kobarashi et al., 1989). Friction at the die-workpiece interface can increase the

deformation force and may result in non-uniform or Iocalized deformation and surface

bulging (Dieter, 1984A; Nickoletopoulos, 2000).

The effect of pre-draw on cold heading will be reviewed in next section.

2.2 Pre-Draw

2.2.1 Pre-Drawing Process

Pre-draw, a common practice in the manufacture of fasteners (Havranek, 1984), is a cold

drawing process performed after annealing and prior to cold heading.

In the wire drawing process, the cross-sectional area of a wire is reduced by pulling

through a die, the geometry of which determines the final dimensions, the cross-sectional

area of the drawn wire, and the reduction in area. To avoid fracture or unstable

deformation during the drawing process, the pulling force cannot exceed the strength of

7

the wire being drawn (Davis, 1988). Figure 2-3 illustrates a procedure for drawing of rod

or wire.

Nib height 10nun

Die

Figure 2-3: Drawing of rod or wire (adapted from Davis, 1988)

Bell alogie Min at-gIe~2>tapplO.ElChat-gle

Back Ielief90

Figure 2-4: Cross section of a typical wire die for drawing 5.5mm (0.218 in.) diameter rod to 4.6mm (0.18 in.) diameter wire (adapted from Davis, 1988)

As one type of forming process, wiredrawing is a complex interaction of such main

parameters as material properties (flow stress, modulus of elasticity, work hardening),

strain rate (drawing speed), lubricant (friction, coatings), reduction in area, die geometry,

8

and temperature during the drawing process (Shemenski, 1999). The resulting mechanical

properties of drawn wire are controlled by the interplay of all these many factors. Hence,

in this work they are all taken into consideration in finite element simulations.

Figure 2-4 shows a typical carbide-drawing die. As can be seen, there are four functional

zones in a drawing die. The first zone, the bell zone, where a lubricant is introduced and

is pulled into the die-wire interface by a moving wire, is the entrance of the die. The

approach zone is the second zone, where the wire is forced to contact with the drawing

cone along the approach angle and is plastically reduced into the dimensions of the third

zone, bearing area. No further reduction occurs in the bearing area, but final dimensional

control and surface finish are established here. The final zone is the exit zone

distinguished by the back relief angle (Davis, 1988).

It is important to notice that there are two key parameters about a drawing die, an

approach angle and drawing reduction in area. The approach angle (2a) is the included

angle between the two sides of the approach zone. This angle is usually expressed by the

semi-angle (a), which is the angle between one side of the approach zone and the

longitudinal axis of the die (Shemenski, 1999). The fraction al drawing reduction in area

may be expressed as: (2.1)

where Ao and A f are the original and finishing cross-sectional area of the rod

respectively.

Wright (1979) pointed out that the die semi-angle (a) and the reduction per die (r)

control deformation of a drawing process because they determine the shape of the

deformation zone in a conical drawing die. Both parameters are incorporated into the tl.

parameter:

9

1

~=(a)[1+(l_r)2]2 r

(2.2)

A low ~ value indicates a long defonnation-zone shape and increased die contact,

resulting in excessive frictional work and heat generation requiring optimum lubrication

and lower coefficient of friction (Shemenski, 1999).

There are many advantages in cold drawing wire before gomg into cold heading.

Pre-draw serves to improve dimension al tolerances and the surface finish of the final

product (Havranek, 1984). Cold drawn wire is stiffer, feeds better and shears cleaner. One

of the more important effects of wiredrawing is its influence on the cold heading limit

(that is, the maximum head diameter to which a wire diameter can be upset), which is an

indication of material formability during cold heading (Gill and Baldwin, 1964).

2.2.2 Effect of Pre-Drawing on Formability during Cold Heading

It has been weIl demonstrated by many researchers that the ductility of a material is a

strain-history dependent parameter (Rogers, 1962). Surface ductility is a function of both

strain history and steel type (Brownrigg et al., 1981). Due to the fact that pre-draw can

change the strain history of CHQ wire steel, it is reasonable to corne up with the idea that

pre-draw should have an effect on cold headability in subsequent cold heading operations.

In fact, many researchers have observed that wire drawing after process annealing can

increase ductility in subsequent cold heading operations.

Gill and Baldwin (1964) carried out tests to investigate the effects of pre-drawing on cold

heading limits by cold heading more than 8000 bolts of AISI 1335, a common cold

heading steel, to a range of head diameter expansions from wire that had been drawn to

various reductions in area with 7°, 15° and 30° dies. During their investigation, first, the

10

wire, which was judged to be of cold heading quality, was fully spheroidized and drawn

into the "ready to finish" diameters listed in Table 2-1. After being re-annealed, the wire

was pre-drawn to different reductions, as shown in Table 2-1, through polished carbide

dies with three different die angles on a drawing machine at 150 ft per min (0.762 mis).

Then the wire was upset in a single blow on a cold header with a hammer speed of

approximately 1000 inch per min (0.423 mis). The cold heading limit was defined as "the

greatest expansion in diameter that could be made on the wire without the appearance of

45 shear cracks in the head" (Gill and Baldwin, 1964).

Ready to Finish Final

Diameter (in) Diameter* (in)

7° die

0.157 (1) 0.139

0.165 (1) 0.139

0.179 (2) 0.139

0.211 (3)0.139

0.241 (4) 0.139

5° die

0.213 (1) 0.211

0.220

0.226

0.241

0.266

0.278

0.304

0.304

0.157

0.165

0.179

0.211

0.241

(1) 0.211

(1) 0.211

(1) 0.211

(1) 0.211

(1) 0.211

(2) 0.211

(3) 0.l79

30° die

(1)0.139

(1) 0.139

(2) 0.139

(3) 0.139

(4) 0.139

Reduction

in Area (%)

23

29

40

57

67

7

12

23

37

42

52

65

23

29

40

57

67

* Numbers ln parentheses indicate number of

drawing passes to go from ready to finish size to

finish size.

Table 2-1: Wire sizes for cold heading (adapted from Gill and Baldwin, 1964)

Il

The results are shown in Figure 2-5. According to this graph, for 15° dies, pre-draw

reductions ranging from 35% to 40% improve heading limits significantly from 2.2 to

2.6; for 7° dies, the greatest cold heading limits are obtained at about 40% reduction in

area; however, drawing through 30° dies reduces the heading limit when reduction in area

is beyond 25%. Gill and Baldwin suggested that a pre-drawing with about 35% reduction

in area, through dies with any die angle in the range of 12° to 20°, should be adopted in

order to obtain a significant improvement in cold heading limits.

2.8

2.6

2.4 ;!::

E 2.2 :::i C)

2 c "0 cu Q) 1.8 J: "0

1.6 (5 ------ ---- -

0 1.4

1.2

0 10 20 30 40 50

Reduction by Drawing, %

-.-7 degree

. -lll-15degree

~30degree

60 70

F' 2 5 C Id h d' l' , head diameter) d ' f 19ure -: 0 ea mg ImIt ( " versus percentage re uchon 0 area Wlre dwmeter

by drawing by 7°,15° and 30° dies (adapted from Gill and Baldwin, 1964)

Luntz (1969/1970) has carried out wire drawing on annealed B.S.311/1 and B.S.311/2

steels. After drawing, the wires that had received three different drawing reductions were

cut into billets with a length-diameter ratio of 1. These billets were then upset at a mean

strain rate of approximately 2 s -J using lubricated platens. The obtained results, true axial

strain at the onset of shear cracking versus reduction in area, are shown in Figure 2-6. It is

12

obvious that Wlre drawing can have a marked effect on ductility during subsequent

upsetting, and ductility increases with increasing drawing reduction in area in terms of

true axial strain at the onset of shear cracking.

1.6 -,----------------------'--..., -o ... 1.4 Q) III c: o Q) C)

~ .5 ... .:.:: 1.0

1.2

... (,) cu cu .50 0.8 cu L-

b cu en Q) 0.6 -~ cu en ')( 0.4 « Q) :::s 0.2 L-I-

0.0 -f-----r----r----r----,....---..,...------l

o 5 10 15 20 25 30

Reduction by Drawing, %

---+- B.8.3111/2

~B.8. 3111/1

Figure 2-6: Plot of fracture true axial strain versus pre-strain by drawing (adapted

from Luntz, 1969/1970)

Tozawa and Kojima (1971) conducted an extensive investigation into the effects of

pre-deformation on the cold headabilities of three alloys: S45C, Mn steel and 17S. The

mechanical properties of these materials are shown in Table 2-2. In this investigation,

drawing dies with three different die angles, 15°,30°, and 60°, were used. After rods were

annealed, they were pre-drawn through the drawing dies to reductions as high as 40% to

get a constant finish diameter of 8 mm. Lubricants were applied during the drawing

process. Then they were cut to aspect ratio of 1.5 and were upset between flat dies

without lubrication. Both the drawing process and the upsetting process were performed

quasi-statically.

13

Yield (MPa) UTS (MPa) Elongation at Reduction in Break, % Area, %

S45C 387.34 655.04 14.2 52

Mn steel 433.43 699.17 19.0 53 17 S 109.83 222.60 17.0 48

Table 2-2: Mechanical properties of three steels (adapted from Tozawa and Kojima, 1971)

83

81 ---- - ----------- -

79

?f? ~ 77 .t: C)

"CD :J: 75

.= r:: 73 0 .. CJ ::l

71 "C QI

0::: -+- Mn steel; 15 degree

69 --.- Mn steel; 30 degree

r--=~--------~----~ r~----~----~---~

--*- Mn steel; 60 degree

67 +-------~----~~--___j -'-178; 30 degree

-+- 845C; 30 degree 65

0 5 10 15 20 25 30 35 40 45

Reduction of Area, %

Figure 2-7: Effect of the reduction of area in drawing, A -A

o ft, on the upsetting Ao

ho -hfT limit, (adapted from Tozawa and Kojima, 1971) h

Figure 2-7 shows the effect of reduction in area by drawing on the upsetting lirnits for the

three rnaterials. For S45C and 17S, wires that are pre-drawn through 30° dies to around

15%-30% and 25%-35% reductions, respectively, have rnarkedly irnproved upsetability.

For the Mn steel, the upsetability of the wire drawn through 15° and 30° dies constantly

14

increases with increasing reduction in area, while the upsetability of the Wlre drawn

through 60° die reaches the lowest value at the reduction of 10% and increases steadily

beyond 20% reduction in area. At constant reduction, the effects of die angles on the

upsetting limits for Mn Steel and S45C are illustrated in Figure 2-8. It can be seen that a

15° die produces the maximum improvement in upsetablility for both materials, which is

in agreement with Gill and Baldwin's findings.

-;;e. .;-

80

78

76

"§, 74 "a;

72 :J: c 70 c 0 68 ~ u 66 :l

"t:J CI)

0::: 64

62

60

0 15 30 45

Die Angle (degree)

60

-0--Mn Steel reduction 10%

- - - - - - Mn steel annealed

-0-S45C reduction 8%

- - - - S45C annealed

Figure 2-8: Effect of approach die angle at constant reduction on the reduction in

height for two steels (adapted from Tozawa and Kojima, 1971)

Figure 2-9 shows the curves of average stress versus average strain in upsetting for the

wire of S45C without pre-draw and with pre-draw of 20% and 40% reductions through a

30° die. It can also be seen that with increased reductions in pre-drawing, initial yield

stress increases, whereas work hardening rates decrease. Figure 2-10 illustrates the

comparison of the curves of average stress versus average strain in upsetting for Mn steel

wire pre-drawn with 40% reduction through 15°, 30° and 60° dies. The result

demonstrates that increasing die angle increases average flow stress, while hardening

rates remain unchanged since the curves are almost parallel.

15

140~----------------------------------------------------------~

120 Ci c.. ~ 100- .... --.-... -.-II) II)

e _ 80 en ni ')( 60 « QI C)

~ 40 -QI > «

20

--e- 40% reduction in area

--- -e-20% reduction in area

-b:- 0% redcution in area o~----~------~----~----~~--~~====~======~

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Average Axial Strain

Figure 2-9: S4SC. Average axial stress versus average axial strain for upsetting with

different reductions of area (adapted from Tozawa and Kojima, 1971)

160

Ci c.. :lE 120 1--II) II) 100 e -en ni 80

~ QI 60 C)

~

----- -.------_.- -_._----- -_._-----

------ .- -----_._- -----_._--- ----------

QI 40 _._-------- --- -_ .. ---_._-

~ -i3-15 degree

20 -~- -.-- -hi- 30 degree

~60degree

o~--~----~----~--~----~----~--~~==~~~ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8


Figure 2-10: Mn steel. Average axial stress versus average axial strain for upsetting

with 40% reduction of area (adapted from Tozawa and Kojima, 1971)

16

2.2.3 The Bauschinger Effect

To study the plastic behavior of metals, it is important to notice that a phenomenon called

the Bauschinger effect occurs in metals whenever there is a reversaI of loading applied

after a previous plastic straining (Chakrabarty, 1987). The Bauschinger effect is a

phenomenon in which a material, after plastic pre-straining, develops a reduction in yield

strength on reverse loading in comparison with that reached on the continuing forward

loading (Chakrabarty, 1987). Figure 2-11 illustrates the Bauschinger effect.

o

Figure 2-11: Schematic Bauschinger effect curve

The solid curve in Figure 2-11 shows the true stress versus strain curve of a specimen of a

typical metal material in simple tension. If after being completely unloaded from a tensile

plastic state, represented by the point A, the specimen is reloaded in simple compression.

The resulting path will follow the curve ABCD, where the yield strength at the new yield

point C is smaller in magnitude than that at A. This phenomenon is called the

Bauschinger effect. In the figure, the line AG is reflected in the reverse quadrant EF so

that permanent softening, ~(), can be defined, which is the indication of the Bauschinger

effect.

17

Havranek (1984) has studied the effect of pre-drawing on strength and ductility in cold

forging of two commercial steels, which were AISI K1020 and AISI K1040. In his

research, rods of the two materials with 13 mm diameter, in both as-rolled and

spheroidize annealed conditions, were cold drawn by 6%, 15% and 29% reductions in

area through dies with included angle of 16°. The drawing was carried out with

lubrication at a drawing speed of 0.27 ms· l. Then specimens, with 10 mm in diameter and

15 mm in height, eut and machined from the drawn rods, were compressed under nearly

homogeneous conditions. In addition, to evaluate the ductility of the materials, a long

cyl indri cal specimen of 10 mm in diameter was expanded by pushing it through a die into

a cavity, because simple compression was insufficient to produce fracture. The true stress

versus true strain curves during the homogenous compression for K1020 in the as-rolled

condition are shown in Figure 2-12, in which the redundant strain has been neglected.

1000 1: .2 1/) 900 1/)

~ Q. 800 E ou ca 1: 0.. 700 "- :i!E 1/)-1/) ~ 600 -en ~ 500 ... 1-

---cr- As rolled

-0-6%

-I::r-15%

-X-29%

400+-~--~--~--~~--~--~~--~--~~

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

True Strain in Compression

Figure 2-12: The effect of pre-drawing on strength in compression. Material K1020.

Only the homogeneous drawing strain is shown (adapted from Havranek, 1984)

The curves in Figure 2-12 were replotted from the same origin in Figure 2-13. Figure

2-14 to Figure 2-16 present the replotted true stress versus true strain curves during the

homogenous compression for K1020 in the spheroidize anneal condition and K1040 in

18

both the as-rolled and spheroidized annealed conditions. It can be seen that a strong

Bauschinger effect occurred in aIl pre-drawn rods as indicated by the permanent softening

~(J'. Moreover, the Bauschinger effect becomes stronger as pre-drawing reduction

increases; therefore, the deformation work represented by the area under the curves

decreases.

'i 1000

::!: -c 900 o in 1/) CI) ... c-E o

C,.)

c

800

700

1/) 600 1/) CI) ... ... en CI) :::l

500

-<>- As rolled

-X-6%

-â-15%

-<>-29%

~ 400+---,---~~--~--~--~--~--~--~--.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Figure 2-13: The effect of pre-drawing on strength in compression. Material KI020.

(adapted from Havranek, 1984)

1000 c .2 1/) 900

ë Co 800 E o

C,.) IV C a.. 700 .- ::!: 1/)-

g: 600 b en CI) 500 :::l ... 1-

-0-As spheroidized

-0-29%

400+---~--~~--~--~--~--~--~--~--~~

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1


Figure 2-14: The effect of 29% pre-drawing on strength in compression. Material

KI020, spheroidized 700°C124h (adapted from Havranek, 1984)

19

_ 1100 CV Il.. ~ 1000 c .2 1/) 900 1/)

f Cl. 800 E o (.) 700 c

~ 600 f ... ~ 500 :::l L-

I-

-o-As rolled

-t::r-6%

--+-15%

~29%

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Figure 2-15: The effect of pre-drawing on strength in compression. Material KI040

(adapted from Havranek, 1984)

900

g 850 ïii 1/) 800 Q)

C. 750 E o -100 (.) cv .!: ~650 1/) -600 1/)

f 550 ... CI) 500 Q)

2 450 1-

-0-As spheroidized

-0-29%

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1


Figure 2-16: The effect of 29% pre-drawing on strength in compression. Material

KI040, spheroidized 700°C/24h (adapted from Havranek, 1984)

Figure 2-17 and Figure 2-18 illustrate the fracture limit diagrams for K1020 and K1040 in

the as-rolled and spheroidize annealed conditions. From these diagrams, Havranek

20

concludes that the fracture limit of a pre-drawn rod is higher than a rod in the original

as-rolled and/or spheroidized conditions, and increases with the increasing pre-drawing

reduction in area.

1.5 -.---------------....,

c ·f .... tn 1.0 1\1 +' C al .. ~ E 0.5 ::::s CJ ... (3

0.0 +-----,.---------r------l 0.0 0.5 1.0 1.5

Axial Strain

~K1020 as-rolled and drawn 29%

-II- K 1020 as-rolled

--fr- K1 040 as-rolled and drawn 29%

-*- K1 040 as-rolled

Figure 2-17: Fracture limits in KI020 and KI040 determined in the support upset

tests (adapted from Havranek, 1984)

C

1\1 ... .... tn 1.0 iii +' c ~ .! E 0.5 ::::s CJ ... (3

0.0 +-----,------.,------1 0.0 0.5 1.0 1.5

Axial Strain

~ K1020 spheroidized and drawn 29%

-II- K1 020 spheroidized

--fr- k1 040 spheroidized and drawn 29%

-*- K1 040 spheroidized

Figure 2-18: Fracture limits in spheroidized KI020 and KI040 determined in the

support upset tests (adapted from Havranek, 1984)

21

2.3 Cold Headability and Ductile Fracture Criteria

Fracture is the failure of a workpiece under load. It can be either brittle fracture or ductile

fracture. In metal forming processes, failures are classified into two typical types of

cracks (Jenner and Dodd, 1981):

(i) Free surface cracks during processes such as upsetting, bending and rolling etc.

(ii) InternaI cracks, such as arrowhead cracks in extrusion and drawing.

AlI of these typical cracks are caused by ductile fracture mechanisms (Jenner and Dodd,

1981), rarely by brittle fracture. Therefore, the occurrence of ductile fracture is a major

limitation governing the limits of formability during metal forming processes. In the cold

heading process, failures can be free surface cracks due to exhaustion of the material

ductility as weIl as internaI cracks caused by the adiabatic shear band (ASB) phenomenon

(Okamoto et al., 1973). The aim of this work is to examine the cold headability or

formability in terms of free surface cracks during the cold heading process. Hence, in the

context of this work, cold headability or formabilty can be defined as the ability of a

material to be cold headed without free surface cracks. The cold headability is a function

ofboth material properties and process parameters (Rao et al., 2003).

It has proved to be difficult to determine a general law for fracture criteria of metals

because the criteria depend on various variables, which are difficult to quantify

experimentally. Therefore, various fracture criteria have been proposed in the literature to

predict the initiation of ductile fracture. According to Rao et al. (2003), the fracture

criteria can be broadly categorized in two groups: empirical and semi-empirical. The

empirical criteria include the strain-based criteria and stress-based criteria. A fracture

locus and a stress formability index are applications of strain-based and stress-based

criteria, respectively. The semi-empirical criteria consist of two approaches: the

22

macromechanical approach (cumulative plastic energy models) and the micromechanical

approach (void coalescence models).

2.3.1 The Macromechanical Approach to Ductile Fracture

The Macromechanical approach involves integrating plastic energy of deformation along

the strain path for the possible failure material points in a workpiece and developing a

ductile fracture criterion by incorporating the calculated plastic energy. The

implementation of the approach requires accurate calculations of stress and strain fields

throughout the deformation process until failure occurs. The development of finÏte

element software for large deformation plasticity has made it possible to accurately

simulate the metalworking processes and predict the required stress and strain

distributions during the deformation processes. Therefore, a ductile fracture criterion

based on the macromechanical approach in conjunction with the use of FEM and the

verifications of workability experiments has become a promising technique in predicting

the occurrence of ductile fracture (Rao et al., 2003).

The typical ductile fracture criteria based on the macromechnical approach postulate that

ductile fracture initiates at a material point in a workpiece when the accumulated plastic

energy reaches a critical value. The criteria can be written in a general form as (Bao and

Wierzbicki,2oo4):

if!

fF(cJij)de P =Co (2.3) o

where

F (0 ij) = a function of actual stress state

Co = material constant

23

l P = equivalent plastic strain

II = equivalent plastic strain to fracture

Many successful criteria belong to the class defined by equation (2.3).

