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Experimental Characterization and Modeling of the Mechanical Response of Titanium for Quasi-Static and High Strain Rate Loads

Permanent Link: http://ufdc.ufl.edu/UFE0022062/00001

Material Information

Title: Experimental Characterization and Modeling of the Mechanical Response of Titanium for Quasi-Static and High Strain Rate Loads
Physical Description: 1 online resource (168 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: anisotropic, material, strength, titanium, twinning
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This dissertation is devoted to the characterization, modeling and simulation of plastic anisotropy and strength differential effects in high-purity, polycrystalline alpha-titanium. A series of uniaxial compression and tension tests were carried out at room temperature under quasi-static conditions to quantify the plastic anisotropy and strength differential effects in the material. Pre- and post-test textures were measured using neutron diffraction techniques and orientation imaging microscopy (OIM). The tests indicated that initially both plates have strong basal textures, one of the plates studied (Plate 1) being orthotropic, whereas the other one (Plate 2) has in-plane symmetry. Significant texture evolution associated primarily with tensile twinning was observed only for Plate 1 when subjected to compression in the rolling direction. Four-point bending tests were performed for validation purposes. Digital Image Correlation techniques were used to obtain the strain field. As a result of the anisotropy and directionality of twinning, qualitative differences were observed between the response of upper and lower fibers in different orientations. Split Hopkinson Pressure Bar tests at strain rates of 400 to 600 sec$^{-1}$ were performed along the axes of symmetry of each plate to characterize the material's strain rate sensitivity. A clear increase in strength with increasing strain rate is observed, the hardening rate remaining practically unchanged for all directions, with the exception of the rolling direction. The dramatic hardening rate increase in the rolling direction was indicative of higher propensity for twinning with increasing strain rate. Taylor cylinder impact tests on specimens cut from Plate 2 were performed at impact velocities in the range of 200 m/s. Based on presented experimental data, it can be concluded that the material has a very complex anisotropic behavior and exhibits tension/compression asymmetry and strain rate sensitivity. A new anisotropic elastic/plastic model was developed. Key in its formulation is an yield criterion that captures strength differential effects. Anisotropy was introduced through a linear transformation on the Cauchy stress tensor applied to the material. An anisotropic hardening rule that accounts for texture evolution associated to twinning was developed. It was demonstrated that the model describes very well the main features of the quasi-static response of high-purity Ti when subjected to monotonic loading conditions. Validation of the model was provided through comparison between measured and simulated strain distributions in bending. In particular, the shift of the neutral axis was well described. An extension of the model that incorporates rate effects was also developed and used to describe the anisotropic high rate behavior of the material. It was shown that the rate dependent model describes well the deformed profiles and final cross section of the specimens.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Cazacu, Oana.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022062:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022062/00001

Material Information

Title: Experimental Characterization and Modeling of the Mechanical Response of Titanium for Quasi-Static and High Strain Rate Loads
Physical Description: 1 online resource (168 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: anisotropic, material, strength, titanium, twinning
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This dissertation is devoted to the characterization, modeling and simulation of plastic anisotropy and strength differential effects in high-purity, polycrystalline alpha-titanium. A series of uniaxial compression and tension tests were carried out at room temperature under quasi-static conditions to quantify the plastic anisotropy and strength differential effects in the material. Pre- and post-test textures were measured using neutron diffraction techniques and orientation imaging microscopy (OIM). The tests indicated that initially both plates have strong basal textures, one of the plates studied (Plate 1) being orthotropic, whereas the other one (Plate 2) has in-plane symmetry. Significant texture evolution associated primarily with tensile twinning was observed only for Plate 1 when subjected to compression in the rolling direction. Four-point bending tests were performed for validation purposes. Digital Image Correlation techniques were used to obtain the strain field. As a result of the anisotropy and directionality of twinning, qualitative differences were observed between the response of upper and lower fibers in different orientations. Split Hopkinson Pressure Bar tests at strain rates of 400 to 600 sec$^{-1}$ were performed along the axes of symmetry of each plate to characterize the material's strain rate sensitivity. A clear increase in strength with increasing strain rate is observed, the hardening rate remaining practically unchanged for all directions, with the exception of the rolling direction. The dramatic hardening rate increase in the rolling direction was indicative of higher propensity for twinning with increasing strain rate. Taylor cylinder impact tests on specimens cut from Plate 2 were performed at impact velocities in the range of 200 m/s. Based on presented experimental data, it can be concluded that the material has a very complex anisotropic behavior and exhibits tension/compression asymmetry and strain rate sensitivity. A new anisotropic elastic/plastic model was developed. Key in its formulation is an yield criterion that captures strength differential effects. Anisotropy was introduced through a linear transformation on the Cauchy stress tensor applied to the material. An anisotropic hardening rule that accounts for texture evolution associated to twinning was developed. It was demonstrated that the model describes very well the main features of the quasi-static response of high-purity Ti when subjected to monotonic loading conditions. Validation of the model was provided through comparison between measured and simulated strain distributions in bending. In particular, the shift of the neutral axis was well described. An extension of the model that incorporates rate effects was also developed and used to describe the anisotropic high rate behavior of the material. It was shown that the rate dependent model describes well the deformed profiles and final cross section of the specimens.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Cazacu, Oana.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022062:00001