Frudenthal (1950) supposed that fracture occurs once the strain en erg y reaches a critical

value, 'CI':

El

fdde r =CI

o (2.4)

Recognizing the coupled role of tensile stress and plastic strain III inducing ductile

fracture, Cockroft and Latham (1968) have suggested that for a given material,

temperature and strain rate, fracture will occur when the work done by the maximum

tensile stress, '0 l " reaches a cri tic al work energy density value, 'C 2 ' •

(2.5)

A modification to Cockroft and Latham criterion (equation 2.5) was proposed by Brozzo

et al. (1972), who point out that the stress function depends on the level of both the

largest principal stress, 0 l ' and the hydrostatic stress, 0 m :

(2.6)

In terms of the ratio of the maximum tensile stress to the equivalent stress, Oh et al.

(1976) introduced a fracture model:

24

(2.7)

2.3.2 The Micromechanical Approach to Ductile Fracture

The micromechanical approach attributes fractures in metalworking to the effects of

inclusions, microvoid nucleation, growth and coalescence. This approach provides

valuable insight into the fundamentals of damage evolution and is verified by

experiments. However, due to the numerical instabilities and difficulties in obtaining the

necessary material constants, the micromechanical approach is difficult to apply (Behrens

et al., 2000).

One of the examples of this type is proposed by Oyane et al., (1980). According to the

Oyane criterion, the development of inclusions into ductile fracture occurs when total

volumetrie strain reaches a critical value, which is material dependent. This criterion is

expressed as:

(2.8)

where

AI and C 5 = fracture criteria constants

o m = hydrostatic stress

o = equivalent stress

In this criterion, the effect of tensile stress triaxiality ( 0 m / 0 ) is taken into account. For

a particular material, two material constants (AI and C 5) can be determined through

25

experiments in conjunction with finite element method simulations of the experiments.

Oyane et al., (1980) has successfully applied this criterion into simple upsetting tests with

different workpiece geometries and friction conditions.

2.4 Constitutive Relations

In continuum mechanics, a constitutive relation is a law that approximates the observed

physical behavior of a real material under specifie conditions of interest (Hozapfel, 2000).

The mathematical expressions of a constitutive law is called a constitutive equation,

which enable us to specify the stress components in terms of other field functions such as

strain and temperature. Therefore, to determine the response of a material of interest, an

adequate constitutive equation for the material is necessary.

The constitutive relations for plastic behavior of a material consist of three parts

(Mendelson, 1983): a yield criterion that defines the initial plastic response of the material,

a flow rule that relates plastic strain increments to stress increments after initiation of the

plastic response and a hardening rule that predicts changes in the yield surface due to the

plastic strain. In general, the yield criterion is demonstrated as a convex surface, which

initially contains the origin of a stress space. For metals, the most common yield surfaces

are those of von Mises and Tresca . The well-defined flow rule is the associated flow rule

due to von Mises. According to different hardening rules postulated in literature, the

constitutive relations can be classified as isotropie hardening material models, kinematic

hardening material models and combined kinematic/isotropic hardening material models.

26

2.4.1 Isotropie Hardening Material Models

Neglecting the anisotropy of a materiaI and the Bauschinger effect developed during

plastic pre-straining, the isotropie hardening rule assumes that the yield surface uniformly

expands without change in shape in stress space, and for the conditions of loading, under

which all strain rate and thermal effects can be neglected, the current flow stress is either

a function of total plastic work per unit volume (the first hypothesis) or a function of a

certain measure of total plastic strain (the second hypothesis) as expressed by the

following equations (Chakrabarty, 1987):

The first hypothesis:

The second hypothesis:

where

a = equivalent stress, effective stress or flow stress = ~~ sijs!)

sij = components of a deviatoric stress tensor = aij -.!.(all + (J22 + (J33)Oij 3

tJij = the Kronecker delta whose vaIue is unit y when i=j and zero when i* j

a ij = components of a stress tensor

dE: = components of a plastic strain increment

de P = ~ de: de: = equivalent plastic strain increment

(2.9)

(2.10)

27

The yield criterion defines the limit of elastic behavior under any possible combination of

stresses. It is independent of a hydrostatic pressure (Chakrabarty, 1987). For isotropie

hardening, the elastic limit is expressed as (Chen, 1994):

j(o,k)=O (2.11)

where the function j is called a yield function, surface of f = 0 is called a yield surface,

° is a stress tensor, and k is a material constant.

For isotropie hardening models, the von Mises criterion is a well-defined and widely used

yielding criterion. The yield surface for this criterion is a cylinder in stress space. It is

mathematically expressed as:

(2.12)

where

y = the yield stress in uniaxial tension (or compression)

With the von Mises yield criterion, the isotropie hardening rule reduces to a description of

the yield surface through the evolution of a single positive scalar quantity, the flow stress

or the equivalent stress (Dabboussi, 2003).

It can be seen that the isotropie hardening rule contradiets the Bauschinger effect, because,

according to the isotropie hardening rule, after plastic pre-straining, the yield strength on

the reverse loading will be the same as that on the continuing forward loading. In Figure

2-18, for example, the stresses represented by points A and C should be the same in

magnitude in terms of the isotropie hardening rule. Moreover, the fact that only squares of

stresses and stress differences are involved in equation (2.12) implies absence of the

Bauschinger effect (Chakrabarty, 1987).

To correctly describe the material behavior of a material under loading conditions during

28

dynamic events such as metal forrning processes, hardening models should take into

account effects of strain, strain rate and temperature. In general, more complicated models

can give a more realistie approximation of material response. However, no models are

universal for every material; different materials need different models to get a better

approximation ofthe material behavior (Johnson and Cook, 1983).

The Johnson-Cook model (Johnson and Cook, 1983) is an isotropie hardening model,

which is primarily intended for the purpose of computations. With the von Mises yield

criterion, the model is expressed as

where

(5 = [A + B(e p r](1 + Cln.f)(1- T"m)

o = von Mises equivalent stress

E P = equivalent plastic strain

-'-p

f* = ~ = dimensionless equivalent plastic strain rate for Éo= 1.0 S-I ë o

t P = equivalent plastic strain rate

normalized temperature

T = CUITent temperature

Tme't = melting tempe rature of the material

To = reference temperature

A , B, n, C and m = five material constants.

(2.13)

In this model, the von Mises equivalent stress is expressed as a function of the equivalent

plastic strain, equivalent plastic strain rate, and temperature, and the three corresponding

terms in the equation are uncoupled. The first term of (2.13) represents the flow stress as

29

a functÏon of strain hardening for É * = 1 and r* = 0, while the second and third terms

represent the strain rate hardening and thermal softening effect, respectively (Johnson and

Cook, 1983).

2.4.2 Kinematic Hardening Material Models

In order to account for anisotropy and the Bauschinger effect exhibited by real materials,

Prager (1955) proposed a hypothesis of a kinematic hardening rule. According to the

hypothesis of a kinematic hardening rule, the yieId surface of an originally isotropic

material is assumed to translate in stress space without any change in its initial shape and

orientation during plastic deformation, and the direction of the incremental translation of

the yieId surface is the same as that of the plastic strain increment, nameIy, the direction

of the exterior normal of the yield surface at the CUITent stress point according to the

associated flow rule.

If the initial yield function is expressed as 1 (0, k) = 0, where k is a material constant,

according to Prager 's hardening ruIe, the current yield function is given by:

1[(0 -a),k] = 0 (2.14)

where a is a symmetric tensor usually called the backstress tensor denoting the

resulting displacement of the yield surface at any stage of the deformation. Because the

deviatoric part of a is not necessariIy zero, the material becomes anisotropic as a result

of the hardening process. Therefore, the kinematic hardening ruIe, due to Prager, assumes

the form expressed as:

30

(2.15)

where c is a material scalar quantity, daij and de: are the incremental components of

the backstress tensor and plastic strain tensor. Due to the plastic incompressibility of

metals, that is de,~ = 0, where i = 1,2,3, we have daii = cdet = 0 , which indicates that

d d dev dev a= a or a=a . (2.16)

where a and a dev are the backstress tensor and the deviatoric backstress tensor,

respecti vel y.

When c is a constant, the plastic hardening becomes linear work-hardening with a

plastic hardening modulus of H = ~c, and the sum of the yield stresses in tension and 2

compression is equal to twice the initial yield stress (Chakrabarty, 1987).

Ziegler (1959) made a modification to the Prager's hardening rule by assuming that the

direction of translation of the yield surface is in the direction of the line connecting the

center of the yield surface to the CUITent stress point. It is expressed as (Chakrabarty,

1987):

(2.17)

where dJ1 is a positive scalar. Due to the condition that the stress point remains on the

yield surface, dJ1 is determined:

dŒijde; df.1 = ----'-----"-

(Œkt - a kl )de{, (2.18)

If the initial yield surface is that of von Mises, the two hardening mIes coincide because

31

the direction of the plastic strain increment coincides with that of the line connecting the

center of the yield surface to the CUITent stress point, and the CUITent yield function

becomes:

f k 3 S dev S dev ) y 3 dev dev ) y 0 2 19) [(a-g),]= 2"c-a ):C-g - = 2"(sij-aij )(sij-aij - = (.

where S and a are the deviatoric stress tensor and the deviatoric backstress tensor

respectively.

Both Prager's and Ziegler's kinematic hardening rules can only roughly describe the

Bauschinger effect with a single material scalar quantity (Chun et al., 2002). Due to

Prager, the translation of the yield surface represents the strain path, while due to Ziegler,

it generally does not. The two hardening rule coincide in uniaxial tension and

compression and in the simple and pure shear (Chakrabarty, 1987).

2.4.3 Combined Kinematic/Isotropic Hardening Material Models

Realizing the limitation of both isotropie hardening and kinematic hardening rules on the

ability to account for the Bauschinger effect, researchers proposed a more realistic

hardening rule combining the two hardening rules to describe the material behavior under

reverse Ioading conditions (Chakrabarty, 1987). According to this hardening rule, the

yield surface simultaneously undergoes both translation and expansion during hardening

processes. Therefore, the combined hardening rule includes two parts: a kinematic

hardening part, which de scribes the translation of the yield surface in stress space through

the backstress, a, which can be linear or nonlinear; the isotropie hardening part, which

describes the expansion of the yield surface and defines the size of the yield surface as a

function of the plastic deformation. According to the combined hardening rule, the current

32

yield surface is given by:

f[(a - a),a 'SO] = 0 (2.20)

where 0 isa is a function of the total plastic deformation defined in equation (2.10) and

represents a measure of the size of the yield surface.

If the yield surface is that of von Mises, then equation (2.20) becomes:

f [ ( a - a_), a ISO] = l (S - a dev) : (S - a dev) - a iso = 0

2 - - --

where

In the case of monotonie simple tension, the combined hardening mIe reduces to:

. 3 da ,sa = da --cde P

2

where do and dt P are the increment of uni axial stress and uni axial plastic strain.

(2.21 )

(2.22)

(2.23)

In uniaxial tension, dE P = dt P • Let do = l h ,where h is a material scalar quantity. dEP 2

Then the plastic hardening modulus is:

(2.24)

where h and c are measures of the isotropie part and the anisotropie part of the plastic

hardening modulus, respectively. When h and c are constant, then the plastic hardening

modulus is constant, implying a linear strain-hardening mIe. Due to the freedom of h

and c, they may be assumed as functions of the variables of plastic deformation,

including arbitrary material constants to be determined from experiments (Ch akrab art y,

1987).

33

Various combined kinematic/isotropic hardening rules have been introduced by

researchers. A combined nolinear kinematic/isotropic hardening model used by

ABAQUS/Standard (HKS inc., 2004) is reviewed here: the evolution of the center of the

yield surface, i.e. the kinematic part of the model, is defined as:

(2.25)

where Cz and Î z are material scalar quantities, and C denotes the rate of change of Cz

with respect to tempe rature and field variables. The first term in equation (2.25) is the

contribution from Ziegler's hardening rule, which defines that the rate of ~ due to plastic

straining to be in the direction of the line connecting the center of the yield surface to the

current stress, namely, 0 - ~; the second term implies that the rate due to temperature

change is toward the origin of stress space; the last term introduces the nonlinearity in the

hardening rule.

The isotropic part of this model defining the size of the yield surface is assumed to be a

function of equivalent plastic strain -gP, temperature T, and field variables 1;:

2.4.4 Flow Rules

The flow rule defines the relationship between the plastic strain increment and the stress

increment. In general, the plastic strain increment is defined in the form:

(2.26)

34

where the function g is called the plastic potential defining the ratios of the components

of the plastic strain increment (Chakrabarty, 1987); dÂ is a positive scalar.

Assuming the identity of the plastic potential g and the yield function l , the associated

flow rule is obtained as:

dê P = al dÂ lJ a O"ij

(2.27)

Based on von Mises' maximum work theorem, the associated flow implies that the

direction of the increment of plastic strain follows that of the exterior normal at the any

point on the yield surface with a uniquely defined normal (Chakrabarty, 1987).

2.5 Numerical Simulations of Metal Forming Processes

With the development of FEA technology, numerical simulations have been used as an

efficient engineering tool to analyze and optimize metal forming processes, resulting in

lower costs, reduced development time, and improved quality. The common industrial

applications of numerical simulations into forming processes involve predicting material

flow, geometry of final products, die stresses and crack formation (Walters et al., 2005).

2.5.1 Numerical Simulations of Drawing Processes

As early as 1982, Brandal et al. developed a computer program based on the finite

element method to calculate the evolution of plastic flow and the stress-strain distribution

in the interior of the wire during passage through the die. Parameters describing

35

stress-strain relations and presenting the drawing process were used as numerical input

data. Renz et al.(l996) studied the effect of die geometry, back-pullloads and geometries

of raw material on material flow and residual stresses during wiredrawing processes using

the finite element method. The results of their study showed that FEM is a helpful tool for

optimization of drawing processes. By conducting several finite element simulations with

the commercial software called DEFORM, Vijayakar (1997) analyzed the influence of

thermal gradients within the wire on residual stresses during the drawing process.

Experimental work proved to be difficult to accurately measure residual stresses, whereas

finÏte element modeling is capable of calculating the stress distribution in detail during

each stage of the drawing process. Zhao et al. (1998) performed an application in

wiredrawing making good use of the finite element method to study the evolution of

voids in the continuous cast rod. In their work, a commercial software package was used

to simulate wiredrawing as an axisymmetric process with the rod modeled as an isotropic

material. The discrepancy between the results of the simulations and those of the physical

experiments and the resulting anisotropy from the drawing process suggested that

modeling the rod as an anisotropie material should give doser results to those of the

physical experiments. Shemenski (1999) investigated the effect of lubrication and die

geometry on wiredrawing process using the DEFORM software package. Numerical

simulation for the drawing of an AISI carbon steel wire in 13 passes to a total 94.5%

reduction in area was carried out. The flow stress of this carbon steel was assumed to be

isotropic and expressed as a function of strain, strain rate, and temperature; frietion

coefficients of 0.10 and 0.30 were selected. The result of this work showed that there

were no major differences in stress levels or resulting gradients for combination of

entrance angle, approach geometry, and coefficient of friction used in the FEA

simulations. He et al. (2003) conducted a study on the residual stress in cold drawn wire

of low carbon steel by means of three-dimensional elasto-plastic fini te element method

(FEM) analysis and X-ray diffraction. Instead of the isotropic von Mises yield criterion, a

texture-based anisotropic yield locus was applied to the three-dimensional mode! to

36

simulate the process and calculate the residual stress. Good agreement between

experimental data and computation al results were reached. Camacho et al. (2005) studied

the influence of back-pull in the drawing process by means of the finite element software,

ABAQUS/Standard. The drawing process for different values of back tension was

simulated under the assumptions that the material behavior of an aluminium alloy is rigid

perfectly plastic, and the Coulomb friction coefficient equals to 0.1. The results from the

simulation are similar to experimental data.

2.5.2 Numerical Simulations of Cold Heading Processes

Numerical simulations have been successfully performed in analysis and optimization of

the deformation process during upsetting by researchers. Roque and Button (2000)

simulated an upsetting process by means of a commercial general fini te element software,

ANSYS. In their work, the stress-strain response, the material flow during the simulated

stage, and the required forming force obtained by experiments were used as numerical

model input data and to validate the numerical models. The blank steel was modeled as the

multilinear el asto-plastic material; contacts between the dies and the blank were modeled

by the Coulomb law with a penalty method. It was demonstrated that numerical prediction

of the final shape, flow stress, and the stress field were in good agreement with the

experimental results. Luo et al. (2000) simulated the upsetting of cylindrical billets between

rough plate dies as a typical thermomechanical coupling process. The effects of upsetting

speed and initial temperature of billets on the upsetting process were demonstrated by the

simulations. To assess the formability of a medium-carbon cold heading steel 1038,

Nickoletopoulos (2001) simulated the drop weight test (an upsetting process) by a

commercial finite element package, FORGE2. The steel was taken to be an

elastic-rigid-plastic material, and the friction parameter was determined to be 0.13 by the

friction ring test. The ductile fracture criterion proposed by Cockroft and Latham was

37

evaluated using FEM simulations, and proved to be valid for upsetting in cold heading. Hu

et al. (2004) studied the pressure distribution on a die surface during an upsetting process

by finite element method simulation. The simulation results were basically consistent with

those of validation experiments. They concluded that the pressure distribution on the die

surface during the upsetting of a cylinder is non-uniform, and the friction factor has an

important influence on the total upsetting force.

2.5.3 Numerical Simulations of the Fastener Manufacturing Process

Petrescu et al. (2002) simulated the fastener manufacturing process, which includes wire

drawing followed by subsequent multistage cold forming. In their work, the finite element

software package FORGE2 was used, and a physical simulation test was developed to

provide the necessary data for the numerical model. To accurately simulate the typical

procedures employed in the fastener manufacturing process, the wire drawing with three

passes was simulated first followed by the simulation of the subsequent cold bulging

process. For both the drawing and bulging processes, the workpiece material, 51B40 steel,

was simulated as an elastic-plastic material, linear in the elastic domain and isotropic in

the plastic domain. Hensell-Spittel's law was used to de scribe the flow stress, and friction

between the material and dies was assumed to obey Coulomb's law. In the drawing

simulation, the material rheology at the end of the simulation of each drawing pass was

saved and then incorporated into the simulation of the next drawing pass. In the bulging

simulation, the material rheology from the last drawing pass was incorporated. The final

results show that the final billet geometry from the numerical model agree weB with that

produced by the bulging test; the location of fracture observed during the bulging test

correlates with the area of high ho op stress in general, which complies with the damage

model developed by Cockroft and Latham.

38

3 Model Development

In this chapter, the mathematical background is presented in five main parts. The first part

describes the fracture criterion determination. The second part presents the selected

Johnson-Cook isotropic hardening model and the identification of the corresponding

material constants. In the third part, a new combined linear kinematic/ nonlinear isotropic

hardening model is proposed and derived. A user subroutine VUMAT, which is used as

an interface to specify the new material model in the finite element software

ABAQUSlExplicit as weIl as user subroutine UMAT, which is used as an interface to

specify a linear elastic material model in the finite element software ABAQUS/Standard,

are developed in the fourth part. Finally, the verification of the VUMAT is presented in

the fifth part.

3.1 Failure Criterion Determination

Successful application of the Cockroft and Latham criterion to the prediction of fracture

in different tests, such as tensile, torsion, bending and extrusion tests, showed that the

Cockroft and Latham criterion is capable of predicting the fracture under complex stress

conditions (Cockroft and Latham, 1968). Jenner and Dodd (1981) concluded that the

Cockroft-Latham criterion is not only accurate enough to predict the onset of surface

cracking in cold upsetting, but also simple and intuitively correct. Research has also

suggested that the Cockroft and Latham failure criterion is amongst the best for practical

applications (MacCormack and Monaghan, 2002). This criterion has been used to predict

fracture in processes such as extrusion, drawing, rolling and upsetting.

In the current work, the macromechanical approach to ductile fracture was used, and the

Cockroft and Latham criterion was chosen. This criterion, equation (2.5), states that for a

given material, temperature and strain rate, fracture will occur when the work done by the

maximum tensile stress accumulates a critical work energy density value, "C2 ".

39

Cf

fC7 deP = c 1 2 (2.5) o

where 0 1 is the maximum tensile principal stress, lP is the equivalent plastic strain, II

is the equivalent plastic strain at failure. When there are only compressive stresses, 0 1 is

set to zero, and fracture does not occur.

For cold heading applications, the critical Cockroft and Latham constant, "C 2 ", is

determined by using a drop weight compression test (DWCT) (Nickoletopoulos, 2000). In

this test, the cri tic al height to fracture for specimens of a material with different aspect

ratios is obtained. Finite element simulations of the experimental tests are performed to

compute the Cockroft and Latham parameter for each specimen at its critical height to

fracture. These values expected to be approximately the same for any partieular material

regardless of the different aspect ratios, are then averaged. The average value is regarded

as the critical Cockroft and Latham constant and used as a threshold to assess the

potential for the initiation of fracture.

3.2 Identification of the Corresponding Material Constants for

Johnson-Cook Hardening Model

In the CUITent work, the widely used Johnson-Cook isotropie hardening model, equation

(2.13), is selected in order to compare the results of fini te element simulations with the

isotropie hardening model and those with the combined isotropic/linear kinematic

hardening model proposed in next section.