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EXPERIMENTAL CHARACTERIZATION AND MODELING OF THE MECHANICAL
RESPONSE OF TITANIUM FOR QUASI-STATIC AND HIGH STRAIN RATE LOADS



















By

MICHAEL E. NIXON


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2008


































2008 Michael E. Nixon































Dedicated to my late father, Rufus Nixon. Although he had little chance to obtain a

formal education, he continually strived to learn new things and urged me to continue

learning all of my life. From an early age he taught me to be strong but it was better to

use your brain than your brawn.









ACKNOWLEDGMENTS

The list of persons who have influenced my life and this research is too long to

properly document but I must attempt to thank a number of people. First I would like

to thank my family, especially my wife and mother. I would also like to acknowledge the

strong support of my emplc.,- ri at the Air Force Research Lab Munitions Directorate, in

particular Dr. Larry Lijewski.

I could not have begun to accomplish the level of effort that went into this research

without the contributions of many of my professional collaborators. First my colleagues

from the Air Force Research Lab. Dr. Brian Plunkett provided many insights into

material modeling and with help in distinguishing what was important and what was less

important. Having him next door to my office proved to be a valuable asset when my

insight failed me or my energies 1 .-.- d. Dr. Martin Schmidt provided me with insights

in how to get through the maze surrounding obtaining a degree and when to finally

i- enough is enough. Thanks to Joel Stewart for his deep philosophical insights. Dr.

Joel House has provided many hours of discussions over several years. Technical topics

including dislocation motion and twinning and other discussions on how to maintain my

sanity when it seems like the whole world has gone crazy. I would also like to thank Joel

and Philip Flater for providing much of the experimental data.

I also had a lot of help from my colleagues at the Los Alamos National Labortory

that not many graduate students get to enjoy. Dr. Ricardo Lebensohn not only provided

valuable insight into the mechanics of deformation in hcp materials and the p .li-, I il i 11*11w

code VPSC but acted as my liaison for much of the quasi-static testing and texture

investigations reported here. His patience and diligence is much appreciated. I would

also like to acknowledge the discussions with Dr. Carlos Tom6 and Dr. George Kaschner

concerning the VPSC code, multi-scale material behavior, and experimental techniques.

Special thanks to Manuel Lovato and Dr. C('!. n Liu for performing the quasi-static









uniaxial tests and four point beam tests that are vital to this research. Thanks to Dr.

Gwenaelle Proust and Dr. Sven Vogel for the beautiful texture and OIM work.

I would like to give a special acknowledgement to Dr. Davy Belk for inspiring me to

expand my universe and pursue my goal of achieving a PhD at my advanced age. And

finally I want to thank my advisor, Professor Oana Cazacu. She is the reason that I was

able to work in an area of research that was of particular interest to me. She made the

work enjo'1--l-, and relative. She is the hardest working person I know and without her I

could not have completed this work.









TABLE OF CONTENTS
page

ACKNOW LEDGMENTS ................................. 4

LIST OF TABLES ....................... ............. 9

LIST OF FIGURES ....................... ........... 10

ABSTRACT . . . . . . . . . . 17

CHAPTER

1 GENERAL INTRODUCTION ........................... 19

1.1 Material Modeling ................... ....... 20
1.1.1 Elasticity ............................. 20
1.1.2 Plasticity ............................. 21
1.1.2.1 Isotropic yield surfaces ......... .......... 22
1.1.2.2 Anisotropic yield surfaces ........ .......... 24
1.1.2.3 Flow rules ............... ........ .. 27
1.1.2.4 Hardening ............... ........ .. 27

2 TITANIUM ................... ........ . .... 30

2.1 Basic Properties ............... ............ .. 30
2.2 Single Crystal Properties ............... ........ .. 32
2.3 Deformation Mechanisms ............... ........ .. 33
2.3.1 Slip .................. ............ .... 33
2.3.2 Twinning. .................. ............ .. 34
2.3.3 Hardening ............... ............ .. 34
2.4 High Purity Titanium Plates . ............. ... 35

3 EXPERIMENTS ........................... . 39

3.1 Quasi-Static Tests ................ . . 40
3.1.1 (C! i '.terization Tests.... ............ ..... .. 40
3.1.1.1 Test description .................. .... .. 40
3.1.1.2 Plate 1 results. .................. .... .. 42
3.1.1.3 Plate 2 results .................. ..... .. 47
3.1.2 Four Point Beam Bend Tests ....... . . .. 50
3.1.2.1 Four point beam bend test results: Plate 1 . ... 53
3.1.2.2 Four point beam bend test results: Plate 2 . ... 57
3.2 High Rate Tests .................. .............. .. 61
3.2.1 ('C! o i.terization Tests .... . . . 61
3.2.1.1 Description of the split Hopkinson pressure bar . 62
3.2.1.2 Test results .................. ..... .. 64
3.2.1.3 Plate 1 HR characterization tests . . ..... 66









3.2.1.4 Plate 2 HR characterization tests . . 68
3.2.2 Cylinder Impact Tests ................ .. ... .. 70
3.3 Texture . ................ ............ .. .. 74
3.3.1 Grain Size .............. . . ... .76
3.3.2 Determination of Rolling Direction .... . 79
3.3.3 Variation of Texture in Through Thickness Direction . ... 81
3.3.4 Texture Evolution .................. ......... .. 86

4 M ODELING .................. .................. .. 92

4.1 Proposed Yield Criterion .................. ......... .. 92
4.1.1 Isotropic Yield Function .................. .... .. 92
4.1.2 Extension to Orthotropy .................. ... .. 94
4.2 Identification of Material Parameters ............... . .. 98
4.2.1 Cost function involving only yield stresses . . ..... 99
4.2.2 Cost function involving Lankford coefficients . . 99
4.2.3 Cost function involving biaxial data ..... . . ..... 99
4.3 Anisotropic Hardening .................. .......... .. 100

5 SIMULATIONS .................. ................. .. 102

5.1 Application to Mg-Li Alloy ................... . .... 102
5.2 Application to High-purity Titanium ............... .. ... 103
5.2.1 Plate 1 . . . . . . . .... 103
5.2.2 Plate 2 . . .. . . ...... 104
5.3 Comparison to Hill's Quadratic Model ............ .. .. 105
5.4 FE Implementation of Proposed Model ............ ... ..110
5.4.1 Elastic-Plastic Model ....... . . ...... 111
5.4.2 Elastic-viscoplastic Extension of the Proposed Model . ... 113
5.4.3 Effective Stress Calculation ...... .......... . .. 115
5.4.4 Derivatives of Yield Function ................ .. ..115
5.4.5 Anisotropic Hardening ................ . .. 116
5.4.6 Parameter Values ............... ....... .. .. 117
5.5 FE Simulations ................ ............. .. 117
5.5.1 Single Cell ............... ........... .. .. 119
5.5.1.1 Plate 1 results. .............. .. 120
5.5.1.2 Plate 2 results .............. 122
5.5.2 Four Point Bend Tests .............. 122
5.5.2.1 Plate 1 results. .............. .. 126
5.5.2.2 Plate 2 results .............. 137
5.5.3 Cylinder Impact Tests ............... 145
5.5.3.1 Hardening ............... ........ 146
5.5.3.2 Finite element mesh ................ ... 148
5.5.3.3 Simulations ............... .... 149










6 CONCLUSIONS .....

6.1 Present Research .
6.2 Future Research .
6.3 Concluding Remarks

REFERENCES .........

BIOGRAPHICAL SKETCH. .


. . . . ... . 16 0

. . . .. . . 16 0
. . . .. . . 16 3
. . .. . . 16 3

. . . . . . . . 16 4

. . .. . . 16 8










LIST OF TABLES


Tabl

1-1

2-1

2-2

3-1

3-2

3-3

3-4

3-5

3-6

3-7


3-8

5-1

5-2

5-3

5-4

5-5

5-6

5-7


e

Phenomenological yield functions . ............

Physical properties of Titanium . ............

C'!, I. II analysis of test m material . ...........

Measurements of deformed beam bend specimens from Plate 1

Measurements of deformed beam bend specimens from Plate 2

Strain rates acheived for tensile SHPB tests . .

Strain rates acheived for tensile SHPB tests . ......

Quasi-static and high rate compressive yield values for Plate 1

Impact velocities from high rate cylinder tests . .....

Ratios of in I i wr to minor final diameters and ratios of final to in
from high rate cylinder tests . ....

Grain size averages at locations shown in Figure 3-50 . .

Model parameters for the yield surface in Figure 5-1 . .

Compressive yield data used to identify Hill48 parameter values

Tensile yield data used to identify Hill48 parameter values .

Parameter values for Hill48 model using Plate 1 data . .

Plate 1 anisotropy coefficient values for discrete strain levels .

Plate 2 anisotropy coefficient values for discrete strain levels .

Johnson-Cook hardening law parameter values . ....


page

25

31

36

57

61

64

64

66

72


itial


lengths


. 73

. 78

. 102

. . 110

. . 110

. 110

. . 117

. . 119

. 48










LIST OF FIGURES


re


Figu

1-1

1-2

2-1

2-2

2-3

2-4

2-5

2-6

2-7

3-1

3-2

3-3

3-4

3-5

3-6

3-7

3-8

3-9

3-10

3-11

3-12

3-13

3-14

3-15

3-16


Results of quasi-static tension and compression tests in TT direction for Plate 1

Hardening in tension and compression in the TT direction for Plate 1 ..

Plate 2 quasi-static in-plane data .. .....................