3.2.1 Typical Procedures to Determine the Corresponding Material

Constants for the Johnson-Cook Hardening Model

In equation (2.13), the constants in the first set of brackets, A, B, and n, are usually

determined from the data of quasi-statie tension tests or quasi-static homogenous

40

compression tests on samples of the material at room temperature. Since quasi-static tests

are performed at a very low strain rate and at room temperature, the effect of the thermal

softening is negligible. Assuming the strain rate is 0.002s-' , we have:

a = [A + B(tPt][1 + Cln(0.002)] = A* + B* CEPt

where A* = A[I + C In(0.002)] and B* = B[I + C In(0.002)].

(3.1)

A* is the yield stress at the strain rate of 0.002 S-I and can thus be determined directly

from the true stress-strain curve obtained from the quasi-static test. Then, B* and n can

be found by either a nonlinear regression analysis of the true stress-strain curve in the

plastic zone, or alternatively a linear regression analysis after re-writing equation (3.1) as:

(3.2)

The curve of ln( a - A *) verses InCE P) is linear with the slope n and intercept ln B* .

The values of A* and B* , which are determined from the quasi-static tests, are for the

strain rate of 0.002 S-I or E * = 0.002 S-I , therefore, the y must be adjusted for the strain rate

of 1.0 S -1 to obtain the values of A and B . Thus

A* A=-----

1 + C In(0.002)

B* B=------

1 + C In(0.002)

where C will be determined later.

(3.3)

(3.4)

To determine the parameter m, quasi-static tensile tests over a range of temperatures

need to be performed. m can be determined by performing a nonlinear regression

analysis on the curve of the thermal softening fraction, KT , versus normalized

temperature, T* , taken from the data of the tensile tests. KT is the ratio of the stress at

elevated temperature to that at the room temperature at the quasi-static strain rate, and

expressed as:

41

K = [A+B(&"pr](l+Clnt*)(l-T*nI) =l_T*nI

T [A + B(&"P)](l + Clnt*)(l- 0) (3.5)

To determine the parameter C, substituting equations (3.3) and (3.4) in equation (2.13),

we have:

- [A* B* (-p)n](1 Cl ·*)(1 T*nI) (j = + ê + nê -1 + C ln(0.002) 1 + C ln(0.002)

(3.6)

The only unknown in equation (3.6) is C, which can be found similarly by a nonlinear

regression analysis about the curve of equivalent stress, 0 , versus strain rate Ë obtained

from the data of dynamic Hopkinson bar tensile tests. Once C is determined, A and B

can therefore be determined.

3.2.2 Determination of the Corresponding Material Constants for the

Johnson-Cook Hardening Model in this Work

110

100 -----

90

N E E -C)

~ 70 -II) II) Cl) 60 ... -en

50

40 ---- -

30 0.0 0.1 0.2 0.3

Strain

-.e.- Mn steel; compression

-+- Mn steel; tension

. -- --a- S45C; compression

-b- S45C; tension

0.4 0.5 0.6

Figure 3-1: Stress versus strain in simple tension and compression tests (adapted

from Tozawa and Kojima, 1971)

42

In the current work, material constants for the Johnson-Cook hardening model for S45C

and Mn steel are determined. The mechanical properties of the two materials (Tozawa

and Kojima, 1971) are shown in Table 2-2. The stress versus strain curves of the

materials in simple tension and in compression (with friction) from the literature are

shown in Figure 3-1. They are replotted in the plastic zone in Figure 3-2.

1100

1000

900

n; 800 C. :::!: -1/) 700 1/) G) ... - 600 U)

500

400

300

0 0.1 0.2 0.3

Plastic Strain

.... ~ Mn steel; compression

-+- Mn steel; tension

-e- S45C; compression

-fs- S45C; tension

0.4 0.5 0.6

Figure 3-2: Stress versus plastic strain in the simple tension and compression tests

materials A* [MPa] B* [MPa] n Coefficient of Multiple

Determination (R 2 )

S45C 387.34 2265.30 0.79 0.98

Mn steel 433.43 1098.02 0.49 0.99

Table 3-1:Values ofmaterial parameters obtained from tension curve fitting

To find values of A* , B* and n for the two materials, the stress versus plastic strain

curves in simple tension tests from Figure 3-2 are considered. Both linear and nonlinear

regression analyses mentioned above are performed, and the nonlinear regression

analyses give a better fit. The values of material parameters obtained from nonlinear

43

regression analyses are given in Table 3-1. Figure 3-3 shows the stress versus plastic

strain curves ca1culated from the Johnson-Cook hardening model with the values of the

material parameters in Table 3-1 superimposed on the experimental ones. It can be seen

that, for the strain within 0.09, a good fit is observed for both materials when comparing

the tension results with those ca1culated with the Johnson-Cook hardening model.

800

700

_ 600 ni

Do :E 500 -fi) fi) 400 ~ -en 300

200

100

o 0.01 0.02 0.03

-- -+- S45C; Calculated trom Johnson-Cook

--- S45C; tension test in the literature

-&- Mn stee;calculated trom Johnson-Cook

---br- Mn steel; tension test in the literature

0.04 0.05 0.06 0.07 0.08 0.09 0.1

Plastic Strain

Figure 3-3: Comparison of the stress versus plastic strain curves calculated from the

Johnson-Cook hardening model with the corresponding values of the material

parameters obtained from tension curve fitting and those from tension tests in the

literature

To examine the tension curve fitting results for the strain beyond 0.09, the plot comparing

the stress versus plastic strain curves ca1culated from the Johnson-Cook hardening model

to those from the compression tests (with friction) in the literature (shown in Figure 3-2)

for both materials is shown in Figure 3-4. It is observed that when strain goes beyond 0.1

for S45C, and beyond 0.17 for Mn steel, the results calculated from the tension curve

fitting are increasingly higher than those from the compression tests. The stress in simple

compression (with friction) is larger than that in a homogeneous compression (without

friction) since the friction at the die-workpiece interface can cause non-uniform plastic

44

de formation and surface bulging, causing an increasing deformation force. Hence, for

strain beyond 0.1 for S45C, and beyond 0.17 for Mn steel, the results from the tension

curve fitting do not fit those from the homogeneous compression. In addition, Figures 2-9

and 2-10 show that the plastic strain at the end ofupsetting is in the range of 1.0 to1.7. To

simulate the process, a good fit in this range is required. There fore , the values of the

material parameters obtained from the tension tests are not suitable.

1600

1400

_ 1200 CG D. :E 1000 -1/)

1/) 800 ~ -U)

600+-;;;li~P:

400

200

o 0.1

- --.- S45C; calculated from Johnson-Cook

. ____ --+- S45C; compression test in the literature

-a- Mn steel; calculated from Johnson-Cook

0.2

-&- Mnsteel;compression test in the literature

0.3

Plastic Strain

0.4 0.5 0.6

Figure 3-4: Comparison of stress versus plastic strain curves calculated from the

Johnson-Cook hardening model with the corresponding values of the material

parameters obtained from tension curve fitting and those from compression tests in

the literature

materials A* [MPa] B* [MPa] n Coefficient of Multiple

Determination (R 2 )

S45C 387.34 784.54 0.40 0.97

Mn steel 433.43 733.29 0.32 0.95

Table 3-2: Values ofmaterial parameters obtained from compression curve fitting

45

Ci Il.. :! -1/) 1/) CI) ... -CI)

1300~------------------·-------------------------------------,

1200 --- --~---- ------ ---~-- ~-- ----- -=...-tlCA!!Fl---

1100

1000

900

800

700

600

500

400

300 0

-~Ad=--/"-------------------- ------------------ - ------

-+- Mn steel; compression test in the literature

-1ir- Mn steel; calculated tram Johnson-Cook

-a- S45C; compression without pre-drawing in the literature

-&- S45C; calculated trom Johnson-Cook

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Plastic Strain

1.1 1.2 1.3

Figure 3-5: Comparison of stress versus plastic strain results from compression tests

in the Iiterature and those calculated from the Johnson-Cook hardening model with

the corresponding values of the material parameters obtained from compression

curve fitting

To obtain more reliable values for A* , B* and n for the two materials, stress versus strain

curves from compression tests without pre-drawing with a larger strain range should be

considered. Since there are no experimental results available in homogeneous

compressions (without friction) for the two materials, the stress versus strain curve in

compression (with friction) without pre-drawing from Figure 2-9 for S45C, and that in

compression from Figure 3-2 for Mn-steel are chosen. Both the linear and nonlinear

regression analysis methods are used to determine the values of A* , B* and n, and still

the nonlinear regression analysis gives a better fit. The values of the material parameters

for both materials obtained from the nonlinear regression analyses are shown in Table 3-

2. The stress versus plastic strain curves calculated from the Johnson-Cook hardening

model superimposed on the experimental ones for both S45C and Mn steel are shown in

Figure 3-5. In the strain range of 0.09 to 1.2 for S45C and 0.09 to 0.5 for Mn steel, a

46

better fit is observed. Although the compressIOn tests considered here are not

homogeneous, these curve fitting results are better than those obtained from tension tests

for plastic strain beyond 0.09. Therefore, the values deterrnined from compression tests

were more reasonable and hence adopted in this work.

The material constants, m and C, taken from (Johnson and Cook, 1983) for comparable

steels are 1.0 and 0.022 for S45C and 1.03 and 0.014 for Mn steel respectively. Finally,

the values of A and B determined according to equations (3.3) and (3.4) are 448.68

(MPa) and 908.79 (MPa) for S45C, and 433.43 (MPa) and 733.29 (MPa) for Mn steel

respectively. The values of material parameters are summarized in Table 3-3.

materials A* (MPa) B* (MPa) A (MPa) B(MPa) n C m

S45C 387.34 784.54 448.68 908.79 0.399 0.022 1.0

Mn steel 433.43 733.29 433.43 733.29 0.32 0.014 1.03

Table 3-3: Values of material parameters for S45C and Mn steel

3.3 A Proposed New Combined Linear Kinematie/Nonlinear

Isotropie Hardening Model

In the CUITent work, forming processes involving reverse loading conditions were

simulated. To account for the possible Bauschinger effect, a new combined linear

kinematicl nonlinear isotropie hardening model for metals subjected to reverse loading is

proposed. The following are the assumptions considered in this model.

The material is assumed to be initially isotropic and harden anisotropically by a

combination of expansion and translation of the yield surface in stress space.

47

Yield criterion:

It is assumed that the material obeys the von Mises yield criterion. After plastie

deformation, the yield criterion takes the form defined in the equation (2.21). From

equation (2.16), the proposed yield function is expressed as

f[ «(5 - a), (5 iso ] = ~ ~ (~ -~) : Œ - a) - (5 Iso = 0 (3.7)

Hardening rule:

It is assumed that the evolution law of this combined model includes two components: a

linear kinematic hardening component, which defines the translation of the yield surface

in stress space through backstress, ~; and an isotropie hardening component, which

describes the change of the size of the yield surface by 0 ISO, defined as a function of

equivalent plastic strain, E P; temperature, T; and strain rate, t P .

The linear kinematic hardening component takes the form of the linear Prager' s hardening

law defined in equation (2.15), re-written in a rate form as:

. 2 H · p a=- t: 3 -

(3.8)

3 where H = -c, as defined previously. H represents the anisotropie part of the plastic

2

hardening modulus.

The isotropie hardening component assumes that (TsO = G(EP ,T,i

p). The function G

can be determined from the true stress- stain curve in uni axial tension or compression. It

can be shown that in the case of a monotonie uni axial tension, the integration of equation

(2.23) is expressed as:

(3.9)

where 0 and é P are the axial stress and axial plastic strain, respectively.

48

Assuming that the stress-strain relation in uni axial tension is described by the form of the

Johnson-Cook model, then

(3.10)

where

.p

Ë * =~. = dimensionless axial plastic strain rate for Eo = 1.0 S-I axial ê O

E P = axial plastic strain rate

T* T -To 1· d = = norrna Ize temperature Tmelt -To

T = CUITent temperature

Tmelt = melting temperature of the material

To = reference temperature

A , B , n, C and m = five material constants.

Substituting equation (3.10) into equation (3.9), we obtain:

(3.11)

In uniaxial tension, the equivalent plastic strain increment, dE P, equals the axial plastic

strain increment, dE P , i.e. dE P = dE P • Therefore, in uniaxial tension,

(3.12)

Substituting equation (3.12) into equation (3.11), we have

(3.13)

Therefore, the isotropie hardening component assumes that

49

where A ,B, C , n and m are the material constants determined according to the

method described in section 3.2. H is the kinematic hardening modulus, which must be

calibrated from cyclic test data. For an example, H can be simply determined from the

stress-strain data of cycles obtained from a symmetric strain-controlled cyclic experiment

with strain range !!.E . Due to the fact that the elastic modulus of a material is large

compared to its plastic hardening modulus, the magnitude of the plastic strain at the yield

point in reverse loading is the same as that in the continuing loading. Figure 3-6 shows

the cycles. H is determined as

H =_1 f 3(a: +aI

C

)

N 1=1 2E/ (3.15)

where N is the number of cycles; a: and at' are the tensile and compression yield

stresses of cycle i, respective1y. E,P is the tensile plastic strain at yield point of cycle i.

ô t n

rr t 2

ôl

~ GP fl GP = Gf - G~

t

oi ~ OC

2

crC n

Figure 3-6: Symmetric straiD cycle experimeDt (adapted from HKS IDe., 2004)

The flow rule:

It is assumed that the plastic deformation obeys the associated flow rule as defined in

equation (2.26). Substituting the yield function (3.7) into equation (2.27), we obtain:

50

dA can be found by substituting equation (3.16) into dEl' = %dë; dë; . Since

In view of equation (3.7), we have

dA = 3dEP

20'sr)

Using (3.18) in (3.16), we obtain the following in a tensor form:

The von Mises yield surface is a cylinder in stress space with a radius of

(3.16)

(3.17)

(3.18)

(3.19)

(3.20)

The normal to the Mises yield surface according to the associated flow mIe can then be

written as

Using (3.21), (3.19) can be written as:

d§.p =#QdEP

(3.21)

(3.22)

We finally get the proposed specific expressIOn of the associate flow mIe for this

combined model after writing (3.22) in a rate form:

51

fP =#Q€P (3.23)

where the equivalent plastic strain rate is expressed as

-'-p ~.P .p E = -E : E 3- -(3.24)

With assumptions about the yield eriterion, hardening rule and flow rule, the new

eombined linear kinematie and nonlinear isotropie hardening model was proposed.

3.4 Implementation of the Proposed Combined Linear

Kinematie/Nonlinear Isotropie Hardening into ABAQUS

In this section, a user subroutine VUMA T implementing the proposed combined material

model in finite element software ABAQUSlExplicit and a user subroutine UMAT,

implementing a linear elastic material model III finite element software

ABAQUS/Standard, are developed.

3.4.1 Overview of User Subroutine

ABAQUS (HKS Inc., 2004) has interfaces that allow the user to implement general

constitutive equations. In ABAQUS/Standard the user-defined material behavior is

specified with a user subroutine UMAT, while in ABAQUSlExplicit the user-defined

material behavior is implemented in a user subroutine VUMAT. Both user subroutines

VUMAT and UMAT are used when none of the existing material models included in the

ABAQUS material library accurately represents the behavior of the material to be

modeled.

User subroutine VUMAT:

• can use and update solution-dependent state variables

52

• can use any field variables that are passed in

User subroutine UMAT:

• can be used with any procedure that includes mechanical behavior

• can use solution-dependent state variables

• must update the stresses and solution-dependent static variables to their values at

the end of the increment for which it is called

• must provide the material Jacobian matrix, a~%~ê for the mechanical

constitutive model

3.4.2 The Governing Equations

A basic assumption of many elastic-plastic material models is that the deformation of a

solid body can be divided into an elastic part and a plastic part. According to this

assumption, a total strain rate can be decomposed according to the additive strain rate

decomposition. Hence

(3.25)

where È is the total mechanical strain rate tensor, t.. e is the elastic strain rate tensor, and

t.. P is the plastic strain rate tensor.

The stress rate is expressed in terms of Hooke's law by

(3.26)

where À and 2f.1 are the Lames constants for the material.

Plasticity of the new combined model is summarized as follows:

• Yield function:

53

• Plastic flow mIe:

• Prager (linear) kinematie hardening

Using (3.23) in (3.8), we obtain

~=~H [3 QÊP = (2 HQÊP 3 V-z- V3-

• Nonlinear isotropie hardening

3.4.3 Integration of the Governing Equations

(3.7)

(3.23)

(3.27)

To solve the governing equations, we first need to integrate them in rate form in order

to transform the differential problem into a solvable algebraic problem. The backward

Euler integration method is unconditionally stable and simple; applying this method

to the flow mIe (3.23) over a time increment gives

(3.28)

Applying this method to the strain rate decomposition equation (3.25) gives

(3.29)

54

The integration of the stress rate equation (3.26) gives the trial elastic stress based on

purely elastic behavior:

(3.30)

where the subscripts old and new refer to the beginning (at time t) and end (at time

t +!1t) of the increment, respectively. If the trial stress does not exceed the yield

stress, the deformation is within the elastic limit, and the new stress is set equal to the

trial stress. If the trial stress is larger than the yield stress, plasticity occurs in the

increment, and according to (3.29) and (3.28), the new stress is expressed as:

_ trial 2 A P _ trial '-;::6 A - P Q () new - () new - f.1D§.. - () new - V 0 f.1Dê (3.31)

Due to the plastic incompressibility, !1ê~ = 0, using (3.31) and the definition of ~new'

we obtain

(3.32)

Applying the backward Euler integration method to the kinematic hardening equation

(3.27) gives

(3.33)

According to the definition of the normal to the yield surface, equation (3.21), at the

end of the increment, we have

(2-;so Q S g:new + V3(}new _ = _new

This cao be expanded using (3.32) and (3.33) as

lIRA - P Q #-;so Q - striai '-;::6 A - P Q g:old+ - Dê + -(}new -_new- vO f.1Dê

3 - 3 - -

55

Taking the tensor product of this equation with Q, we obtain

.Q [2UA-PQ.Q [2-iSOQ.Q_ s tr;al.Q 176 A-PQ.Q a old . _ + ,r"311uE _. _ + v30'new _. _ - _new . _ - ,\/OfluE _._ (3.34)

During the active plastic loading the stress must remain on the yield surface, so that

~Q:Q =1

Therefore, the equation (3.34) reduces to

. Q [2 UA -P [2-;'0 _ striai. Q 176 ~-P (lold . _ + V 3 11UE +v3

0'new - _new . _ -'\/Ofl E

{3 -P(2 2 H _ ( trial ). Q [2-iso V2~E fl +3 ) - ~new -aold . _ - v30'new

L . j:/rial striai h h b . b ettmg ~new = _new - (lold ,t en t e a ove equatlOn ecomes

SeP - [2 1 [(;:trial:;:tr;al)~_ [2O',~O]=O V3 2 ~new ~new V3 new (2fl +-H)

3

where

O'iSO = [A + B(eP )n ][1 + Cln(e(ew )][1- (Tnew - To)m] - HeP new new - T _ T, new

Eo melt 0

~-P

where e::,w = _E_ by using the backward Euler scheme. M

(3.35)

(3.36)

(3.37)

At the beginning of an increment, çtrial can be ca1culated, andTnew is a function of SK P, _new

which will be shown in the next section. Therefore, equation (3.36), is actually a

nonlinear equation of a single variable b..-e p• Solving for b..-e P numerically, we can then

determine 0 new , 0. new and e::'w' which is expressed as e::,w = et:d + ~eP by using the

backward Euler scheme.

56

3.4.4 Derivation of Temperature Increment for an Adiabatic Analysis

An adiabatic analysis is perfonned when an extensive inelastic defonnation is occurring

extremely rapidly, and the heat caused by mechanical dissipation associated with plastic

straining has no time to diffuse (HKS Inc. 2004).

An adiabatic analysis assumes that plastic straining gives rise to a heat flux per unit

volume of

(3.38)

where yP is the heat flux that is added into the thermal energy balance; 1] is the plastic

heat fraction, which is assumed to be constant; a is the stress tensor; and fP is the rate of

plastic straining.

The component form of (3.38) is as follows

(3.39)

Since sC = 0, (3.39) becomes

Then we have:

P .p y

ê =-1]!i

(3.40) I.e.

Using (3.40) in (3.24), we have

-'-p F·p .p ê = -ê :ê = 3- -

Therefore

(3.41)

57

Using the backward Euler scheme to integrate the plastic strain in (3.41), we obtain the

approximated value of yP at the end of the increment as

A -P P

_ilê Y =17a -

~t

The heat equation solved at each integration point is

(3.42)

(3.43)

where p is the material density; C(7) is the specifie heat; t is the rate of temperature.

Using (3.43) in (3.42), we obtain

Integrating for temperature in the above equation, we obtain

~T = 17(f~&P = ~ (f~&P pC(T) C (7') P

(3.44)

where af),.& P is the dissipated inelastic specifie energy per unit mass. It is clear that ~T p

is a function of ~E P , and hence, Tnew = Tald

+ ~T is indeed a function of ~E P •

3.4.5 Flow Chart and Code of VUMAT and UMAT

The flow chart of the subroutine VUMA T for the proposed combined hardening model is

shown in Figure 3-7. According to the flow chart, 2-dimensional and 3-dimensional

VUMA Ts were developed. 2-dimensional VUMA T is shown in Appendix A. The code of

the subroutine UMAT for isotropie isothermallinear elasticity is taken from ABAQUS.