Plate 2 comparison of in-plane quasi-static tension versus compression data .

Results of quasi-static tension and compression tests in TT direction for Plate 2

Plate 2 comparison of TT quasi-static tension versus compression data .....


Elastic coefficients required for various < i --il symmetries . ...

Projection of Tresca yield surface . .................

Variation of Ti single < i 1--I elastic modulus . ..........

Titanium (i -I 1J structure . . . . . . .

Active twinning systems in Ti . ..................

Titanium plates . ..........

Micrograph of high purity Titanium plate material . .......

Plate 1 pole figure with center in TT direction . .........

Plate 1 pole figure with center in RD . ...............

Geometry and dimensions of the through-thickness tensile specimen .

Quasi-static compression specimens.... . .....

Geometry and dimensions of quasi-static in-plane specimens for tension .

Definition of the specimen orientations... . ......

Results of quasi-static compression tests along the RD on Plate 1 .

Orientation Imaging Microscopy map at 10' strain . .

Orientation Imaging Microscopy map at 211' i strain . .

Results of quasi-static tensile tests along the RD conducted on Plate 1 .

Results of quasi-static tensile tests along the TD conducted on Plate 1 .

Hardening during uniaxial tension and compression in the TD for Plate 1


page

. 21

. 23

. 32

. 33

. 34

. 37

. 37

. 38

. 38

. 40

. 41

. 41

. 41

. 42

. 43

. 43

. 44

. 45

45










3-17 Orientation definition for four point beam test specimen

3-18 Four point beam test jig . ............

3-19 Typical Load vs Displacement curve for bend tests .

3-20 Beam grid pattern used in DIC to compute strain field

3-21 Plate 1 experimental axial strain (e,) fields for Case 1


Plate 1 experimental axial

Plate 1 experimental axial

Plate 1 experimental axial

Deformed cross section of

Deformed cross section of

Measurement locations on

Plate 2 experimental axial

Plate 2 experimental axial

Plate 2 experimental axial

Plate 2 experimental axial


strain (a.) fields for Case 2

strain (Fy) fields for Case 3

strain (Fy) fields for Case 4

beam from Plate 1 for Case 1

beam from Plate 1 for Case 3

deformed four point beam tes

strain (a.) fields for Case 1

strain (e,) fields for Case 2

strain (Fy) fields for Case 3

strain (Fy) fields for Case 4


3-22

3-23

3-24

3-25

3-26

3-27

3-28

3-29

3-30

3-31

3-32

3-33

3-34

3-35

3-36

3-37

3-38

3-39

3-40

3-41

3-42

3-43


. 60

. 60

. 61

. 62

. 63

. 65

. 65

. 67

. .. 67

. 68

. 69

. 70


Deformed cross section of beam from Plate 2 for Case 1 and 2 . .

Deformed cross section of beam from Plate 3 for Case 3 and 4 . .

High rate test specimens ..... . .

Failed surface from high rate tension test specimen . .

Schematic of Split Hopkinson bar apparatus . .....

Experimental compression results showing anisotropy of Plate 1 . .

Experimental compression results for Plate 2 . .....

Plate 1 High rate TD data ................. . .....

Comparison of compressive high rate to quasi-static TT data for Plate 1

Plate 1 High rate RD data ................. . .....

Plate 2 High rate in-plane data ............... . .....

Plate 2 experimental high rate tension data . ...........


s . . 51

.. . 52

. . 52

. . . 53

. . . 54

. . . 54

. . . 55

. . . 55

and 2 . ... 56

and 4 . ... 56

t specimens . .. 57

. . . 58

. . . 58

. . . 59

. . . 59









3-44 Plate 2 experimental high rate through thickness compression data


3-45 Taylor cylinder impact test setup .................. ....... .. 71

3-46 High rate cylinder test results .................. ......... .. 73

3-47 High rate cylinder impact test specimens ................ .... 74

3-48 Measured i, i i" and minor profile data from test number 107 . .... 75

3-49 Measured deformed footprint from test number 107 .............. 75

3-50 Micrograph locations for Plate 1 ... ............ ..... .. 76

3-51 Optical microscopy (50X) at locations 1 and 2 .............. .. 77

3-52 Optical microscopy (50X) at locations 3 and 4 .............. .. 77

3-53 Optical microscopy (50X) at locations 5 and 6 .............. .. 78

3-54 Optical microscopy (50X) at locations 7 and 8 .............. .. 78

3-55 Plate 1 with 20 coupons removed ............... ....... .. 79

3-56 Definition of sample orientation from sectioned coupon ........... .80

3-57 Initial (0002) pole figures for Plate 1 ................ ..... 80

3-58 Plate 1 and Plate 2 with pole figures superimposed to determine RD . 81

3-59 Position of scan locations for through thickness texture measurements . 82

3-60 Bulk texture measurement of Plate 1 ................ ..... 82

3-61 Plate 1 pole figures from positions 1 and 2 ................ .... 83

3-62 Plate 1 pole figures from positions 3 and 4 ................ ... 83

3-63 Plate 1 pole figures from positions 5 and 6 ................ .... 83

3-64 Plate 1 pole figures from positions 7 and 8 ................ .... 84

3-65 Plate 1 pole figures from positions 9 and 10 .............. . 84

3-66 Plate 1 pole figures from positions 11 and 12 .............. .. 84

3-67 Plate 1 pole figures from positions 13 and 14 ..... . . ... 85

3-68 Plate 1 pole figures from positions 15 and 16 ..... . . ... 85

3-69 Plate 1 pole figures from position 17 ................ .... ... .. 85

3-70 Plate 1 Initial texture from three perspectives .............. .. 86










3-71 Plate 1 (0001) pole figure for specimens loaded
in transverse direction .. ..........

3-72 Plate 1 (0001) pole figure for specimens loaded
strain in transverse direction ....

3-73 Plate 1 (0001) pole figure for specimens loaded
in through thickness direction .. .......

3-74 Plate 1 (0001) pole figure for specimens loaded
strain in through thickness direction .....

3-75 Plate 1 (0001) pole figure for specimens loaded
in rolling direction .. ............

3-76 Plate 1 (0001) pole figure for specimens loaded
strain in rolling direction . .


3-77

3-78

3-79

4-1

4-2

4-3

5-1

5-2

5-3

5-4

5-5

5-6

5-7

5-8

5-9

5-10

5-11


in compression to 10 and 20


in compression to 30 and 40


in compression to 10 and 20


in compression to 30 and 40


in compression to 10 and 20


in compression to 30 and 40


Texture evolution for compressive loading in the rolling direction

Texture evolution for compressive loading in the transverse dir

Texture evolution for compressive loading in the through thick

Plane stress yield locii for various rations of T/c . .


)n

action

less c


Comparison with pcl.i v i -I illii: simulations . ........

Arbitrary angle definition, x is rolling direction . ......

Projection in the plane a3=0 for Mg-Li alloy sheet . .

Theoretical model compared to experimental data for Plate 1 .

Average experimental in-plane compression data for Plate 2 .

Theoretical model compared to experimental data for Plate 2 .

Comparison of Hill's criterion to proposed criterion for Plate 1 data

Theoretical yield curves for Plate 1 . .............

Theoretical yield curves for Plate 2 . .............

Single cell computational configuration . ..........

Single cell simulation results for Plate 1 A) RD tension B) RD comp

Single cell simulation results for Plate 1 A) TD tension B) TD comp

Single cell simulation results for Plate 1 A) TT tension B) TT comp


. 89

. 90

S . 91

direction 91

. 93

. 94

. 97

. 103

. . 104

. 105

. . 106

. 111

. 118

. 118

. 120

session 121

)ression 121

ression 122


.........










5-12

5-13

5-14

5-15

5-16

5-17

5-18

5-19


5-20 Case 1: Long axis in RD, loading in TD .. ..................


5-21 Plate 1, Case


5-22

5-23

5-24

5-25

5-26

5-27

5-28

5-29

5-30

5-31

5-32

5-33

5-34

5-35

5-36

5-37

5-38


Plate 1,

Plate 1,

Case 2:

Plate 1,

Plate 1,

Plate 1,

Case 3:

Plate 1,

Plate 1,

Plate 1,

Case 4:

Plate 1,

Plate 1,

Plate 1,


Case

Case

Long

Case

Case

Case

Long

Case

Case

Case

Long

Case

Case

Case


1: Comparison of axial strain countours . . .....

1: Axial strains (ca) versus height at centerline: x=RD, y=TD

1: Comparison of cross sections from experiment and simulation

axis in RD, loading in TT . . .....

2: Comparison of axial strain countours . . .....

2: Axial strains (a.) versus height at centerline: x=RD, z=TT

2: Comparison of cross sections from experiment and simulation

axis in TD, loading in RD . . .....

3: Comparison of axial strain countours . . .....

3: Axial strains (Fy) versus height at centerline: x=RD, y=TD

3: Comparison of cross sections from experiment and simulation

axis in TD, loading in TT . . .....

4: Comparison of axial strain countours . . .....

4: Axial strains (Fy) versus height at centerline: y=TD, z=TT

4: Comparison of cross sections from experiment and simulation


Comparison of cross sectional area for Plate 2 . . .....

Plate 2 Isotropic simulation versus model for Case 1 and 3 . . ..

Plate 2 Isotropic simulation versus model for Case 2 and 4 . . ..


128

129

130

130

131

131

132

133

133

134

134

135

135

136

136

137

138

138


Plate 2 In-plane single cell simulations... . ......

Plate 2 TT single cell simulations...... . .....

Symmetrical brick arrangement for tetrahedral elements . .....

FE Computational mesh for beam bending tests . .

Typical deformed mesh showing plane of symmetry . .

Comparison of cross sectional area from simulations of Plate 1 . .