58

Initialization of parameters (E, v, A, B, n, C, m, C(T), 11, Tmelt, To, to, H)

If step time = 0

Yes

Calculate stress and elastic strain assuming the material pure elasticity

No

Calculate the magnitude of the deviatoric trial stress difference (dsmag) and calculate the radius (radiusold) of the yield circle in deviatoric plane

If dsmag > radius?

Yes

Solving the equation for equivalent plastic strain increment using Bisection iteration

update stress and solution dependant state variables

1

No

Set equivalent plastic strain equal to zero

Figure 3-7: Flow chart for VUMAT

3.5 Verification of the User Subroutine VUMAT

When developing a user subroutine, it is strongly recommended to test them thoroughly

on single element models with prescribed loading conditions before attempting to use

them in real analysis work. In order to verify the VUMAT developed in this work, one

element tests with an 8-node brick element (C3D8R) under uniaxial loading conditions

and reversed loading conditions were performed. S45C was chosen as an example to test

the user subroutine VUMA T specifying the proposed combined hardening material

model.

59

3.5.1 One-Element Tests under Uniaxial Loading Conditions

140°r-------~J;~~~~--------------------------1

(;' Il.

1200

~ 1000 VI VI ~ û) 800 ... 1: G)

iij .~ 600 ::l C" W

~ 400 VI

~ 200

~~~--------------------

-e-- H=1 00 (MPa); Strain Rate 2 -f--Isotropie Hardening; Strain Rate 2

----6- H=1 00 (MPa); Strain Rate 20 --*-Isotropie Hardening; Strain Rate 20

-H=100 (MPa); Strain Rate 200 -e- Isotropie Hardening; Strain Rate 200

O __ ~----~------~------~------~------~------r-----~ o 0.5 1.5 2 2.5 3 3.5

Equivalent Plastic Strain

Figure 3-8: S4SC. Mises stress versus equivalent plastic strain results from uniaxial

tension simulations with H = 100 (MPa) and isotropie hardening

The purpose of the tests under uniaxial loading conditions (uniaxial tension and uni axial

compression) is to verify the proposed combined model by comparing it to the Johnson

Cook plasticity model built into ABAQUS with equivalent plastic hardening. The one

element testing models are described as follows: the nodes at the two ends of an element

were given equal and opposite prescribed velocities (v, ramping up from 0 tovmax ) in the

z -direction. The original length of each side of the element is unit length. The nominal

strain rate is, therefore, 2v, with its maximum value being approximately 2v max. This

analysis was run with maximum nominal strain rates of 2, 20, and 200 S-I and with a

properly prescribed time step to generate a uniaxial strain of approximately 3.0 in

magnitude. In addition, the heat generated by the plastic deformation was taken into

account, and the testing value of the kinematic hardening modulus, H, was chosen to be

100 (MPa).

60

1600

1400

liS Il.

1200 :i!: -1/) 1/) Q) 1000 ... -en -r::: 800 ~ cu .~ ::l 600 cr

W 1/) Q) 400 1/)

~ 200

0

0 0.5

-e- H=1 OO(MPa); strain rate 2

-&- H=1 00 (MPa); strain rate 20

-H=100 (MPa); strain rate 200

1.5 2

-I--Isotropie hardening; strain rate2

~ Isotropie hardening; strain rate 20

-B-Isotropie hardening; strain rate 200

2.5 3 3.5


Figure 3-9: S4SC. Mises stress versus equivalent plastic strain results from uniaxial

compression simulations with H = 100 (MPa) and isotropie hardening

Figure 3-8 contains plots of Mises stress versus equivalent plastic strain at different strain

rates from tension testing models with the combined hardening model with

H = 1 00 (MPa) and the isotropie hardening model. It can be observed that the results

with the combined hardening model are almost identical to the corresponding results with

the isotropie model. Figure 3-9 shows plots of Mises stress versus equivalent plastic strain

at different strain rates from compression simulations with the combined hardening model

with H = 100 (MPa) and the isotropie hardening model. Although sorne discrepancy is

noticed for the maximum nominal strain rate of 200 S-l and at strains greater than 2, this

does not affect the application of this combined hardening model in this work since the

nominal strain rate in this work is weIl below 200 S -) .

61

3.5.2 One-Element Tests under Reverse Loading Conditions

The reverse loading tests are performed to test the accuracy of the algorithm of the

VUMA T to account for the Bauschinger effect. The testing value of the kinematic

hardening modulus, H, is chosen to be 0 and 100 (MPa). When H equals zero, the

combined model should reduce to the Johnson-Cook model. In this testing model, the

nodes at one end of an element (C3D8R) were constrained from moving in the z -

direction, meanwhile, the nodes at the opposite end of the element were given a

prescribed velocity of 2 mfs (ramping up from 0 to 2 mfs) in the z -direction to stretch the

element. After plastic deformation, the nodes at one end of the element were still

constrained from moving in the z- direction, while the nodes at the opposite end were

given an equal and opposite prescribed velocity of 2 mfs in the z -direction to compress

the element. The originallength of each side of the element is unit length. This analysis

was run with maximum strain rates of 2 sec -1 , and the heat generated by the plastic

deformation was taken into account.

-+-Isotropic hardening; strain rate 2

-II- H= 0; strain rate 2 ··1000- ___ A (1.79, 1230)

-cu a.. ··~----500·

~ -·3 -2 -1 2

~-----~-5G0 .------

B (1.76, -1200) ----~~ --100& ---

Axial Plastic 5train

Figure 3-10: S4SC. Axial stress versus axial plastic strain results from reverse

loading testing models with H = 0 and isotropie hardening

62

Figure 3-10 shows the eurves of axial stress versus axial plastic strain of the reverse

loading testing models with H = 0 and the isotropie hardening. The two eurves are

essentially identieal, whieh indieates that when H equals zero, the eombined hardening

model reduces to the Johnson-Cook mode!. Yield stresses at points A (1.79, 1230) and B

(1.73, 1200) are 1230 (MPa) and -1200 (MPa), respeetively. The differences in

magnitude between them are eaused by the numerieal ea1culation error. Therefore, the

eombined hardening model is identieal to the Johnson-Cook isotropie hardening model

when H equals zero, as expeeted.

1/)

ë -cn-ni ·x cC

-+-- H=1 00 (MPa); strain rate 2

C (1.79,1230)

------- -5

-3 -2 -1 2

D (1.76, -840)

Axial Plastic Strain

Figure 3-11:S45C. Axial stress versus axial plastic strain results from reverse

loading testing models with H = 100 (MPa)

The axial plastic strain versus axial stress eurves of the reverse loading testing model with

H = 100(MPa) is illustrated in Figure 3-11. A strong Bausehinger effeet was manifested

in the eurve sinee the magnitude of the yield stress at point C (1.79, 1230) is about one

and an half times that at point D (1.76, -840). In addition, by substituting ea1culated

results from the reversed loading models for the eorresponding variables in equation

(3.7), the equation is satisfied. Therefore, the algorithm of the VUMAT is verified to be

correct.

63

4 Numerical Simulations

Numerical simulations in this work are performed using ABAQUS v6.4, which includes

ABAQUS/Standard and ABAQUSlExplicit. ABAQUS/Standard is a general-purpose

finite element pro gram using automatic incrementation based on the full Newton solution

method, whereas ABAQUSlExplicit is an explicit dynamic fini te element program based

on the implementation of an explicit integration mIe together with the use of diagonal or

lumped element mass matrices. The characteristics of implicit and explicit procedures

determine which method is appropriate for a given problem. For a smooth non-linear

response, the Newton's method can give a rapid quadratic rate of convergence. However,

if the model contains highly discontinuous processes, such as contact and friction al

sliding, quadratic convergence may be lost and a large number of iterations may be

required. Therefore, ABAQUSlExplicit is recommended to resolve complicated contact

problems (HKS Inc., 2004).

In this chapter, numerical simulations of physical tests from two papers in the literature,

used to investigate the effect of pre-drawing on the formability during cold heading, are

described. The two papers are Tozawa and Kojima's (1971) and Gill and Baldwin's

(1964) as reviewed in chapter 2.

4.1 Numerical Simulations of Tozawa and Kojima's Tests

4.1.1 Experimental Procedure

The process of Tozawa and Kojima's experimental work is illustrated in Figure 4-1.

[ DRA WING 1 .. [ CUTTING 1 .. 1 UPSETTING 1

Figure 4-1: The procedure of Tozawa and Kojima's test

After initial annealing, cylindrical rods were first drawn in the drawing process to

different reductions in area through dies with different angles to a fixed radius of 4 mm,

64

with lubricants applied. Then, in the cutting process, they were cut to 12 mm long with a

radius of 4 mm. Finally, in the upsetting process these pre-drawn rods 12 mm long, with a

radius of 4 mm, were compressed between two flat dies without lubrication until surface

fracture occurs. Both the drawing process and the upsetting process were quasi-static.

Three materials were tested, and two of them, S45C and Mn steel, are modeled in this

work. The chemical composition of the two materials is shown in Table 4-1. Table 4-2

shows the pre-drawing reductions in area and die angles (approach angle, 2a) used for the

two materials. The experimental results are shown in Figures 2-8 to 2-11.

C Mn Si P S

S45C 0.49 0.82 0.28 0.022 0.015

Mn steel 0.35 1.74 0.30 0.023 0.023

Table 4-1: Chemical composition of materials used (adapted from Tozawa and Kojima, 1971)

Materials Reductions in area Die approach angle 2a

S45C 10% 20% 40% 30°

Mn steel 10% 20% 40% 15° 1

30° 1

60°

Table 4-2: Pre-drawing reductions in area and die approach angle for S45C and Mn steel

4.1.2 Description of Simulation

According to the experimental procedure illustrated in Figure 4-1, three numerical models

were created: a drawing model, a cutting model and an upsetting model.

4.1.2.1 Description of the Drawing Mode)

The drawing process is a quasi-static process and involves contact and frictional sliding.

Rence, as recommended by ABAQUS, the numerical analysis of this process was

performed using ABAQUS/Explicit v6.4. The finite element model is axisymmetric, and

65

only half of the cylindrical rod and a die were considered because the geometry of the rod

and the die, and the loading conditions of the drawing process are axisymmetric. The

reductions in area and die approach angles used in this model (Figure 4-2) are the same as

shown in Table 4-2.

Geometry:

Figure 4-2 shows the schematics of the drawing model.

Figure 4-2: Geometry and mesh for FEM drawing mode}

With the different reduction in area, the original geometry of a cylindrical rod before pre

drawing corresponding to a final geometry of 12 mm long, with a radius of 4 mm, is

calculated as follows:

From equation (2.1), we have

66

AI 1CR} R} r=I--=I--=I--

Ao 1CR; R;

where Ra and Rf are the original and finishing radius of the rod, respectively

Then, we have

Rf' R = ' o ~

Since the material is assumed to be incompressible, we have

lo1CR; = If1CR~

R} 10 = If' - = If (1- r)

'R 2 , o

(4.1)

(4.2)

(4.3)

where final length, 1 f' equals to 12 mm; lais the original length corresponding to the

finishing length 12 mm. Table 4-3 shows the results calculated according to equation

(4.2) and (4.3)

Geometry Reduction Reduction Reduction 10% 20% 40%

10 (mm) 10.80 9.60 7.20

Ro(mm) 4.22 4.47 5.16

Table 4-3: Original geometry of the rod corresponding to finallength 12 mm

Geometry Reduction Reduction Reduction 10% 20% 40%

3Ia(mm) 32.40 28.80 21.60

Ra (mm) 4.22 4.47 5.16

Table 4-4: Original geometry of the rod for the FEM drawing model

To eliminate the end effect, the originallength of the rod in the drawing model was taken

to be three times of 10 calculated above. Then, in the cutting process, the two ends were

cut off, and only the middle part, 10 long, which is one-third of the whole rod length, is

67

considered for upsetting. Table 4-4 shows the original geometry of rods used in the

drawing model.

The actual geometry of the dies in the drawing test is unknown. The bearing length for aIl

dies was chosen to be 1.5 mm (National Machinery).

Mesh:

In the drawing model, the cylindrical rod was modeled as a deformable solid body while

the die was assumed to be rigid. In ABAQUSlExplicit, a rigid body can be modeled with

either an analytical rigid surface or a discrete rigid element. An analytical rigid surface is

more efficient than a discrete rigid element since it does not involve element calculation.

Therefore, in this model, an analytical rigid surface is used for modeling the die.

An axisymmetric solid element was used in the simulation to reduce the problem size.

The initial cylindrical rod was meshed using element type CAX4R, a 4-node quadrilateral

with reduced integration and hourglass control to control spurious mechanisms caused by

the fuIly reduced integration. The reason for choosing this type of element is that it is

relatively inexpensive for problems involving nonlinear constitutive behavior since the

material calculations are only done at one single point in each element; in addition, a first

order triangle element is overly stiff and exhibits slow convergence with mesh

refinements.

Various mesh refinement rates were tested for mesh convergence before final selection of

the mesh density. Mesh refinement rates were first tested for the upsetting model, and

then the mesh density for the drawing model was decided thereafter. In these preliminary

simulations for the upsetting model, compression of a rod, 12 mm long, with a radius of 4

mm, was performed between two flat dies. The mesh refinement rates were judged in

terms of the value of the principal stress on the surface of the rod at the end of the

simulation because the principal stress is a decisive factor in the Cockroft and Latham

ductile fracture criterion as indicated by equation (2.5). The accuracy of the FEM solution

68

improves as the mesh is refined further. However, the number of elements is limited by

the computational cost. At a certain point, the solution will become similar from one

refinement to the next further refinement, indicating that the mesh refinement at this point

can be taken as the final selection. Therefore, the mesh for the upsetting model was

selected to be 22x54 (1188). Accordingly, the rod for drawing was discretized into 2904

elements with finer mesh in the middle and coarser mesh at the ends to reduce the number

of elements and the computational time.

(a) (b)

Figure 4-3: The end shape of the eut rod (a) with adaptive mesh and (b) without adaptive mesh

In addition, to get better end shape of the cut rod after the cutting process, automatic

rezoning using adaptive meshing was applied to the region of the two-thirds of the rod in

the middle. Figure 4-3 shows the end shape of the cut rod with and without adaptive

meshing.

Material models:

As discussed in previous chapters, a material model should be able to adequately de scribe

the response of a material under specific conditions of interest. In this work, an elastic-

69

plastic model was assumed for both S45C and Mn-steel. The elastic behavior of the

material was assumed to be linear and isotropie; Young's modulus is 200 (GPa), and

Poisson's ratio is 0.29. The plastic behavior of the material was assumed to be described

by the new combined linear kinematic/ nonlinear isotropie hardening model proposed in

Chapter 3 since both materials were tested under reversed loading conditions. In addition,

for the purpose of comparison, the Johnson-Cook isotropic hardening model was also

applied for both materials to the drawing simulation model.

Sinee the drawing proeess is quasi-static, the temperature increase caused by the heat

generated by mechanical dissipation associated with plastic straining was negleeted, and

the temperature of the rod was assumed to remain at constant room temperature (25°).

The strain rate for the quasi-statie proeess was assumed to be 0.002 s -1 , then aeeording to

equation (3.1), Johnson-Cook isotropie hardening model beeomes:

ForS45C:

Cf = [A + B(e P r][1 + Cln(0.002)] = 387.34 + 784.54(e P )0.399

For Mn steel:

a = [A + B(e P r ][1 + C In(0.002)] = 433.43 + 733.29(e P )032

Renee, for the eombined kinematielisotropie model, the nonlinear isotropie hardening

becomes:

For S45C:

For Mn steel:

Therefore, both the isotropie hardening model and the combined kinematic/isotropic

model are simplified to rate-independent material models. The rate-independenee of the

70

materials eases the simulations of a quasi-static process since increasing speed does not

affect the material behavior.

In the current work, there is no cyc1ic test available to calibrate the kinematic hardening

modulus, H. Rence, H was determined by performing simulations of tests

corresponding to Figure 2-9 and Figure 2-10 with different values of H. The one, with

which the results of the simulations best fit those shown in Figure 2-9 and Figure 2-10,

was chosen as the value of H for S45C and Mn steel, respectively.

Interface behavior:

The contact between the lateral surface of the rod and the rigid die was modeled with the

*CONT ACT PAIR option, and the friction between them was assumed to obey

Coulomb's law, with a friction coefficient of 0.12 (Petrescu et al, 2002) and with no limit

for a shear stress. The interface was assumed to have no conductive properties, and the

die was assumed to be at constant room temperature. The heat dissipated as a result of

friction was neglected since the drawing process is a quasi-static process.

Boundary conditions:

Simulating quasi-static problems usmg ABAQUSlExplicit reqUlres sorne special

considerations. A static solution is defined as a long-term solution. It is usually

impossible to simulate a quasi-static process in its natural time sc ale as this would require

an excessive number of small time increments. To obtain a computationally economical

solution, the quasi-static process must be accelerated in sorne way, in which inertial

forces caused by the acceleration remain insignificant. A general rule for selecting a

properly increased speed for a quasi-static process is that the kinetic energy of the

deforming material should not exceed a small fraction (typically 5% to 10%) of its

internaI energy throughout most of the process (RKS Inc., 2004). In this work, different

drawing speeds were applied to drawing simulation models, and the drawing speed of 0.8

mis was found to be the appropriate speed for the drawing simulation. Figure 4-4 shows

71

an ex ample of the ratio of kinematic energy to internaI energy of the rod throughout the

whole drawing simulation process.

- 1.6E-03 ~ ~ >- 1.4E-03 ~ CI) s:::::

1.2E-03 CI)

cu s::::: 1.0E-03 "-CI) -s::::: 8.0E-04 :.:::: >-~ 6.0E-04 CI) s::::: CI)

() 4.0E-04 ~ cu E 2.0E-04 CI) s:::::

~ O.OE+OO

0 10 20 30 40 50 60 70

Time (ms)

Figure 4-4: History of ratio of kinematic energy to internai energy

The kinematie boundary condition is symmetric on the aXIS of the rod, having

ur = 0 described, and u z = 0 was prescribed on the bottom of the rod. The u z -

displacement of the rigid die was described using a displacement boundary condition

whose value was ramped up over step time to ensure the rod goes through the die totally

at the constant drawing speed of O.8m/s. The radial and rotation al degrees of freedom of

the rigid die were constrained.

4.1.2.2 Description of the Cutting Model

The cutting process was simulated using ABAQUS/Standard by importing the deformed

mesh and its associated material state of one-third of the whole drawn rod in the middle

from the final increment of the drawing simulation to this cutting simulation mode!. Then

the imported part underwent self-relaxation since no die or external loading is involved.

Therefore, the cutting model is actually a statie simulation without external loading or

contact. The reason for using ABAQUS/Standard in this model is that it can obtain a

72

static solution in just a few increments while ABAQUSlExplicit must solve a static

problem by obtaining a dynamic solution over a time period that is long enough for the

solution to reach a steady state; in addition, ABAQUS only provides the capability to

transfer results of simulations from ABAQUS/Standard into ABAQUSlExplicit or

ABAQUS/Standard and vice versa; ABAQUS cannot transfer results between

ABAQUSlExplicit (HKS Inc., 2004). Figure 4-5 shows the configuration of the drawn

rod at the final increment of the drawing simulation and the initial configuration of the

imported part of the rod in the cutting simulation.

o

(a) (b)

Figure 4-5: (a) Configuration of the drawn rod at the final increment of the drawing simulation and (b) The initial configuration of the rod in cutting model

It is important to notice that to import the material state from the drawing model

involving the Johnson-Cook isotropie hardening material model, the Johnson-Cook

hardening model has to be input into the drawing model in a discretized tabular format,

which both ABAQUSlExplicit analysis and ABAQUS/Standard analysis accept, since the

73

built-in Johnson-Cook hardening material model exists only in ABAQUSlExplicit's

materiallibrary, and not in ABAQUS/Standard's. In the case of the drawing model with

the Johnson-Cook hardening material model, no material definition need to be specified

in the cutting model as the material definition from the drawing model will be imported

as weB.

The initiaBy imported rod is approximately 12 mm long, with a radius of about 4 mm.

The imported mesh is the same type as in the drawing model. Since only elastic

springback was assumed to happen in this cutting process, the material model here was

assumed to be linear elastic for both materials with the same values of Young' s modulus

and Poisson' s ratio as in the drawing model. u = 0 was described on the axis of the rod

to ensure the symmetric kinematic boundary condition there, and the node at the bottom

of the axis of the rod was encastered to constrain the movement of the rod.

4.1.2.3 Description of the Upsetting Model

For a similar reason as the drawing model, the upsetting model was developed using

ABAQUSlExplicit. In this model, the deformed mesh and its associated material state

was imported from the final increment of the cutting simulation, and then was

compressed between two flat rigid dies. Figure 4-6 shows the initial configuration of this

model.

Figure 4-6: Initial configuration of the upsetting mode}

74

Geometry and model:

The imported initial rod is approximately 12 mm long, with a radius of about 4 mm; the

imported mesh type is CAX4R with 1188 elements in total. The material models defined

here are the same as in the drawing model since the upsetting process is also a quasi-static

process. However, when the drawing model is specified with the Johnson-Cook

hardening material model, no material definition needs to be specified, since the material

definition in the cutting model, which was imported from the drawing model, is imported

into this upsetting model. A Coulomb friction model with a friction coefficient of 0.13

(Nickoletopoulos, 2000) was used to model the friction between the top, bottom and

lateral surfaces of the rod and two flat rigid dies. The heat generated by the friction was

neglected since the process is quasi-static. The heat transfer between the dies and the rod

were also neglected, and the temperature of the dies and the rod were fixed to 25°C.

Boundary conditions:

The compression speed of this quasi-static process was chosen to be 0.5 mis according to

the procedure similar to the drawing model. The boundary condition is symmetric on the

axis of the rod i.e. u = o. The bottom die was encastered; the top die was constrained to

have no rotation and ur -displacement, and its Uz -displacement was prescribed using a

displacement boundary condition who se value was ramped up over time step until the

height of the deformed rod reached the height to fracture ensuring a constant compression

velocity of 0.5 mis. The heights to fracture (shown in Table 4-5) for the two materials

A -A were deterrnined according to Figure 2-7, w hich is the curves of 0 fr X 100% versus

Ao

ho - h fr -_.....::.- x 100%, where Ao and A ji- are the initial cross-sectional area and the crossho

sectional area at fracture of the rod, respectivel y; ho and h fr are the initial height and the

height at fracture of the rod, respectively.

75

Reduction in area 0% 10% 20% 40% Heights to fracture

3.81 3.38 3.03 3.47 (mm)

(a) For S45C

Height at fracture Height at fracture Height at fracture Reduction in area with 15° approach with 30° approach with 60° approach

angle (mm) angle (mm) angle (mm) 0% 3.00 3.00 3.00 10% 2.49 2.83 3.33 20% 2.31 2.63 2.80 40% 2.18 2.31 2.58

(b) For Mn steel

Table 4-5: Heights of compressed rods to fracture (a) for S45C and (b) for Mn steel

4.2 Numerical Simulations of Gill and Baldwin's Tests

4.2.1 Experimental Procedure

C Mn Si P S

AISI1335 0.33-0.38 1.60-1.90 0.20-0.35 0.035 max. 0.040 max.

Table 4-6: Chemical composition of AISI 1335 (adapted from EAD Inc., 1977)

Gill and Baldwin's Tests follow the same procedure as illustrated in Figure 4-1, and a

detailed description was reviewed in Chapter 2. In the drawing process, the wire of AISI

1335 was drawn at drawing speed of 0.76 mis, to reductions as high as 67% on poli shed

dies with 7°, 15° and 30° approach angles. In the subsequent cold heading process, the

pre-drawn wire was upset at a speed of 0.42 mis, without lubrication. The wire sizes for

cold heading are shown in Table 2-1, and the experimental results are shown in Figure 2-

5. The chemical composition of AISI 1335 is shown in Table 4-6.

76

4.2.2 Description of Simulations

According to the experimental procedure illustrated in Figure 4-1, Gill and Baldwin' s

tests were also modeled with three numerical models: a drawing model, a cutting model

and an upsetting model, which are aIl similar to the simulation models of Tozawa and

Kojima's tests respectively. Based on the assumption that the Cockroft and Latham

constant is a material property, which is independent of the geometry of the tested

specimens, the geometry of the initial wire in the simulations of Gill and Baldwin's tests

was chosen to be the same as in the simulations of Tozawa and Kojima's tests. The

aspects in the simulation models of Gill and Baldwin's tests, which are different from

those of Tozawa and Kojima's tests, will be described in the following.

In the drawing model, the process was simulated as an adiabatic-dynamic process since it

takes place in a short time of about 50ms. Both the Johnson-Cook isotropic and combined

kinematiclisotropic material models, which account for the effect of strain, strain rate and

temperature as presented in Chapter 3, were applied. Hence, the heat generated by the

plastic deformation was taken into account.

Since the chemical composition of AISI 1335 shown in Table 4-6 is similar to that of Mn

steel shown in Table 4-1, we make an approximation to assume that the values of the