Comparison of tension versus compression data for RD and TD in Plate 1

Comparison of cross sections from Plate 1 beam simulations . .


. 123

. 123

. 124

. 125

. 125

. 126

127

. 127









5-39 Plate 2, Case 1: Comparison of axial strain (a.) countours . . .. 139

5-40 Plate 2, Case 1: Axial strains (e,) versus height at centerline: x=RD, y=TD 139

5-41 Plate 2, Case 1: Comparison of cross sections from experiment and simulation 140

5-42 Plate 2, Case 2: Comparison of axial strain (a.) countours . .... 141

5-43 Plate 2, Case 2: Axial strains (a.) versus height at centerline: x=RD, z=TT 141

5-44 Plate 2, Case 2: Comparison of cross sections from experiment and simulation 142

5-45 Plate 2, Case 3: Comparison of axial strain (Fy) countours . . 142

5-46 Plate 2, Case 3: Axial strains (Fy) versus height at centerline: x=RD, y=TD 143

5-47 Plate 2, Case 3: Comparison of cross sections from experiment and simulation 143

5-48 Plate 2, Case 4: Comparison of axial strain (Fy) countours . . 144

5-49 Plate 2, Case 4: Axial strains (Fy) versus height at centerline: y=TD, z=TT .. 144

5-50 Plate 2, Case 4: Comparison of cross sections from experiment and simulation 145

5-51 Deformed elliptical footprint obtained in high rate cylinder test. . ... 146

5-52 High rate compresive data with linear fit used in parameter identification 147

5-53 Comparison of yield values obtained from J-C law to experimental data ..... 148

5-54 Initial FE mesh for Taylor impact simulations .............. 149

5-55 Cylinder impact simulation results using isotropic von Mises and J-C hardening 150

5-56 Comparison of profiles from isotropic simulation ..... . . 151

5-57 Cylinder impact simulation results using anisotropic elastic/plastic model and
J-C hardening without rate effects .................. ..... 152

5-58 Comparison of deformed cylinder profile from two locations for simulation using
proposed anisotropic model with no rate effects ................. .. 153

5-59 Comparison of deformed cylinder profile from two locations for simulation using
proposed anisotropic viscoplastic model .................. ...... 153

5-60 Cylinder impact simulation results using anisotropic parameters for proposed
criteria using J-C hardening with rate effects .............. 154

5-61 Comparison of i i' i" profiles obtained using the different models ...... ..155

5-62 Comparison of minor profiles obtained using the different models ...... ..156

5-63 Comparison of the predicted foot print obtained using the different models 156









5-64 Comparison of deformed impact surface: FE simulations with viscoplastic model
to experimental data .................. .............. 157

5-65 Comparison of 1ni i' '. axis radial strain versus height predicted by the anisotropic
model and experimental data ............... ........ 158

5-66 Comparison of minor axis radial strain versus height predicted by the anisotropic
model and experimental data ............... ........ 158

5-67 Comparison of ratio of 1, ii' 'r to minor diameters versus heightpredicted by the
anisotropic model and experimental data ................ .... 159









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

EXPERIMENTAL CHARACTERIZATION AND MODELING OF THE MECHANICAL
RESPONSE OF TITANIUM FOR QUASI-STATIC AND HIGH STRAIN RATE LOADS

By

Michael E. Nixon

May 2008

C'!: i': Oana Cazacu
Major: Aerospace Engineering

This dissertation is devoted to the characterization, modeling and simulation

of plastic anisotropy and strength differential effects in high-purity, polil i-I 11 lw:

a-titanium.

A series of uniaxial compression and tension tests were carried out at room

temperature under quasi-static conditions to quantify the plastic anisotropy and strength

differential effects in the material. Pre- and post-test textures were measured using

neutron diffraction techniques and orientation imaging microscopy (OIM). The tests

indicated that initially both plates have strong basal textures, one of the plates studied

(Plate 1) being orthotropic, whereas the other one (Plate 2) has in-plane symmetry.

Significant texture evolution associated primarily with tensile twinning was observed only

for Plate 1 when subjected to compression in the rolling direction. Four-point bending

tests were performed for validation purposes. Digital Image Correlation techniques were

used to obtain the strain field. As a result of the anisotropy and directionality of twinning,

qualitative differences were observed between the response of upper and lower fibers in

different orientations.

Split Hopkinson Pressure Bar tests at strain rates of 400 to 600 sec-1 were performed

along the axes of symmetry of each plate to characterize the material's strain rate

sensitivity. A clear increase in strength with increasing strain rate is observed, the

hardening rate remaining practically unchanged for all directions, with the exception of









the rolling direction. The dramatic hardening rate increase in the rolling direction was

indicative of higher propensity for twinning with increasing strain rate. Taylor cylinder

impact tests on specimens cut from Plate 2 were performed at impact velocities in the

range of 200 m/s.

Based on presented experimental data, it can be concluded that the material has

a very complex anisotropic behavior and exhibits tension/compression .,i-v i i, I1 ry and

strain rate sensitivity. A new anisotropic elastic/plastic model was developed. Key in its

formulation is an yield criterion that captures strength differential effects. Anisotropy was

introduced through a linear transformation on the Cauchy stress tensor applied to the

material. An anisotropic hardening rule that accounts for texture evolution associated

to twinning was developed. It was demonstrated that the model describes very well the

main features of the quasi-static response of high-purity Ti when subjected to monotonic

loading conditions. Validation of the model was provided through comparison between

measured and simulated strain distributions in bending. In particular, the shift of the

neutral axis was well described. An extension of the model that incorporates rate effects

was also developed and used to describe the anisotropic high rate behavior of the material.

It was shown that the rate dependent model describes well the deformed profiles and final

cross section of the specimens.









CHAPTER 1
GENERAL INTRODUCTION

The importance of accurately modeling the deformation of materials has become

essential in the design and analysis of most manufactured products. It is often not good

enough to make the assumption that a material is isotropic and remains so even under

large plastic deformations. All materials can exhibit anisotropic behavior due to strong

textures related to a particular manufacturing process. Some materials, such as those with

a hexagonally close packed (hcp) i 1--I structure, exhibit strong anisotropic behavior

both at the single < i v-I 1 and pc.l-, i--I I1 level. Furthermore, such materials may display a

strength differential, or non-symmetry between tensile and compressive strengths.

This study involves both an in-depth experimental characterization of a high purity

a phase titanium and the development of a theoretical model to account for the observed

anisotropic behavior and for the clear .,-vmmetry between tensile and compressive

strengths. The proposed model was implemented into an explicit finite element code and

verified and validated against experimental data.

A brief overview of material modeling including the description of classical mathematical

theory of plasticity is given and a general description of titanium, its alloys, its uses

as well as a more detailed description of the ( i --I 1 structure and its effects on the

overall behavior. An extensive experimental effort was done to characterize the a phase

titanium material and provides validation data for the model implementation. The

experimental program includes uniaxial loading in both tension and compression for

different orientations. The experimental tests were carried out at room temperatures for

both quasi-static and high loading rates. A theoretical model is proposed and described

in detail as well as the integration procedure used to implement the model into an

explicit Finite Element (FE) code. Then the model implementation is verified against

uniaxial data using a single hexagonal computational cell. Next, the model is used to

perform quasi-static validation simulations of a four point bending test. The very good









agreement between simulations and experimental results demonstrate the model's ability

to capture not only the anisotropic behavior but the strength .,-i-iiiii., ry as well. Finally,

a viscoplastic form of the model is used in validation simulations of the classic Taylor

cylinder impact tests. The agreement between simulation and experiment show the ability

of the model to account for rate effects during anisotropic deformation.

1.1 Material Modeling

A constitutive model describes the deformation of a particular material when

subjected to a set of applied loads. All constitutive models idealize the real behavior

to some degree, i.e. the mathematical description models only certain aspects of the

behavior. Hooke's Law describes the behavior of an ideal elastic material, for which there

is a one-to-one relation between the applied loads (stresses) and the deformation (strains).

At higher levels of stress, the internal structure of the material is changed in such a

way that not all of the energy goes into the deformation and it can not be recovered

upon removal of the applied loads, i.e. plastic deformation occurs. Classic plasticity,

elasto-plasticity and visco-plasticity account for various aspects of this behavior. As usual,

current research builds on the foundations laid by these classic approaches and attempts

to refine existing models to incorporate more and more physical mechanisms as these

mechanisms are identified through physical and/or computational experimentation.

1.1.1 Elasticity

The generalized Hooke's law relates the nine stress components to nine strain

components by a linear homogeneous relationship aci = C ,.ikl. In general, there

are 81 constants Cijkl but this number can be reduced based on symmetry considerations.

Assuming that both the stress and strain tensors are symmetric the number of independent

constants is reduced to 36. With the further assumption that the material is elastically

isotropic there are only two independent constants and the law takes the familiar form:


aij = [ Aij1kl + P(6ik6jl + 6il6jk)lEkl









where A and p are Lame constants and 6ij is the Kronecker delta.


A concise presentation of elasticity matrices for various symmetries can be found in

Hosford (1993). A subset of these are shown in Figure 1-1. For example, for an orthotropic

material the tensor C has nine independent coefficients: C11, C22, C33, C12 C21, C13

C31, C23 C32, C44, C55, and C66.

triclinic orthorhombic hexagonal
****** ** * *
SS@.. S.





.. . .




eqc~ Equal values of c or s
S. + 2(s1- s2) or (c1- C)/2



Figure 1-1. Elastic coefficients required for various crystal symmetries


1.1.2 Plasticity

When materials deform beyond a certain amount the deformation is said to become

inelastic or plastic. In the material, the internal structure has changed in a fundamental

way by dislocation motion and/or twinning and the resulting deformation is no longer

adequately described by linear elasticity. The history of deformation must be accounted

for and there is no one-to-one correlation of a given stress state to a given strain field.