corresponding material constants of the Johnson-Cook isotropic hardening model and the

combined isotropic/linear kinematic hardening model are the same for both materials.

Therefore, in this work, the values of the material constants for AISI 1335 were taken to

be the same as the corresponding material constants for Mn steel. The material properties

of Mn steel were summarized in Table 4-7.

The drawing die was prescribed a velocity of 0.76 mfs in the z-direction while the other

two degrees of freedom were constrained. The approach angles of the die used here are

7°, 15° and 30°, and drawing reductions in area are 20%, 40% and 60%.

77

The upsetting model was analyzed as an adiabatic-dynamic process since it takes place in

a short time of about 30 ms. The material models are the same as in the drawing model.

The top die was prescribed a constant velocity of 0.42 mis in the z-direction while the

other two degrees of freedom were constrained. According to the ratios of head diameter

(diameter of a rod to fracture after compression) to wire diameter (the initial diameter of

the rod at the beginning of the upsetting simulation) shown in Figure 2-5, the

corresponding heights of the compressed rods to fracture were ca1culated based on the

assumption that the volume of the rods does not change during plastic deformation.

Therefore, in the upsetting simulations, rods were compressed to the ca1culated heights

shown in Table 4-8.

Material properties Mn steel

Material density p ( % 3) 0.00787 mm

Young' modulus E (MPa) 200000 Poisson's ratio v (MPa) 0.29

A (MPa) 474.73 B (MPa) 803.17

n 0.32 C 0.014 m 1.03

Plastic heat fraction Tl 0.9

Specific heatC(T) (XgoC) 472

Melting temperature of the material Tme/! (OC) 1519.85

Reference temperature To (OC) 25

Table 4-7: Material properties of Mn steel

Approach Reduction in Reduction in Reduction in Reduction in angles area of 0% area of 20% area of 40% area of 60%

(mm) (mm) (mm) (mm) 70 2.47 2.19 2.09 2.64 15° 2.47 1.90 1.79 2.13 30° 2.47 2.41 2.69 3.34

Table 4-8: Corresponding beigbts at fracture from GiII's paper

78

5 Numerical Results and Discussion

Simulation models defined in Chapter 4 were created using both ABAQUSICAE v6.4, a

preprocessor to create models graphically, and a text editor to create ABAQUS input files

directly. The proposed combined linear kinematicl nonlinear isotropic hardening material

model used in the drawing and upsetting models, and the linear elastic material model

used in the cutting model were implemented with user subroutine VUMA T in

ABAQUSlExplicit v6.4 and with user subroutine UMAT in ABAQUSIStandard v6.4,

respectively. For the purpose of comparison, the Johnson-Cook isotropic hardening model

was incorporated in the simulation models in a tabular format as described earlier. The

only unknown to be determined was the kinematic hardening modulus, H, for both

materials .. Once it was determined, and simulations were performed and completed,

ABAQUS/Viewer v6.4 was used to evaluate the simulation results interactively.

5.1 Determination of the Kinematic Hardening Modulus, H

The kinematic hardening modulus, H, for materials S45C and Mn steel was determined

according to the average axial stress versus average axial strain curves in Figure 2-9 and

Figure 2-10, respectively.

5.1.1 Determination of the Kinematic Hardening Modulus, H , for S45C

To determine the kinematic hardening modulus, H, for the material S45C, simulations of

tests corresponding to Figure 2-9 were performed with different values of H. The three

tests corresponding to Figure 2-9 are: upsetting of a rod without being pre-drawn and

upsetting of rods which were pre-drawn to 20% and 40% reductions in area through a die

with an approach die angle of 30°. The detailed description of the physical processes and

the simulation modeling definitions are presented in Chapter 4.

79

As the first step, the upsetting of a rod pre-drawn to 20% reduetion in area was simulated

with the eombined hardening material model with H taken to be 200, 300, 400 and 600

(MPa) in order to find the value of H , with whieh the average axial stress versus average

axial strain eurves ealculated from the force versus displacement resuIts of the

simulations give the best fit to the eorresponding eurves in Figure 2-9. After the value of

H was determined, it was verified by the simulations of the other two processes. In

addition, the Johnson-Cook isotropie hardening model was also used to simulate the

processes to make a eomparison.

350

300

250

-Z 200 ~ -11) (.) ~

0 150 LI..

100

50

0

0

<> Isotropie Hardening

o H=200 (MPa)

l>. H=300 (MPa)

x H=400 (MPa)

Il H=600 (MPa)

2 3 4 5 6 7 8 9 10

Displacement (mm)

Figure 5-1: Force versus displacement curves for the simulations of upsetting after

20% pre-drawing by a 30° die for S45C

The resuIts of the simulations used to ealeulate the average axial stress versus average

axial strain eurves are the force versus displacement results. Figure 5-1 shows the force

versus displacement plot from the simulations of the upsetting after 20% pre-drawing. It

is evident that inereasing the value of H deereases the die reaetion force; the simulation

with the Johnson-Cook isotropie hardening model gives the largest force, while the

80

simulation with H = 600 (MPa) produces the lowest. In addition, for aIl the curves, there

is a long graduaI rise followed by a steep rise to the maximum load.

160

- 140 --- --------------- - ------- -------------

ni Il.. :E - 120 -------

1/) 1/) CI) "--CI)

"iü .;( 80 cC CI) C)

60 ----ni • Experiment from literature "-CI)

> ~ H=200 (MPa) cC 40 -- - -- ------------ ----- -. --------

o H=300 (MPa)

20 A H=400 (MPa) Il H=600 (MPa)

0 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6


Figure 5-2: Simulation and experimental average axial stress versus average axial

strain curves for upsetting after 20% pre-drawing by a 30° die for S45C

Based on the force versus displacement curves, average axial stress versus average axial

strain relations were calculated, which are defined as follows (Tozawa and Kojima,

1971):

Average axial stress = ~, where F is the force applied by the die; Ac is the CUITent Ac

cross-section al area of the rod.

According to the assumption of plastic incompressibility of metals, we have:

hcAc = hoAo, where Ao is the original cross-sectional area of the rod; he and ho are the

current and original heights of the rod.

81

· F Fhc Hence, average axIal stress = - = -- .

Ac hoAo

In addition, average axial strain = _ rhe dh = ln( ho ) . Jho h h

c

Figure 5-2 shows the plot eomparing the average axial stress versus average axial strain

results from the experiment in the literature and those ea1culated from the simulations. A

good agreement between the results from the experiment and from the simulation with

H equal to 300 (MPa) is observed. As expeeted, with the inereasing value of H, the

average axial stress is increased. Among the curves, the lowest average axial stress was

produced by the combined hardening mode with H equal to 600 (MPa), while the highest

was ealculated by the Johnson-Cook isotropie hardening model. Since H equal to 300

(MPa) gives the best fit between the result from the simulation and the experiment, 300

(MPa) was ehosen as the most feasible value for H . However, before the final decision is

made, this value has to be verified by the other two simulations.

Simulations for the other two tests were performed with the eombined hardening model

with H taken to be 200, 300, 600 (MPa), as weIl as the Johnson-Cook isotropie

hardening model. Figure 5-3 and 5-4 are force versus displacement plots from simulations

of the upsetting without pre-drawing and after pre-drawing of 40% reduetion in area,

respectively. The plots eomparing the average axial stress versus average axial strain

results from the experiments in the literature and those ealculated from the force versus

displaeement results (shown in Figures 5-3 and 5-4) are presented in Figures 5-5 and 5-6,

respeetively.

It ean be seen that as the value of H inereases, both force and average axial stress

inerease. Simulations with the isotropie hardening give the highest force and average

axial stress. The average axial stress versus average axial strain eurve for the simulation

with H equal to 200 (MPa) for 40% reduetion in area has a better agreement with that

from the literature than that for the simulation with H equal to 300 (MPa). But in general,

the simulation with H of 300 (MPa) has a better agreement than that with H equal to 200

82

(MPa). Hence, the kinematic hardening modulus, H , for S45C was taken to be 300 (MPa)

in this work.

350 x Isotropie hardening

300 o H=200 (MPa) ------------ --

Il H=300 (MPa)

250 <> H=600 (MPa) ------------------ - -------------- ------ ----

-Z 200 ----- - --------- ------------~ -Q) (,) ... 150 0

LI.

100 ~ -~ ~~ - -----

50

0 0 2 3 4 5 6 7 8 9

Displacement (mm)

Figure 5-3: Force versus displacement curves for the simulations of upsetting

without pre-drawing for S45C

350,-~------~---r-------~---------~-----------------------~ x Isotropie hardening

300 Il H=200 (MPa)

o H=300 (MPa)

250~â~H~==6~O~O~(~M~P~a)~--s-------------------------------~~_1

-~ 200 -Q)

~ o 150 +---~-~~-~~~~---~--------~~ LI.

100 r=i!:~-~-----~~~~~-~ 50!!!!

o 2 3 4 5 6 7

Displacement (mm)

8 9


40% pre-drawing by a 300 die for S45C

83

180r-----------------------·--------,--·------------------------, 160 Ci a.. 140 +-----:2 -1/) 120 1/) QI ... .... 100 -------------

CI)

CV '>( 80 <C QI 60 C) cv ... QI 40 >

<C 20

0

0 0.2 0,4 0,6 0,8 Average Axial Strain

<> Isotropie hardening

o H=200 (Mpa)

Â H=300 (MPa)

o H=600 (MPa)

• Experiment from literature

1,2 1,4


strain curves for upsetting without pre-drawing for S45C.

160

Ci "" a.. 140 :2

èi) 100 r--,.',000 00 Il. v

cv '>( 80 r::-

~ 60r:::~_~--~::--~~~~~~~~~~-Î t! x Isotropie hardening

<C~ 40 ~-~ CI H 200 (MPa) f' 6 H=300 (MPa)

20 f---._~- <> H=600 (MPa) o Experiment from literature

O __ ------,--------r------~------~~----~r_------~----~ o 0.2 0,4 0.6 0.8 1.2 1,4



strain curves for upsetting after 40% pre-drawing by a 30° die for S45C

84

5.1.2 Determination of the Kinematic Hardening Modulus, H, for Mn

steel

The kinematie hardening modulus, H, for Mn steel was determined aeeording to a

similar procedure as that for S45C. The simulations of three tests eorresponding to Figure

2-10 were performed with different values of H. The three tests are upsetting of rods

after being pre-drawn to 40% reduetion in area through a die with three different

approaeh die angles of 15°, 30° and 60°. AIl simulations were also performed with the

Johnson-Cook isotropie hardening model for the purpose of eomparison.

First, for the simulations of upsetting after 40% pre-drawing by a 30° die, H was taken to

be 150,200,300 and 470 (MPa). The force versus displacement plot from the simulations

is shown in Figure 5-7. The eurves have the same trend as the eurves in Figure 5-1. With

inereasing values of H, the force deereases; the eurve for the simulation with the

isotropie hardening model has the highest force. Aeeording to the eorresponding force

versus displacement results shown in Figure 5-7, the resulting average axial stress versus

average axial strain results were ealculated and eompared to those from the experiment in

Figure 5-8. The eurve with H equal to 200 (MPa) has the best agreement with the

experimental result from the literature. Therefore, it was taken and verified by the other

two simulations.

Figure 5-9 and Figure 5-10 show the force versus displacement plots from the other two

simulations of upsetting with 40% pre-drawing by 15°and 60° dies, respeetively. The

simulations were performed with H equal to 200, and 300 (MPa) as weIl as the isotropie

hardening model. Figure 5-11 and 5-12 present the plots eomparing the average axial

stress versus average axial strain eurves from the experiments in the literature with those

ealculated from the eorresponding force versus displacement results shown in Figures 5-9

and 5-10, respeetively. In general, as H inereases, both force and average axial stress

inerease regardless of the differenee of the approaeh die angles. Simulations with the

isotropie hardening model give the highest force and average axial stress. The average

axial stress versus average axial strain eurves for the simulations with H equal to 200

85

(MPa) have a good agreement with those from the experiments in both cases. Hence, the

kinematic hardening modulus, H, for Mn steel was taken to be 200 (MPa) in this work.

600

500

400 -Z ~ -CI) 300 (.) ... 0

LL

200

100

0 0

~ Isotropie hardening

o H=150 (MPa)

o H=200 (MPa)

x H=300 (MPa)

A H=470 (

2 4 6

Displacement (mm)

-------_.

~

8 10 12


40% pre-drawing by a 30° die for Mn steel

250,---------------------r-------------------------------~

ni 200 a.. :2 -

~ Isotropie hardening )( H=150 (MPa) o H=200 (MPa) A H=300 (MPa) o H=470 (MPa)

---~~~~-----~--~---~--~ -----~ ~~~-----

0 __ ----,-----~----_r----~----~------~----~----~----4 o 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8



strain curves for upsetting after 40% pre-drawing by a 30° die for Mn steel

86

600

500

400 -Z .:.:: -CI) 300 u ... 0 LL

200

100

0 0

<>. Isotropie hardening

A H=200 (MPa)

o H=300 (Mpa)

~----------- ~ ----------_.

2 4

<>. ~-----~~~~~~--- ~----<>.~----~--~~~

~~-~~-----~

6

Displacement (mm)

8 10 12



-Z .:.:: -CI) u ... 0

LL

600,----------y-------------------------------------------,

500

400

300

200

<>. Isotropie hardening

D H=200 (MPa)

o H=300 (MPa) ---.----- -------~--- ~~------i

~~--------~ ------~----------~~

6 7 8 9 10

Displacement (mm)



87

250~---------------------r------------

-~ 200 :!: -If) If)

o Isotropie hardening

o H=200 (MPa)

A H=300 (MPa)

-Experiment tram literature

~ 'Wl..~-.-.:-:--:-: ... ~-":.~--:-:--:-.-.. -~~.-~~--~--~ .. ::::::-:-:--:-:;-;-~-~.-;;;;;:: cr: 100 Q) C)

~ Q)

> 50 cr:

o 0.2 0.4 0.6 0.8 1.2


1.4 1.6 1.8



"iù 200 a.. :!: -

o

o Isotropie hardening

o H=200 (Mpa)

A H=300 (MPa)

-Experiment tram literature

0.2 0.4 0.6

--

.------- - ---_.- ----~._----_._._-~

0.8 1.2 1.4 1.6 1.8




88

5.1.3 Comments

The comparisons of the simulation and experimental average axial stress versus average

axial strain curves for S45C and Mn steel in sections 5.1.1 and 5.1.2 shows that

simulations with the Johnson-Cook isotropic hardening material model result in higher

average stresses than those from simulations with the combined hardening material

model; and with increasing values of H, the corresponding average stress decreases.

These observations are in accordance with the characteristics of the isotropie hardening

model and combined kinematiclisotropic hardening model reviewed in Chapter 2.

Eliminating the anisotropy of a material and the Bauschinger effect developed during

plastic pre-straining, the isotropic hardening model assumes that after pre-straining, the

yield surface expands uniformly in the stress space without changing shape and direction.

As a result, after pre-straining the material described with the isotropic hardening model

hardens in aIl directions with equal increased magnitude. Therefore, the increased

reaction force of the die in the subsequent upsetting is expected.

On the other hand, the combined hardening model takes the Bauschinger effect into

consideration, assuming that after pre-straining, the yield surface simultaneously

undergoes both translation and expansion during the hardening processes. Consequently,

after pre-straining the material hardens in the forward direction, while softening in the

opposite direction, and increasing the kinematic hardening modulus, H , will enhance this

phenomenon. Therefore, after being pre-drawn, the materials hardened in the pulling

direction and softened in the opposite compressive direction; and increasing the kinematie

hardening modulus, H, increases the Bauschinger effect leading to the decreased reaction

force of the die and the decreased average axial stress.

When H equals 300 and 200 (MPa) for S45C and Mn steel, respectively, the

corresponding average axial stress versus average axial strain curves show a good

agreement with those obtained from the experimental works in the literature. Therefore,

the kinematic hardening modulus, H, for S45C and Mn steel were determined to be 300

89

and 200 (MPa), respectively. In addition, the good agreement between the simulation and

experimental results can provide evidence to validate the numerical simulations and the

proposed combined hardening material model.

5.2 Results of Simulations of Tests in Tozawa and Kojima's

Paper

Tests by Tozawa and Kojima (1971) for both S45C and Mn steel were performed quasi

statically. In simulations of the tests, the effect of strain and strain rate was considered,

while the effect of temperature was ignored.

5.2.1 Results of Simulations for S45C

Simulations for S45C defined in section 4.1 were performed with the combined hardening

model with H equal to 300 and 600 (MPa) and the Johnson-Cook isotropie hardening

model.

5.2.1.1 Contour Plot Illustration

The contour plots of the equivalent plastic strain and the maximum principal stress of the

rod obtained from the simulations with H equal to 300 (MPa) for the process with pre

drawing of 20% reduction in area are shown in Figures 5-13 and 5-14, respectively.

It is observed from Figure 5-13 that at the end of the drawing stage, the distribution of the

equivalent plastic strain across the cross-section of the rod is inhomogeneous, being the

lowest in the center to the highest on the exterior, which is caused by the subsurface

redundant deformation; at the cutting stage, the equivalent plastic strain of the imported

part remains unchanged since the imported part undergoes elastic springback, and no

90

further plastic deformation occurs; at the end of upsetting, the highest plastic strain is

reached in the center of the deformed part.

3D\J5 (Av.,. C~it.: 75~)

+5.5n.,-01. +5.13-6.,-01. +4.700.,-01. +4.2-64.,-01. +3.827.,-01. +3.391..,-01. +2:.954.,-01. +2:. 51.8.,-01. +2:. 081..,-01. +1.. -645.,-01. +1..209.,-01. +7.722.,-02 +3.357.,-02

At the end of drawing

3D\J5 (Av.,. Cxit.: 75~)

+4.229.,-01. +4.0-68.,-01. +3.908.,-01. +3.747.,-01. +3.587.,-01. +3.42-6.,-01. +3.2-6-6.,-01 +3.1.05.,-01 +2.945.,-01. +2.784.,-01 +2.-624.,-01. +2.4-64.,-01. +2.303.,-01

At the end of cutting

3D\J5 (Av.,. exit.: 7S~)

+2.438.,+00 +2.275.,+00 +2.11.3.,+00 +1..951..,+00 +1.. 789.,+00 +1.-627.,+00 +1..4-64.,+00 +1..302 .. +00 +1..1.40 .. +00 +9.779 .. -01. +8.1.57 .. -01 +-6.535 .. -01.

- + 4.913 .. -01.

At the end of upsetting

Figure 5-13: Contour of the equivalent plastic strain of the rod obtained from the

simulations with H equal to 300 (MPa) for the process with 20% pre-drawing by a

30° die for S45C

From Figure 5-14, it can be seen that at the end of the drawing stage, the lowest

maximum principal stress is in the center with a negative value indicating that the stresses

there are compressive in aIl directions, while the maximum principal stress at the exterior

surface reaches the highest positive value; during the cutting stage, the distribution of the

residual stresses over the cross-section of the wire maintains a similar distribution trend

with little change in magnitude due to the removal of the two ends of the rod; at the end

of the upsetting stage, the maximum principal stress at the equatorial surface reaches the

91

highest positive value, while the interior obtains the lowest negative value, implying

compression in aIl directions.

The distribution trends of the equivalent plastic strain and maximum principal stress

observed at the drawing and upsetting stage are in accordance with the observations from

simulations of drawing and upsetting processes in literature.

3, K=. PI: inc ip .. l (Av ... Crit.: 75~)

+>5.788,,+02 +5.844,,+02 +4.899,,+02: +3.955,,+02 +3.01.1. .. +02 +2.0157,,+02 +1..1.23 .. +02 +1.. '185,,+01. -'1.155'1,,+01. -1.. '11.0,,+02 -2.>554,,+02 -3.598,,+02 -4.542,,+02:

At the end of drawing

3, K=. Princip .. 1 (Av ... Crit.: '15~)

+>5.1.00,,+02 +5.259,,+02 +4.41.9 .. +02 +3.5'18 .. +02 +2. '138,,+02 +1.. 897,,+02 +1.. 057,,+02 + 2.1.>5>5,,+01. -15.238,,+01. -1.. 4>54 .. +02 -2.305 .. +02 -3.1.45,,+02 -3.98>5,,+02

At the end of cutting

3, K=. Pr inc ip .. l (Av". Crit.: 75~)

+1.. 03'1,,+03 +8.4'14,,+02 +>5.582,,+02 +4.>591.,,+02 +2. '199,,+02 +9.073,,+01. -9.844,,+01. -2.87>5,,+02 -4.7>58 .. +02 ->5 .>559,,+02 -8.551.,,+02 -1.. 044,,+03

~ -1..233,,+03

Region of highest maximum principal stress

At the end of upsetting

Figure 5-14:Contour plot of maximum principal stress (MPa) of the rod obtained

from the simulations with H equal to 300 (MPa) for the process with 20% pre

drawing by a 30° die for S45C

92

5.2.1.2 Results for the Material Point with the Highest Principal Stress

on the Exterior Surface of the Upset Rod

In this work, the Cockroft and Latham criterion (2.5), which assumes that external cracks

result from large tensile circumferential stresses, was chosen to evaluate external cracking

during the upsetting of the rod. Therefore, the element with the highest principal stress at

the end of the upsetting stage is of interest, which is on the exterior of the compressed rod

as indicated in Figure 5-14. To find such a material point, elements on the ex te ri or of the

compressed rod at the end of upsetting stage, were queried for their principal stress

values.

As defined in Chapter 4, the simulation models are axisymmtric, and the boundary

condition on the nodes of the elements at the exterior surface of the rod is free during the

simulation process (drawing, cutting, and upsetting) except when the elements are passing

through the die during the drawing process when contact at the interface causes a non

zero boundary condition on the surface. Therefore, the shear ( a rz) and radial (0 rr) stress

components at the integration points of the elements at the exterior surface should be non

zero only during the time when the elements are contacting with the die. The history of

stress components of the elements with highest principal stress from the simulations with

the combined hardening model with H equal to 300 (MPa), and the isotropic hardening

model are shown in Figures 5-15 and 5-17, respectively. The history of shear (orJ and

radial (0 rr) stress components in both figures shows that the simulation results are in

accordance with the expected behavior. Figure 5-16 and 5-18 presents the history of hoop

(eto) and axial (e~) plastic strain components of the elements with highest principal

stress from the simulations with the combined hardening model with H equal to 300

(MPa), and the isotropie hardening model, respectively.

It can be seen from Figures 5-15 to 5-18, that the drawing process involves five sub

stages. During sub-stage 1, the element approaches the drawing die, and only elastic

de formation occurs at the material point. At the beginning of sub-stage 2, the element

93

cornes into contact with the die, and separates from the die in the end. This is the most

important stage since plastic deformation during drawing takes place at this stage. AIl the

stress components were non-zero, and ho op and axial plastic strain components show a

negative and positive value, respectively. These simulation results are consistent with the

physical process since at this moment, the element is being compressed in the radial

direction and stretched in the axial direction. At the beginning of sub-stage 3, the shear

( a rz) and radial (a rr) stress components became zero since the element is through the

die. The plastic strain components remain unchanged implying no further plastic

deformation occurs. In addition, both the hoop stress caused by residual stress and the

axial stress caused by the drawing force remain stable. At sub-stage 4, as the end of the

rod approaches the die, the axial stress decreases since less drawing force is required, and

other components of stress and strain are unchanged. At sub-stage 5, the rod separates

from the die and undergoes elastic relaxation without further plastic deformation.

The cutting process 1S actuaIly a continuation of the sub-stage 5, and only elastic

relaxation occurs here. In the following upsetting process, the rod is compressed axially,

and consequently expands circumferentially. As a result, the axial plastic strain changes

from a positive value to a negative value, while the hoop plastic strain changes from a

negative value to a positive value, clearly indicating that reverse plastic deformation

occurs in both axial and circumferential directions. Therefore, the Bauschinger efJect

must be considered.

Figures 5-19 and 5-20 are plots of the history of the maximum principal stress, hoop

stress and axial stress of the elements with the highest principal stress from simulations

with the combined hardening model with H equal to 300 (MPa) and isotropie hardening

model, respectively. It is evident that during the drawing process except the sub-stage 2,

the maximum principal stress is the axial stress, while during the upsetting process it is

the hoop stress. Therefore, the principal stress changes ifs direction from the axial

direction to the circumferential direction during the en tire process (drawing, cutting and

upsetting).

94

1500 _ Shear Stress (MPa) A· Radial Stress (MPa) - Axial Stress (MPa) - Hoop Stress (Mpa)

1000

500 --

l 2 3 ·····--~-~iiiiiiiiiiii~

1 i 1 . . .. --------T- /.J;.;;i. ......... - .-·iiiiiiiiiiiiiiiiliiiiiiiiiiiijiiiiiiiiiiiiM ~---_ ....... -._.

~ 0 "'-__ IIII1"IIII--p'IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII~IIIIIIIIIIII"IIIIIIIIIIII"jIII .1IIIIIIIIIII_1IIIIIIIIIII1IIIIIIIIIII .. ~---1 ~ 10 40 1 50i t/) 1 1 ~ û) -500----L -- t

1 i 1 1 1 ---L-I i 1 1 1

1 ________ -··-_·-_t_-___ ~I !<II-t--.. --.-.-- UpsettiJ:lg

":.~

-1000

-1500 Drawing

-2000 -L-_______________ _ Cutting

Time(ms)

Figure 5-15: History of hoop (0 88 ), axial (0 zJ, shear (0 rJ and radial stress (0 rr)

components of the element with the highest principal stress from the simulation with

H equal to 300 (MPa)

--Axial Plastic Strain

-*-Hoop Plastic Strain 0.6

0.4 l 2

t: 0.2 'Iii ... ..... CI) 0

10 20 i

-0.2 . _________ ---.L __ J 1 1 1 1

-0.4

••.. p:rawing -0.6.l......-----

3 4 1 .5 +

1

Time (ms)

80

Figure 5-16: History of hoop (e:8 ) and axial (e~) plastic strain components of the

element with the highest principal stress from the simulation with H equal to 300 (MPa)

95

Cil Il.

~ ln ln I!! .....

CI)

1500 _ Hoop Stress (MPa) _ Axial Stress (Mpa) -1>- Radial Stress (MPa) - Shear Stress (Mpa)

1000 1

500 --_. __ ._-----

0

10

-500

-1000

-1500 -Drawing

2

3

i 30 40 1 1 1

.. -tl-1 1 1 1 1 ········1 i i 1 1

SOi 1 1 i 1 (

1 1

5 Il ii-· ii

1------- 1 , ----II --------------------------~.II~~~-----

Cutting -2000

L-________________________ . _________ __

Time (ms)

Upsetting

Figure 5-17: Time history of hoop (0 ee), axial (0 zJ, shear (0 rz) and radial stress

(0 rr) eomponents of the element with the highest principal stress from the

simulation with the isotropie hardening

0.8

0.6

0.4

r::: 0.2 ïi ... ... U) 0

-0.2

-0.4

-0.6

~~~~~~~~--------~.---------------------------------, --*-Axial Plastic Strain

- Hoop Plastic Strain

1 2 3 5 1 --_._----+---------1

10

Drawing

--IT-----

i 1 i i 1 • ••

-- ---------11-Il

--------I.~i ~ Upsetting

Time (ms) Cutting

80

Figure 5-18: History of hoop (êto) and axial (ê~) plastic strain eomponents of the

element with the highest principal stress from the simulation with isotropie

hardening

96

li' a. ~ VI VI ~ û)

1500

1000

500

0

-500

~ Max. Principal Stress (MPa) - Axial Stress (Mpa) - Hoop Stress (MPa)

1 ,2 1 3 ! 4 ... -------I-p· 1Blllll1IIIIIiIIIII.-.-•. ~~--.

1 1 1

Il ir----·

10

Il

------ -i --- -r.L;;.liiiijiiiijiiililii·-.----.-.--.· -jiiiijiiil-· ifiiijiiiijiiiilijiil" 1

30 40

1 Il 1 Il 1 il

50i i~ 1 Il

..- ... \. Il

1 i i ~ ..... ""..,.

-1000 ---- : 1: : ~ i i 1 Il

-1500 . -- Drawing

1 Il Iii Il . I~--- 1

-------------I.~! !<III!.I---- Upsetting

Cutting -2000 .1....-____________________ .....•............... _ ..•........ --------'

Time (ms)

Figure 5-19: History of the maximum principal stress, hoop stress and axial stress of the element with the highest principal stress from the simulations with H equal to

300 (MPa)

1500 --+- Maximum Principal Stress (MPa) - Hoop Stress (MPa) - Axial Stress (MPa)

1000 1 Il

·fIfS---_IIk--j---I-t---- -1 Il

500

li' 0 a.

~ 10 VI VI ~ -500 -fi)

-1000

, " 40 50! !~

1 1 Il ·-·~--~-----ti i .: ii

3 i 4 5 ii*--...... -----i--- j- ·-ii----

i i ii i i ii

-1500 -- Drawing --'----ii------- - -- Upsetting -----t.~, :..1 •• 1----

-2000 2 Cutting ---------' Time (ms)

Figure 5-20: History of the maximum principal stress, hoop stress and axial stress of the element with the highest principal stress from the simulations with isotropie

hardening

97

Figures 5-21 to 5-23 present plots of history of the maximum principal stress for the

elements with the highest principal stress from simulations with the isotropie hardening

and the combined hardening model with H equal to 300 and 600 (MPa) for pre-drawing

of 10%, 20% and 40% reductions in area, respectively. It is clear that in general, the

isotropie hardening model gives the highest maximum principle stress, while H equal to

600 (MPa) generates the lowest maximum principal stress, implying that increasing H

decreases the corresponding maximum principal stress regardless of different reductions

in area.

Figure 5-24 shows the history of equivalent plastic strain of the elements with the highest

maximum principal stress for simulations with H equal to 300 (MPa). lt is observed that

after the drawing, the element obtained the largest equivalent plastic strain for simulation

with 40% reduction in area, while the element for simulation with 20% reduction in area

had the smallest equivalent plastic strain. The order in terms of the magnitude of the

equivalent plastic strain maintained throughout upsetting stage. Therefore, increasing the

reduction in area increases the equivalent plastic strain of the element with the highest

principal stress.

Figure 5-25 illustrates the history of maximum principal stress from simulations with

H equal to 300 (MPa). It can be seen that at the end of the drawing stage, the maximum

principal stress decreases with increasing reduction in area, and the maximum principal

stress for 40% reduction in area has the lowest value at the upsetting stage.

Figures 5-26 to 5-28 show the comparison of maximum principal stress versus equivalent

plastic strain results among the simulations with the combined hardening model with

H of 300 and 600 (MPa), and the isotropic hardening model. In general, increasing the

value of H decreases the maximum principal stress, and the simulations with the

isotropie hardening mode! have the highest maximum principal stresses when the

corresponding reductions in area are the same.

98

1500r-------------------------------------------------------,

ni Il.. 1000 :! - ------------------- Gi----=---==----5~ ln ln ~ 500 -CI)

ni .g- 0 ... ___ --..

c 10 20 1: Il.. E -500 :::J E ·x ~ -1000 -

30 40

---Drawing

50 i i 60

Il

70 80

-------.ji---- ----- Upsettîng ----I--,..~ . ........ t----Il

_________ Cutting -Isotropie hardening

-El- H=300 (MPa)

- H=600 (MPa) -1500L----------------------------------------~====~~==~==~

Time (ms)

Figure 5-21: History of maximum principal stress for simulations with 10% pre

drawing

- 1500 -.--.------------------ Cutting cu a.. ~ - 1000 tn tn ~ -tn

cu Q.

o C .~

Q.

E ::::s

500

10 30

Drawing --~~! ~-Upsetting Il

,---------- --it---- ----------.,AJ"'------

Il

-- ----~----~

40 50 70 80

E -500 -l----------------' ..... ---------------------------+-t--l-Isotropie hardening .~ 1 1 __ H=300 (MPa)

~ -~OO( -1 000 L __________________ .....::::::::::::::::::::::::::~::::;;:::::::::::::r

Time (ms)


drawing

99

1500~--~~~~~~~----------------~·(=uttin~-------------·~ -Isotropic hardening

_ --H=300 (MPa) ft!

-Drâ~itig ----I~ .. !·~Ûp~~Ùing Il - .

Il. - H=600 (MPa) ~ 1000 +------ -._----------- ._-'II-------'-=---.,.,.

ii ~--- --++---- .-.- --------1-----.,-----

1/) 1/)

e ... 1/)

Cü .~ 500 c .;: c.. E

i i Il

Il

:J E O .... ~~--~~~---~----~---~~-~----~--~ .;< ft! ~Q 70 50 80 40

::E Il

-500~----------------------------------------------------~

Time (ms)


drawing

1.6,-----------,--- .-----------------.-----, --*- Reduction 10%

1.4 - Reduction 20% --

- Reduction 40% .!: ~ 1.2 ...

CIJ C.) 1 -;; 1/)

.!!! Il. 0.8 Drawing ... c -! 0.6 +----

.2: g. 0.4 W

0.2+-----

o 10 20 30

------------ - -- -----

--·~y:tting ____ ··::i::,;~i)'

1 1 ~i i ----'-+. - . ----.iIIP'----c

1 1 1 1

1 1 j--j. 1 1

40 50 60 70

Time (ms)

'.

Upsetting

80 90

Figure 5-24: History of equivalent plastic strain for simulations with H equal to 300

(MPa)

100

, .. ./... CpttiIlg Drawing--+! ~Upsetting

1500 ,..-------'

Ci a.. 1000 ,..--.... .----t1------ . :li! -

30 40 50

-1000-- - -

Time (ms)

Il

tll !!

70 80

--------I-t-- -- .- -----------

Il

-10% reduetion in area

- 20% reduetion in area

- 40% reduetion in area

Figure 5-25: History of maximum principal stress for simulations with H of 300

(MPa).

1500,-------------------------------------------------------,

-~ 1000 :li! -

---------------j,. ... -.,-"--------- -

0.8 1.0 1.2 1 4

-----------------1

-Isotropie hardening

-&- H=300 (MPa)

~ H=600 (MPa) -1500 L ____________________ -=====~;;;;;;.;;;;=="


Figure 5-26: Maximum principal stress versus equivalent plastic strain for

simulations with 10% pre-drawing

101

1500

li Il. 1000 :!E -1/1 1/1 G) ~ ...

500 en ni c. 'u s:: ';: 0 Il.

E o. ~

E ')( -500 ni

:!E

-1000

-_ .. _-----.---

0.60 0.80 1.00 1.20 1. 0

.... -Isotropie hardening

-e- H=300 (MPa)

- H=600 (MPa)




1400

1200 li Il. 1000 :!E -1/1 800 1/1 G) ~ ...

600 en cu c. 400 'u s:: ';: 200 Il.

E 0 ~

E ')( -200 ni

:!E -400

-600

..... -----II-.J_

0.8 1.2 ,--_L4.. __ ..l...b~_~


-Isotropie hardening

... -e- H=300 (MPa)

-iIE- H=600 (MPa)



102

5.2.1.3 Calculations of Reduction in Height from Simulation Results

Before determining reduction in height from simulation results, which is defined as

h -h o Ir xl 00% , where ho and h 'r are the original height and the height at fracture of the

h J' o

compressed rod, the Cockroft and Latham constant was first determined numerically.

Then this value was used as a benchmark to predict the initiation of the fracture of the

rod. Finally, using the heights to fracture, the reductions in height of the compressed rods

were calculated.

1200

cv Q.. 1000 ~ 1/1 1/1 CI) 800 ----... ....

Cf)

ni Cl. 600 u c 'C Q..

400 E :::l

E 'x ni

200

~

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Equivalent Plastic 5train

Figure 5-29: Maximum principal stress versus equivalent plastic strain curve of the

element with the highest maximum principal stress for the simulation with the

combined hardening model with H of 300 (MPa) without pre-draw

In this work, the Cockroft and Latham constant for S45C was determined from the results

of the simulation for upsetting without pre-draw on the rod. At the end of the upsetting

simulation, elements on the exterior surface of the rod were queried for their principal

stress values. The maximum principal stress versus equivalent plastic strain result for the

element with the highest maximum principal stress was extracted and shown in Figure 5-

103

29 for the simulation with the combined hardening model with H of 300 (MPa) without

pre-draw. The value of the Cockroft and Latham constant, which is the area between the

curve and the x-axis, was computed from equation (2.4) to be 313.99 (MPa). The values

of the constant were also ca1culated in the similar way for the combined hardening model

with H of 600 (MPa) and the isotropie hardening model to be 314.32 (MPa) and 310.07

(MPa), respectively.

Similarly, the accumulated Cockroft and Latham parameter "C2 " in (2.5) from the

beginning of the drawing stage to the end of the upsetting stage were ca1culated. During

the ca1culations, the negative areas between the maximum principal stress versus

equivalent plastic strain curve and the x-axis were not included since compressive

maximum principal stresses do not contribute to the ductile fracture. The evolution of the

accumulated "C2 " for simulations with the combined hardening model with H of 300

and 600 (MPa), and the isotropic hardening model are plotted in Figures 5-30 to 5-32,

respectively.

700 Cil """*- 10% reduction in area D.. ~ 600 -El- 20% reduction in area f------------------~I ... CI) -A- 40% reduction in area ... CI)

500 E e cv

D.. 400 E cv oC

300 ... cv

...J "C c 200 cv ~ 0 ... 100 ~ (,,)

0 0

0 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6


Figure 5-30: Evolution of the accumulated Cockroft and Latham parameter for

simulations with the combined hardening model with H equal to 300 (MPa)

104

700r--------------------,---------------------------------------. ~ 10% reduction in area Ci

n. ~ 600 ......... 20% reduction in area

; -e- 40% reduction -CI) 500 E f! ca n. 400 E ca :5 300 ca ...J '0 ~ 200

::: e ..lIC: 100 o o (J

o 0.2 0.4

-- --------- -------------- ----

0.6 0.8 1.2 1.4


Figure 5-31: Evolution of the aeeumulated Coekroft and Latham parameter for

simulations with the eombined hardening model with H equal to 600 (MPa)

1.6

700,-----------r-----~--------------------------------_, Ci ~ 10% reduction in area n. ~ 600 -- 20% reduction in area "CI) - -e-40% reduction

CI) 500 E f! ca n. E ca .c -ca ...J '0

400 -------

300

~ 200 --

~ ..lIC: o o (J

100 _._-----~--------:::------o 0.2 0.4 0.6 0.8


1.2 1.4

Figure 5-32: Evolution of the aeeumulated Coekroft and Latham Parameter for

simulations with isotropie hardening model

1.6

105

The predicted heights to facture and the corresponding reductions in height of the

compressed rods corresponding to the accumulated Cockroft and Latharn parameters

which are equal to the Cockroft and Latham constants calculated previously, are shown in

Tables 5-1 to 5-3 for the simulations the with the combined hardening model with H of

300 and 600 (MPa), and the isotropic hardening model, respectively. According to Figure

2-7 and Table 4-5 (a), the heights to fracture and the reductions in height from the

experiments in the literature are obtained and shown in Table 5-4. The comparisons of the

heights to fracture and the reductions in height between the simulation and experimental

results are shown in Table 5-5 and 5-6.

Without pre- 10% pre- 20% pre- 40% pre-drawing drawing drawing drawing

Predicted heights 3.81 4.33 4.45 5.07 to fracture (mm) Reductions III 68 64 63 58 height (%)

Table 5-1: Predicted heights to fracture and reductions in height for simulations

with H of 300 (MPa)


Predicted heights 3.81 3.56 3.62 3.03 to fracture (mm) Reductions III 68 70 70 75 height (%)


with H of 600 (MPa)


Predicted heights 3.81 5.14 5.40 6.19 to fracture (mm) Reductions in 68 57 55 48 height (%)


with isotropic hardening model

106


Experimental heights 3.81 3.38 3.03 3.47 to fracture (mm)

Reductions in height 68 (%)

72 75 71

Table 5-4: Heights to fracture and reductions in height from the experiments in the literature

10% pre-drawing 20% pre-drawing 40% pre-drawing Predicted Experimental Predicted Experimental Predicted Experimental

(mm) (mm) (mm) (mm) (mm) (mm) H=300

4.33 4.45 5.07 (MPa) H=600

3.56 3.38 3.62 3.03 3.03 3.47 (MPa)

Isotropic 5.14 5.40 6.19 hardening

Table 5-5: Comparisons of heights to fracture between the simulation and

experimental results


H=300 64% 63% 58% (MPa)

H=600 70% 72% 70% 75% 75% 71% (MPa)

Isotropic 57% 55% 48% hardening

Table 5-6: Comparisons of reductions ID height between the simulation and

experimental results

The differences of reduction in height between the predicted values from the simulations

and the corresponding values from literature, which is defined as

literature - predicted . . . -----"'-----xlOO%, are shown III Table 5-7. It IS observed that dlfferences for

literature

simulations with the combined hardening mode! with H of 300 (MPa) are in the range of

Il % to 19%, and for simulations with the isotropic hardening model, the y are in the range

of 21 % to 32%, which indicates that the combined hardening mate rial model is more

107

aeeurate in predieting material behavior in this reverse loading proeess (drawing and

cutting followed by upsetting) than the isotropie hardening model. In addition, the

differences between the experimental results and those from the simulations with the

combined hardening model with H of 600 (MPa) are in the range of 2% to 7%, implying

that the eombined hardening material model has the potential to eorreetly prediet the

material behavior in the reverse loading proeess.

10% pre-drawing 20% pre-drawing 40% pre-drawing

H= 300 (MPa) 11% 16% 19%

H= 600 (MPa) 2% 7% 5%

Isotropic hardening 21% 26% 32%

Table 5-7: Differences of reduction in height

As indicated in Figures 5-16 and 5-18, reverse plastic deformation definitely occurred in

both axial and circumferential direction. Therefore, due to the Bauschinger effect in this

mutiaxial state of stress, the combined hardening model generally predicts lower

maximum principal stress for the element with highest maximum principal stress than the

isotropic hardening model does; and also increasing the kinematic hardening modulus H ,

the predicted maximum principal stress decreases, as illustrated in Figures 5-21 and 5-23.

The lower maximum principal stress leads to a smaller accumulated Cockroft and Latham

parameter for the same amount of plastic deformation. Hence, when the accumulated

Cockroft and Latham parameter reaches the value of the Cockroft and Latham constant,

the reduced height of the compressed rod for the simulation with the combined hardening

model is larger than that for the simulation with the isotropic hardening mode!.

Accordingly the predicted height to fracture for the simulation with the combined

hardening model is smaller than its counterpart for the simulation with the isotropic

hardening model as shown in Table 5-5; moreover, the reduction in height for the

simulation with the combined model is larger than its counterpart for the simulation with

the isotropic hardening model, as shown in Table 5-6.

However, although the combined hardening model is much better than the isotropic

hardening model in predicting surface crack during upsetting after pre-plastic-straining,

108

there are still Il % to 19% discrepancies of reduction in height between the predicted

results from the simulations with the combined hardening model with H of 300 (MPa) and

the results from the experiments in the literature. As reviewed in Chapter 1 and Chapter 2,

the forming process is limited by a complicated interplay of many factors, among them

the possible factors that may contribute to the discrepancies are discussed as follows:

• Material property factor

In this work, the material parameters of the Johnson-Cook material model, A, Band

n, were determined from the average axial stress versus average axial plastic strain

results of inhomogeneous compression test, which should be different from those

determined from homogeneous compression test. In addition, material constants, C

and m, obtained from the comparable steels in the literature may be also different

from the values determined experimentally.

• Friction factors

Since the actual friction conditions are unknown, friction factors for both drawing and

upsetting process are taken from the literature. Possible differences between the

values used in the simulations and actual values may exist.

• Cockroft and Latham criterion

The successful applications of the Cockroft and Latham criterion in the literature are

on monotonic forming processes such as tension, compression, extrusion and

drawing. Throughout the se processes, the direction of the maximum principal stress at

the potential fracture material point is unchanged. However, in this work, the

direction of the maximum principal stress of the element with highest maximum

principal stress on the surface of the rod was changed from the axial direction during

drawing to the circumferential direction during upsetting, as illustrated in Figures 5-

19 and 5-20. This may result in the deviation of the accumulated Cockroft and

Latham parameter.

• Material model

The proposed combined hardening material model is linear kinematic and nonlinear

isotropie. Although the linear kinematic hardening model is simpler than the nonlinear

kinematic hardening model, it is "stiffer" than the nonlinear kinematic hardening

109

model. The "stiffness" of the linear kinematic hardening model can be a factor

contributing to the discrepancies in this work.

5.2.2 Results of Simulations for Mn Steel

Simulations for Mn steel are defined as in section 4.1 and are performed with the

combined hardening model with H equal to 200 (MPa) and the Johnson-Cook isotropic

hardening model. The simulations with pre-drawing by a 30° die are also performed with

the combined hardening model with H equal to 470 MPa to see the effect of increasing

H.

5.2.2.1 Results for the Material Point with the Highest Principal Stress

on the Exterior Surface of the Upset Rod

The history of equivalent plastic strain plots comparing simulations results among

different approach die angles for the simulations with the combined hardening model with

H of 200 (MPa) are shown in Figures 5-33 to 5-35 for the cases with pre-drawing of

10%, 20% and 40% reductions in area, respectively. It is evident that for fixed reduction

in area, increasing approach die angle increases the equivalent plastic strain; moreover,

as the reduction in area increases, the difJerences of the equivalent plastic strains

between difJerent approach die angles for the fixed reduction in area diminish.

The maximum principal stress versus equivalent plastic strain plots comparmg

simulations results among different approach die angles for the simulations with the

combined hardening model with H of 200 (MPa) are shown in Figures 5-36 to 5-38 for

the cases with pre-drawing of 10%, 20% and 40% reductions in area, respectively. These

curves were used to calculate the accumulated Cockroft and Latham parameter. It is

observed the maximum principal stress does not change significantly with the different

approach die angles in terms of the maximum magnitude.

110

2.5 .,-------r------------ClJfting -&-60 degree l' ••..

--*- 30 degree Drawing -----t~~! !.--llpS~~t~rig 2 -+-15dgree ~-------------------~--------_.~-----_4

c 1,~ èi) 1.5 o ~ ln CV il: ... c Q)

cv ,~ 0.5-::J 0" W

10 20 30 40

l' --------W--l' ,1 l' ,1 l' -----'-!

50

-0.5 -1....... ______________ _

Time (ms)

60 70 80

Figure 5-33: History of equivalent plastic strain for simulations with pre-drawing of

10% reduction in area with the combined hardening model with H of 200 (MPa)

2.5 ~--------L _____________ • ____________ ___,

-fIII- 60 degree

~30degree

,!: 2 -a- 15 degree E -(J)

o :; 1.5 ln ~ Q. -c ~ cv ,~ ::J 0" W 0.5

o 10 20

Cutting ___ ... ~j ~ Upsetting .. ' Drawing .... 1.....-----

1 1 1 1 1 --j

-l1li-------- - ----,

30 40 50

Time (ms)

60 70 80 90


20% reduction in area with the combined hardening model with H of 200 (MPa )

111

2.5.-----------r----------------------------------------------~ --60 degree Cutting - 30 degree Il

"= f! 2 --15 rlon,yoo r-----~ Drawing .! I~ Upsetting

..... en ~ 1.5 -1--... ----.-.. -... ---.. -1/) CU ii: ..... c: CI) +-_. __ .. _ ...... _ .. _------_ .. __ ..

1 1 .... -~----.-. r--·-----·---· 1 1

ni "~ ::l C" W 0.5 -1--.-_.--.-.. ---&;=._. __ ._=----=._.-.;;.. . ._ ... _._._ .. _ .. __ .

0 ____ •

o 10 20 30 40 50

Time (ms)

60

" " 70 80 90 100


40% reduction in area with the combined hardening model with H of 200 (MPa)

1500

êi 1000 Q.

:iE -1/) 1/)

500 CI) ... ..... en ni Cl. 0 "0 c: ";: Q.

E -500 ::l E "x CU -1000 :iE

-1500

1.5 2 25 ---+---_. __ ... _ .. _-----------_ .. _ ... _-.. _-------_ ..


____ -e-15 degree

--*- 30 degree

-I:s- 60 degree

Figure 5-36: Maximum principal stress versus equivalent plastic strain results for

simulations with pre-drawing of 10% reduction in area with the combined

hardening model with H of 200 (MPa)

112

1500

li 1000 Q..

:E -1/) 1/)

500 f .. CI)

ii C- O "u t: ï: Q..

E -500 ::l

E ">< ni -1000 :E

-1500

1.2 1.4 1.6 1.8

_____________________ j -e- 15 degree

""""*- 30 degree -b- 60 rio,nro,o





li Q.. :E 1000 -t----------------------h ... ----lIf----

500 1IIi_---------

1.2


1.4 1.6 1 8

-e- 15 degree

""""*- 30 degree -b- 60 rio,nro,o




113

5.2.2.2 Calculations of Reductions in Height from Simulation Results

Following the sarne procedure as used in section 5.2.1.3, the Cockroft and Latham

constants, the accumulated Cockroft and Latham parameters, the predicted heights of the

rods to fracture, the reductions in height and the differences between the predieted

reduction in height and the reduction in height from the experiments in the literature were

calculated. In addition, the heights to fracture for the experiments in the literature are

presented in Tables 4-5 (b), and the corresponding reductions in height is obtained

according to Figure 2-7 and is shown in Table 5-8.

Without pre- 10% reduction 20% reduction 40% reduction drawing

15° approach angle 75% 79% 81% 82% 30° approach angle 75% 76% 78% 81% 60° approach angle 75% 72% 77% 78%

Table 5-8: Reductions in height for the experiments in the literature

Table 5-9 shows the calculated Cockroft and Latham constants from simulations without

pre-drawing. Tables 5-10 to 5-12 present the comparison of the predicted and

experimental heights to fracture for the processes with 15°, 30° and 60° approach angles,

respectively. The comparisons of the corresponding reductions in height between the

predicted and experimental results are shown in Tables 5-13 to 5-15. Finally, the

differences between the predieted and experimental reductions in height are shown in

Tables 5-16 to 5-18.

It is evident that results from simulations with the combined hardening material model are

much closer to those from the experiments in the literature than simulations with the

Johnson-Cook isotropie hardening material model although clearly the agreement for the

case of a 40% reduction is not as good as that for 10% and 20% reductions. The reasons

for the discrepancies between the simulation and experimental results can be concluded

similarly to section 5.2.1.3.

114

H=200 (MPa) Isotropic hardening H=470 (MPa)

Cockroft and Latham constant 461.50 461.47 459.61

(MPa)

Table 5-9: Calculated Cockroft and Latham constants


(mm) (mm) (mm) (mm) (mm) (mm) H=200 3.32 3.43 4.53 (MPa)

Isotropic 2.49 2.31 2.18

hardening 3.54 3.64 5.21

Table 5-10: Comparison of the predicted and experimental heights to fracture for

the process with 15°approach angle


(mm) (mm) (mm) (mm) (mm) (mm) H=200

3.53 (MPa) 3.72 4.14

H=470 2.83 2.83 2.63 2.31

(MPa) 3.00 3.13

Isotropic 3.94 4.16 5.01 hardening


the process with 30oapproach angle


(mm) (mm) (mm) (mm) (mm) (mm) H=200 4.20 4.18 5.55 (MPa)

Isotropic 3.33 2.80 2.58

hardening 5.25 5.36 7.06


the process with 60oapproach angle

115

10% pre-drawing 20% pre-drawin& 40% 2fe-drawing Predicted Experimental Predicted Experimental Predieted Experimental

H=200 72% 72% 62% (MPa)

79% 81% 82% Isotropie hardening

71% 69% 57%

Table 5-13: Comparisons of reductions in height between the simulation and

experimental resuIts for the process with 15° approach angle


H=200 71% 69% 66% (MPa)

H=470 76% 76% 75% 78% 74% 81% (MPa)

Isotropic 67% 65% 58% hardening


experimental resuIts for the process with 300 approach angle


H=200 65% 65% 54% (MPa)

Isotropic 72% 77% 78%

hardening 56% 55% 41%


experimental resuIts for the process with 60° approach angle

10% reduction 20% reduction 40% reduction H= 200 (MPa) 9% 11% 24%


Table 5-16: Differences of reduction in height between results from simulations and

literature for 15° approach angle

116

10% reduction 20% reduction 40% reduction H= 200 (MPa) 7% 12% 19% H=470 (MPa) 0% 4% 9%




10% reduction 20% reduction 40% reduction H= 200 (MPa) 10% 16% 31%




5.3 ResuIts of Simulations of Tests in Gill and Baldwin's Paper

Tests by Gill and Baldwin (1964) are performed dynamically. Hence, the effect of strain,

strain rate and temperature were taken into account when simulating this dynamic

process. As defined in Chapter 4, the simulations were only performed for Mn steel using

the combined hardening material mode1 with H equal to 200 (MPa) and the Johnson

Cook isotropic hardening material model.

5.3.1 Contour Plot Illustration

The contour plots of equivalent plastic strain, the maximum principal stress and

temperature of the rod obtained from the simulations with pre-drawing of 20% reduction

in area are shown in Figures 5-39 to 5-41, respectively. The distributions of equivalent

plastic strain and the maximum principal stress have the same trends as those in Figures

5-13 and 5-14 for S45C.

117

3D'U5 (Av",. C"it.: ?5~)

+5.52:0",-01 +5.090",-01 +4."/h50",-01. +4.229",-01. +3.?99",-01 +3.3-.59",-01 +2.939",-01. +2.509",-01. +2.0?9",-01 +1.-.548",-01 +1.218",-01. +?880",-02 +3.5?9",-02

Drawing

3D'U5 (Ave. exit.: 1,5~)

+4.1.80",-01. +4.031.",-01. +3.882",-01. +3.133",-01 +3.583",-01 +3.434",-01 +3.285",-01 +3.13-.5",-01 +2.98-.5",-01 +2.63?",-01. +2.-.588",-01 +2.,539",-01 +2.390",-01.

Cutting

3D'U5 (Av",. C"it.: 15t)

1 :~: t~~::gg +2.383",+00 +;2: H 192e:+OO +2.000",+00 +1.. 809",+00 +1..-.51. 1",+00 +1_ 42:"'I5e+OO +1..234",+00 +1.043",+00 +8.51.0",-01. +-.5.595",-01 +4.-.560",-01.

Upsetting

Figure 5-39: Contour plot of equivalent plastic strain for 20% reduction

3~ ~~_ Princip~l

(Ave. Crit.: "15%)

1 .. '.> •. +>5.6'18",+02 0/- +S_872:e+02

+4.8-6?e+02: +3 _ 8-.52:e+02: +Z.857e+02 +1.SS:1e+Oz. +8 _ 4Sge+O:l. -:1. 594e+O:1. -1..1.>55",+02 -;2 _ :1.?Oe+02

i1®': -:).:::t..7Se+02 -4_1..8:l.e+02 -5 _ :.1..S..se+02

Drawing

:3 t" M~ _ pz:: inc ip.:a.l (Ave. Crit_: ?5:fe;)

124. !~: ii-g~:g~

+4_ 270-=.+02 +3_42:2e+02: +2: _ 574e+02 +:1..725e+02: +S.773of!:+O:l. +;2: _ 907e+OO -8. 191..e+O:1. -:1..-5..s7e+02: -;2: _ 51.5e+02: -3 _ 3-.54e+02: -4.2::1..2:of!:+02:

Cutting

s~ H~_ Principal (Ave. Crit.: 7S:fe;)

1+9.3'15'",+02

, +7.-.512e+02 +S_845e+02:: +4.078e.+02: +2 _ 312e+02 +5. 450e+O:l. -:1.2:2:2:e+02: -2.S"88e+02: -4.75Se+02 -p.522e+OZ -S.2:88e+02: -:1._ 00-6e+03 -:l...l.82e+03

Upsetting

Figure 5-40:Contour plot of maximum principal stress (MPa) for 20% reduction in

area

118

SDU:l:5 (Ave_ C~it_: 7S~)

1+:,--21."1,,+02 +1_ 12:0e+02 +:1._ 02::J:e+02 +9 _ 2-63e+01 +3 _ 2: SJ'4e+ 0:1. +7 _ :J:2:-Se+O:l. +-.5 _ 3 S..se+O:1. +5 _ 38:7e+0:1. +4_ 41.8 e+ 0:1. +3_44S'e+O:1. +2_4$Oe+01 +:1._ 5:l.1e+O:1. +S_41."7e+OO

Drawing

SDV1.-.5 (Ave_ C~it_: 75_)

1 !~: ~~~:!gI +S_:l.44e+01 +?_808e+O:l. +7 _ 473e+01 +'i'_:l.38e+O:1. +-6_802e+01 +-.5_4'.57e+01 +"15 _ 1.31oe'+O:1. +5 _ ?S'-oe+01 +5 _ 4.50e+O:l. +5_:l.25e+O:l. +4_ 7SS'e+01

Cutting Upsetting

Figure 5-41:Contour plot of temperature eC) for 20% reduction in area

It is observed from Figure 5-41 that the distribution of the temperature across the cross

section of the rod follows the same trend as the distribution of equivalent plastic strain.

This is in agreement with the fact that the heat generated from plastic deformation

increases with the increasing equivalent plastic strain.

5.3.2 Calculations of Cold Heading Limit from Simulation Results

Similar ta section 5.2.1.3, the Cockroft and Latham constant was determined first from

the simulation without pre-drawing, which was taken to be the accumulated Cockroft and

Latham parameter from the beginning of the drawing stage to the point, where the ratio of

the current diameter to the initial diameter of the rod reaches the cold heading limit of 2.2

for upsetting without pre-drawing as shown in Figure 2-6. The values of Cockroft and

Latham constant computed from simulations without pre-draw with the combined

hardening model and Johnson-Cook isotropic hardening model are 549.37 and 587.05

(MPa), respectively. Then the se two values were used as benchmarks to evaluate the

119

accumulated Cockroft and Latham parameters for simulations with different approach

angles using the combined hardening model and the Johnson-Cook isotropie hardening

model, respectively. When the accumulated Cockroft and Latham parameters reach the

corresponding Cockroft and Latham constant, fracture is assumed to occur. The radius of

the rods to fracture are shown in Tables 5-19 to 5-21 for approach angles of 7°, 15° and

30°, respectively. The corresponding ratios of the fracture radius to the initial radius of

the rod were calculated, and those from the experiments in the literature were obtained

according to Figure 2-5. The comparisons of the predicted and experimental ratios are

shown in Tables 5-22 to 5-24. The differences between the predicted and experimental

. h· h· d f· d literature - predicted 10001 h . T bl 5 25 5 ratlos, w IC IS e me as x ~/o, are s own m a es - to-literature

27.

Without 20% reduction 40% reduction 60% reduction reduction (mm) (mm) (mm) (mm)

H= 200 (MPa) 8.80 8.43 6.88 6.91 Isotropic 8.80 7.90 6.28 5.90 hardening

Table 5-19: Radius ofthe rods to fracture for 7° approach angle


H=200 (MPa) 8.80 8.36 7.30 8.12 Isotropie 8.80 7.96 6.02 6.36 hardening

Table 5-20: Radius of the rods to fracture for 15° approach angle


H=200(MPa) 8.80 8.20 7.71 7.39 Isotropie 8.80 7.50 6.60 5.52 hardening

Table 5-21: Radius of the rods to fracture for 30° approach angle

120

20% reduction 40% reduction 60% reduction Predicted Experimental Predicted Experimental Predieted Experimental

H=200 2.11 1.72 1.73 (MPa)

Isotropie 2.34 2.40 2.13

hardening 1.98 1.57 1.48

Table 5-22: Comparisons of ratios of the fracture radius to the initial radius of the

rod between the simulation and experimental resuIts for 7° approach angle

20% reduetion 40% reduetion 60% reduction Predicted Experimental Predicted Experimental Predieted Experimental

H=200 2.09 1.83 2.03 (MPa)

2.51 2.59 2.37 Isotropie hardening

1.99 1.51 1.59



20% reduction 40% reduction 60% reduction Predicted Experimental Predicted Experimental Predieted Experimental

H=200 2.05 1.93 1.85 (MPa)

Isotropie 2.23 2.11 1.90

hardening 1.88 1.65 1.38



20% reduction 40% reduction 60% reduetion H= 200 (MPa) 10% 28% 19%

Isotropie hardening 15% 35% 31%

Table 5-25: Differences between the predicted and experimental ratios for 7°

approach angle

121

20% reduetion 40% reduetion 60% reduetion H= 200 (MPa) 17% 29% 14%



approach angle

20% reduetion 40% reduetion 60% reduetion H= 200 (MPa) 8% 9% 3%



approach angle

It is obvious that in general, for the dynamie proeesses, results from simulations with the

combined hardening material model are mueh doser to those from the experiments in the

literature th an simulations with Johnson-Cook isotropie hardening material model. The

agreement for the case of 30° approaeh angle is the best with the differences in the range

of 3% to 9%. However, for the similar reasons to section 5.2.1.3, larger differences still

exist in the case of 7° and 15° approaeh angle.

122

6. Conclusions and Future Work

6.1 Conclusions and Summary

One of the most common industrial cold forging processes is cold heading of steel wire or

rod to pro duce screws, bolts, nuts and rivets. The forming process is limited by a

complicated interplay of many factors. The amount of cold work (pre-drawing) is one of

the factors. Although several investigations into effects of pre-drawing on the formability

of metal materials during cold heading processes have been conducted, so far no attention

has been given to the numerical simulations of this phenomenon. Most numerical

simulations of bulk forming processes in literature are limited to a single process such as

drawing, extrusion, or upsetting.

In this work, physical tests investigating the effects of pre-drawing on the formability of

three metals, S45C, Mn steel and AISI 1335, from two papers in literature (Gill and

Baldwin, 1964; Tozawa and Kojima, 1971) are simulated with finite element software

ABAQUS v6.4. Since the chemical composition of Mn steel and AISI 1335 are similar,

they are treated as the same material. The tests are simulated with three successive FE

numerical models: the drawing model, the cutting model and the subsequent upsetting

model. The drawing and upsetting models were performed using the finite element

software ABAQUS/Explicit v6.4 package, while the cutting model was performed using

finite element software ABAQUS/Standard v6.4 package. The cutting process was

modeled by the "import" function in ABAQUS, which imports the material state of the

one-third of the drawn rod in the middle from the last increment of the drawing

simulation. Then after elastic springback in the cutting simulation, it is imported into the

upsetting model to be upset between two flat dies.

A new combined linear kinematic/nonlinear isotropie hardening constitutive model is

proposed and derived to account for the Bauschinger effect existing in reverse plastic

deformation. It is implemented into the ABAQUS/Explicit v6.4 by a user subroutine

VUMAT, which is used as an interface to specify a new material model in the

123

ABAQUS/Explicit v6.4 package. The VUMAT is verified by single-element tests under

tension, compression and reverse loading conditions.

An elastic-plastic model is assumed for both S45C and Mn steel. The elastic behavior of

the material model is assumed to be linear and isotropic, while the plastic behavior of the

material model is described by the new combined linear kinematic/ nonlinear isotropic

hardening model since both materials in the literature undergo reverse loading conditions.

In addition, for the purpose of comparison, the Johnson-Cook isotropie hardening model

is also applied for both materials.

The material constants of the Johnson-Cook isotropie hardening model are determined

using the resuIts of the simple compression test, while the kinematic hardening modulus

H was determined by fitting the average axial stress versus average axial strain curves

from the resuIts of simulations with the proposed combined hardening model to those

from the experimental resuIts in Tozawa and Kojima's paper (1971).

The good agreement between the simulation and experimental average axial stress versus

average axial strain curves indicates that the Bauschinger effect is important in this

process and also provides evidence to validate the FE numerical models and the proposed

combined hardening material model.

The Cockroft and Latham criterion (Cockroft and Latham, 1968) is employed to predict

the surface fracture.

After examining the simulation resuIts and comparing them to the resuIts from the

experiments in the literature, the following points are concluded for both the quasi-static

and dynamic processes:

• Reverse plastic de formation occurs in both axial and circumferential directions.

Therefore, it is correct that the Bauschinger effect was accounted for in this work.

124

• The principal stress changes its direction from the axial direction to the

circumferential direction during the entire process (drawing, cutting and

upsetting).

• Increasing the kinematic hardening modulus, H , decreases the corresponding

maximum principal stress regardless of different reductions in area.

• Increasing the pre-drawing reduction in area increases the equivalent plastic strain

of the element with the highest principal stress in the entire process.

• Increasing the approach die angle increases the equivalent plastic strain; the

differences of the equivalent plastic strains between the simulations with different

approach die angles for the fixed reduction in area diminish

• The proposed combined hardening material model is more accurate in predicting

material behavior in this reverse loading process (drawing and cutting followed by

upsetting) than the Johnson-Cook isotropic hardening model.

• The proposed combined hardening material model has the potential to correctly

predict the material behavior in the reverse loading process.

• This work successfully examined the effects of pre-drawing on formability during

cold heading through numerical simulation.

6.2 Future Work

This work is the first step for investigation into effect of pre-drawing on formability

during cold heading through numerical simulations. Further study can be focused on the

applications of combined nonlinear kinematic hardening/nonlinear isotropic hardening

material models. The effect of the friction factor may be of great importance, and

therefore needs to be studied further. In addition, the validity of Cockroft and Latham

criterion in this type of processes needs to be investigated since the maximum principal

stress of the element of interest changes its direction from the axial direction to the

circumferential direction during this process, which is not accounted for in the criterion.

125

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129

APPENDIXA 2-DVUMAT

C USER SUBROUTINE VUMAT SUBROUTINE VUMAT (

C

C

C

C

C

C

READ ONLY -* * * * * *

NBLOCK, NDIR, NSHR, NSTATEV, NFIELDV, NPROPS, LANNEAL, STEPTIME, TOTALTIME, DT, CMNAME, COORDMP, CHARLENGTH, PROPS, DENSITY, STRAININC, RELSPININC, TEMPOLD, STRETCHOLD, DEFGRADOLD, FIELDOLD, STRESSOLD, STATEOLD, ENERINTERNOLD, ENERINELASOLD, TEMPNEW, STRETCHNEW, DEFGRADNEW, FIELDNEW,

WRITE ONLY -* STRESSNEW, STATENEW, ENERINTERNNEW, ENERINELASNEW

INCLUDE 'VABA PARAM.INC'

DIMENSION COORDMP(NBLOCK,*), CHARLENGTH(NBLOCK), PROPS(NPROPS), 1 DENSITY(NBLOCK), STRAININC(NBLOCK,NDIR+NSHR), 2 RELSPININC(NBLOCK,NSHR), TEMPOLD(NBLOCK), 3 STRETCHOLD(NBLOCK,NDIR+NSHR) , 4 DEFGRADOLD(NBLOCK,NDIR+NSHR+NSHR) , 5 FIELDOLD(NBLOCK,NFIELDV), STRESSOLD(NBLOCK,NDIR+NSHR), 6 STATEOLD(NBLOCK,NSTATEV), ENERINTERNOLD(NBLOCK), 7 ENERINELASOLD(NBLOCK), TEMPNEW(NBLOCK), 8 STRETCHNEW(NBLOCK,NDIR+NSHR), 9 DEFGRADNEW(NBLOCK,NDIR+NSHR+NSHR), 1 FIELDNEW(NBLOCK,NFIELDV) , 2 STRESSNEW(NBLOCK,NDIR+NSHR), STATENEW(NBLOCK,NSTATEV), 3 ENERINTERNNEW(NBLOCK), ENERINELASNEW(NBLOCK)

CHARACTER*80 CMNAME DIMENSION INTV(2) PARAMETER (ZERO O.DO, ONE = 1.DO, TWO = 2.DO, THREE = 3.DO,

* FOUR = 4.DO, THIRD = ONE / THREE,PLU=l.DO, * HALF = O.5DO, TWOTHDS = TWO / THREE,DR=O.002DO, * OP5 = 1.5DO,BESECTION=20,TOLER=1.D-6)

C CHECK THAT NDIR=3 AND NSHR=l. IF NOT, EXIT. C

C

INTV(l) = NDIR INTV(2) = NSHR IF (NDIR .NE. 3 .OR. NSHR .NE. 1) TH EN

CALL XPLB_ABQERR(l, 'SUBROUTINE VUMAT IS IMPLEMENTED '// * 'ONLY FOR PLANE STRAIN AND AXISYMMETRIC CASES '// * '(NDIR=3 AND NSHR=l)' ,O,ZERO,' ,)

CALL XPLB_ABQERR(-2,'SUBROUTINE VUMAT HAS BEEN CALLED '// * 'WITH NDIR=%I AND NSHR=%I' ,INTV,ZERO,' ')

CALL XPLB EXIT END IF

E PROPS(l) XNU PROPS(2) A PROPS(3) B PROPS(4) C PROPS(5)

130

C

C

D PROPS(6) F PROPS (7) FRACTION PROPS(8) SHEAT = PROPS(9) TMELT = PROPS(10) TTRANSITION = PROPS(ll) EDOT PROPS(12) H PROPS(13)

TWOMU ALAMDA TERM

E / ( ONE + XNU TWOMU * XNU / ( ONE - TWO * XNU ONE / ( TWOMU + TWOTHDS * H )

C IF STEPTIME EQUALS TO ZERO, ASSUME THE MATERIAL PURE ELASTIC C AND USE INITIAL ELASTIC MODULUS C

IF ( STEPTIME .EQ. ZERO) THEN DO K = 1, NBLOCK

C TRIAL STRESS

C

TRACE = STRAININC(K,l) + STRAININC(K,2) + STRAININC(K,3) STRESSNEW(K,l) = STRESSOLD(K,l)

* + TWOMU * STRAININC(K,l) + ALAMDA * TRACE STRESSNEW(K,2) = STRESSOLD(K,2)

* + TWOMU * STRAININC(K,2) + ALAMDA * TRACE STRESSNEW(K,3) = STRESSOLD(K,3)

* + TWOMU * STRAININC(K,3) + ALAMDA * TRACE STRESSNEW(K,4)=STRESSOLD(K,4) + TWOMU * STRAININC(K,4)

ELASTIC STRAIN STATENEW (K, 6) STATENEW (K, 7) STATENEW (K, 8) STATENEW (K, 9)

END DO ELSE

STATEOLD(K,6) + STRAININC(K,l) STATEOLD(K,7) + STRAININC(K,2) STATEOLD(K,8) + STRAININC(K,3) STATEOLD(K,9) + STRAININC(K,4)

CONST = SQRT(TWOTHDS) DO K = 1, NBLOCK

C TRIAL STRESS

C

TRACE = STRAININC(K,l) + STRAININC(K,2) + STRAININC(K,3) SIG1 STRESSOLD(K,l) + TWOMU*STRAININC(K,l) + ALAMDA*TRACE SIG2 STRESSOLD(K,2) + TWOMU*STRAININC(K,2) + ALAMDA*TRACE SIG3 STRESSOLD(K,3) + TWOMU*STRAININC(K,3) + ALAMDA*TRACE SIG4 STRESSOLD(K,4) + TWOMU*STRAININC(K,4)

TRIAL STRESS MEASURED FROM THE BACK STRESS SI SIG1 - STATEOLD(K,l) S2 SIG2 - STATEOLD(K,2) S3 SIG3 - STATEOLD(K,3) S4 SIG4 - STATEOLD(K,4)

C DEVIATORIC PART OF TRIAL STRESS MEASURED FROM THE BACK STRESS 8MEAN = THIRD * ( 81 + 82 + 83 ) DS1 SI - SMEAN DS2 = S2 - SMEAN DS3 = S3 - SMEAN

C MAGNITUDE OF THE DEVIATORIC TRIAL STRESS DIFFERENCE

C C

DSMAG = SQRT ( DS1*DS1 + DS2*DS2 + DS3*DS3 + TWO*S4*S4

CHECK FOR EQPLASOLD STATEOLD(K,5)

131

C

C

DEQPSOLD = STATEOLD(K,14) DTOLD = STATEOLD(K,15) IF (DTOLD.EQ.ZERO) DTOLD = ONE SSl = DEQPSOLD/DTOLD SS = SSl/EDOT IF(SS.LT.ONE) SS = ONE

TEMOLD = STATEOLD(K,16) TOLD = (TEMOLD-TTRANSITION)/(TMELT-TTRANSITION) IF (TOLD.LT.ZERO) TOLD = ZERO IF (TOLD.GE.ONE) TOLD = ONE

SIGOLD = (A+B*(EQPLASOLD)**C)*(l+D*LOG(SS»*(l-TOLD**F) KSIGOLD = H*EQPLASOLD IF( SIGOLD.LE.KSIGOLD) SIGOLD = KSIGOLD YIELDOLD = SIGOLD-KSIGOLD

RADIUSOLD = CONST * YIELDOLD IF ( DSMAG - RADIUSOLD .GT. ZERO) TH EN

C SOLVE FOR EQUIVALENT PLASTIC STRAIN INCREMENT USING BISECTION ITERATION

C SOLVE FOR FA AA=ZERO DEQPS=AA SIGO = (A+B* (EQPLASOLD)**C) * (l-TOLD**F)

KSIGO = H*EQPLASOLD IF( SIGO.LE.KSIGO) SIGO = KSIGO YIELDO =SIGO-KSIGO

FA = CONST*TERM*(DSMAG-CONST* YIELDO)-DEQPS C SOLVE FOR FB

BB = DR+CONST*SQRT ( (STRAININC(K,l)**2 +STRAININC(K,2)**2 + * STRAININC(K,3) **2 +TWO * STRAININC(K,4)**2 ) )

DEQPS=BB SIG11 SIG1-(TWOMU/CONST)*DEQPS*(DS1/DSMAG) SIG22 SIG2-(TWOMU/CONST)*DEQPS*(DS2/DSMAG) SIG33 SIG3-(TWOMU/CONST)*DEQPS*(DS3/DSMAG) SIG12 SIG4-(TWOMU/CONST)*DEQPS*(S4/DSMAG) MEANSTRESS = THIRD * (SIG11 + SIG22 + SIG33 ) SIl SIG11- MEANSTRESS S22 SIG22- MEANSTRESS S33 SIG33- MEANSTRESS S12 SIG12 SIGMISE = SQRT ( OP5 * (Sll**2 +S22**2 +S33**2 +

* TWO *S12**2 ) ) DTEMP= FRACTION/(SHEAT*DENSITY(K»)*DEQPS*SIGMISE TNEW = (TEMOLD+DTEMP-TTRANSITION)/(TMELT-TTRANSITION) IF (TNEW.LT.ZERO) TNEW = ZERO IF (TNEW.GE.ONE) TNEW = ONE SIGTEMP 1- TNEW**F SIGEQPS = A+B*(EQPLASOLD+DEQPS)**C SSS = DEQPS/DT/EDOT IF(SSS.LT.ONE) SSS = ONE SIGRATE = l+D*LOG(SSS) SIGNEW= SIGEQPS*SIGRATE*SIGTEMP KSIGNEW = H*(EQPLASOLD+DEQPS) IF( SIGNEW.LT.KSIGNEW) SIGNEW = KSIGNEW YIELD = SIGNEW-KSIGNEW FB = CONST*TERM*(DSMAG-CONST* YIELD)-DEQPS

132

C BISECTION ITERATION

*

DO BI=l,BESECTION P=(BB+AA)/2 DEQPS=P SIG11 SIG1-(TWOMU/CONST)*DEQPS*(DS1/DSMAG) SIG22 SIG2-(TWOMU/CONST)*DEQPS*(DS2/DSMAG) SIG33 SIG3-(TWOMU/CONST)*DEQPS*(DS3/DSMAG) SIG12 SIG4-(TWOMU/CONST)*DEQPS*(S4/DSMAG) MEANSTRESS = THIRD * (SIG11 + SIG22 + SIG33 ) SIl SIG11- MEANSTRESS S22 SIG22- MEANSTRESS S33 SIG33- MEANSTRESS S12 SIG12 SIGMISE = SQRT ( OP5 * (Sll**2 +S22**2 +S33**2 +

TWO *S12**2 ) ) DTEMP= FRACTION/(SHEAT*DENSITY(K))*DEQPS*SIGMISE TNEW = (TEMOLD+DTEMP-TTRANSITION)/(TMELT-TTRANSITION) IF (TNEW.LT.ZERO) TNEW = ZERO IF (TNEW.GE.ONE) TNEW = ONE SIGTEMP = 1- TNEW**F SIGEQPS = A+B*(EQPLASOLD+DEQPS)**C SSS = DEQPS/DT/EDOT IF(SSS.LT.ONE) SSS = ONE SIGRATE = l+D*LOG(SSS) SIGNEW= SIGEQPS*SIGRATE*SIGTEMP KSIGNEW = H*(EQPLASOLD+DEQPS) IF( SIGNEW.LT.KSIGNEW) SIGNEW = KSIGNEW YIELD = SIGNEW-KSIGNEW FP = CONST*TERM*(DSMAG-CONST* YIELD)-DEQPS IF ((((BB-AA)/2) .LT.TOLER) .OR. (FP.EQ.ZERO)) GOTO 2 IF ((FA*FP) .GT.ZERO) TH EN

AA=P FA=FP

ELSE BB=P

END IF END DO WRITE(*,l) K

1

* FORMAT(//,30X,'***WARNING - PLASTICITY ALGORITHM DID NOT',

'CONVERGE AFTER ',13,' ITERATIONS') 2 LL=l C CALCULATED INCREMENT IN GAMMA

c

DGAMMA = DEQPS/CONST ELSE

DEQPS = ZERO DGAMMA = ZERO DSMAG = DSMAG + ONE

END IF

C UPDATE EQUIVALENT PLASTIC STRAIN AND EQUIVALENT PLASTIC STRAIN RATE

C STATENEW(K,14) DEQPS STATENEW(K,15) DT STATENEW(K,5) STATEOLD(K,5) + DEQPS

C DIVIDE DGAMMA BY DSMAG SO THAT THE DEVIATORIC STRESSES ARE C EXPLICITLY CONVERTED TO TENSORS OF UNIT MAGNITUDE IN THE FOLLOWING

133

C CALCULATIONS DGAMMA = DGAMMA / DSMAG

C UPDATE BACK STRESS FACTOR = TWOTHDS * H * DGAMMA STATENEW(K,l) STATEOLD(K,l) + FACTOR * DSl STATENEW(K,2) STATEOLD(K,2) + FACTOR * DS2 STATENEW(K,3) STATEOLD(K,3) + FACTOR * DS3 STATENEW(K,4) STATEOLD(K,4) + FACTOR * S4

C UPDATE THE STRESS

C

DGAMMA FACTOR = TWOMU * STRESSNEW (K, 1) STRESSNEW(K,2) STRESSNEW(K,3) STRESSNEW(K,4)

SIG1 - FACTOR * DS1 SIG2 - FACTOR * DS2 SIG3 - FACTOR * DS3 SIG4 - FACTOR * S4

UPDATE THE ELASTIC STATENEW (K, 6) STATENEW(K,7) STATENEW(K,8) STATENEW(K,9)

STRAIN STATEOLD(K,6) STATEOLD(K,7) STATEOLD(K,8) STATEOLD(K,9)

+ STRAININC(K,l)-DGAMMA * + STRAININC(K,2)-DGAMMA * + STRAININC(K,3)-DGAMMA * + STRAININC(K,4)-DGAMMA *

C UPDATE THE PLASTIC STRAIN STATENEW(K,10) STATEOLD(K,10) + DGAMMA * DS1 STATENEW(K,ll) STATEOLD(K,ll) + DGAMMA * DS2 STATENEW(K,12) STATEOLD(K,12) + DGAMMA * DS3 STATENEW(K,13) STATEOLD(K,13) + DGAMMA * S4

C UPDATE THE SPECIFIC INTERNAL ENERGY STRESS POWER = HALF * (

DS1 DS2 DS3 S4

* *

( STRESSOLD(K,l)+STRESSNEW(K,1) ( STRESSOLD(K, 2) +STRESSNEW(K,2) ( STRESSOLD(K,3) +STRESSNEW(K, 3) ( STRESSOLD(K,4)+STRESSNEW(K,4)

* STRAININC(K,l) + * STRAININC(K,2) +

* *

ENERINTERNNEW(K) = ENERINTERNOLD(K)

* + STRESSPOWER / DENSITY(K)

* STRAININC(K,3) ) + * STRAININC(K,4)

C UPDATE THE DISSIPATED INELASTIC SPECIFIC ENERGY SMEAN = THIRD *

* ( STRESSNEW(K,l) + STRESSNEW(K,2) + STRESSNEW(K,3) EQUIVSTRESS = SQRT ( OP5 * (

* ( STRESSNEW(K,l) - SMEAN )**2 + * ( STRESSNEW(K,2) - SMEAN )**2 + * ( STRESSNEW(K,3) - SMEAN )**2 + * TWO * STRESSNEW(K,4)**2 ) )

PLASTICWORKINC = EQUIVSTRESS * DEQPS ENERINELASNEW(K) = ENERINELASOLD(K)

* + PLASTICWORKINC / DENSITY(K) C UPDATE THE TEMPERATURE

DTEMP = (PLASTICWORKINC / DENSITY(K»*(FRACTION/SHEAT) TEMPNEW(K) =TEMPOLD(K) +DTEMP STATENEW(K,16) = STATEOLD(K,16)+DTEMP

END DO END IF RETURN END

134

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