The dependence on the history of deformation is usually accounted for by way of internal

variables, which may or may not have a clear physical meaning. Examples include the

strain rate or a measure of the accumulated plastic strain. Internal variables that are not

directly measurable may also be introduced. It is generally assumed that the total strain









can be decomposed into an elastic, fully recoverable part, and a plastic part.
e e +
c j +71


The elastic portion of the strain can be described by Hooke's law. The description of

the plastic behavior involves: (1) the definition of the yield surface that separates the

elastic and plastic domain (2) a plastic flow rule that describes how the material flows as

it deforms plastically and (3) a hardening rule that describes how the yield surface evolves.

The yield surface characterizes the onset of plastic deformation. It is defined in the stress

space as

O(Jij)= Qo,

where Qo is a constant. For metals, plastic deformation is primarially the result of

dislocation and twinning, both shear mechanisms. Thus, it is usually assumed that

the yield surface is independent of the hydrostatic pressure, which implies that the yield

criterion depends only on the deviatoric stress defined as


Sij rij J ij.


1.1.2.1 Isotropic yield surfaces

For rate-independent materials, the stress can never be outside the yield surface and

the material flows only when the stresses are on the surface. In the simplest approaches,

i.e. rigid-plastic models, the yield surface is fixed. However real materials can harden

or soften. Thus, it is necessary to account for the distortion of the yield surface with

accumulated deformation. Isotropic hardening models that allow the surfaces to expand

(or contract) equally in all directions have been proposed. More complex hardening

models such as kinematic hardening models allow for the yield surface to translate but

remain at a fixed size; mixed hardening models allow for both expansion and translation.

For anisotropic materials the surface may also distort or anisotropically harden.









Tresca(1864) proposed the first widely accepted yield surface. It assumes that there is

plastic deformation if the maximun shear reaches a threshold If r denotes the yield stress

in pure shear, the Tresca criterion is

( 1 0-2 (72 (73 03 1 \
max ( 3 3 a3 1- 7 (1-1)


where a1, a2, o3 are the principal values of the Cauchy stress a. Figure 1-2 shows the

biaxial projection of the surface.


02






OY






Figure 1-2. Projection of Tresca yield surface on the plane a3 = 0


Saint-Venant built on the work of Tresca and Levy (1870) and laid the foundations

for the mathematical theory of plasticity. Huber (1904) proposed a relationship for the

constant distortion energy criterion which was latter proposed by von Mises (1913) as

an approximation of Tresca. This has been the most widely used yield surface due to

its simplicity and accuracy for many materials. The Hubner-Mises yield criterion can be

written as


(ayy a)2 + (a a)2 + (a oyy)2 + 6(< + x + T) 2,7 (1 2)

where au is the yield stress in uniaxial tension. This criterion involves only the second

invariant of the deviatoric stress J2. Drucker (1949) proposed including effects of the third









invariant of the stress deviator, J3, on yielding


J cJ3 = 6


A fairly versatile extension, written in principal stress space, was proposed by Hershey

(1954).

a aai a o
1a1 721' + I2-3a 3 o- I1 Sl1 + IS2 S3l + I3 Sl 2o7 (1-3)


where si, 2, S3 are the principal values of the stress deviator. With appropriate assumptions

for the exponent a, this criterion reduces to either Tresca (a = oo) or Hubner-Mises

criterion (a = 2)

1.1.2.2 Anisotropic yield surfaces

All real materials exhibit some anisotropic behavior and when this behavior is

significant enough models for that behavior must incoprorate the anisotropic nature of

yielding. A great number of descriptions have been proposed with varying degrees of

success. A thorough treatment is given by Zyczkowski (1981) where more than 200 yield

surface descriptions are discussed. Only a few significant models will be discussed here.

The most widely used anisotropic yield description was presented by Hill (1948) and is

given by


F(a- a,)2 + G(7, a,)2 + H(a, ay)2 + 2L' + 2Mr2 + 2N'r 1 (1 4)


where the coefficients F,G,H,L,M, and N are constants and x, y, and z are the orthotropic

axes. Hill later presented the following criterion for material with low r values


a = Fa2 a3 + GCa3 a1 + HNa1 a1 +

L|2ai a2a3 m + M|2a a3 a1 m + N2a73 a1 m7 (1-5)









Table 1-1. Phenomenological yield functions
Yield Criterion Type
Tresca Isotropy
Von Mises Isotropy
Hill (1948) Planar Anisotropy
Hershey (1954) Isotropy
Hosford (1972)
Gotoh (1977) Planar Anisotropy
Bassani (1977) Planar isotropy
Hill (1979) Planar Anisotropy
Logan and Hosford (1980) Planar Anisotropy
Budianski (1984) Planar isotropy


Shear


yes


yes

no
no
no


Dimension


6


3

2

2


where the coefficients F,G,H,L,M,N and m are material constants and a-, -2, and a3 are

the principal stresses coincident with the orthotropic axes. A 1n i, i" limitation of this

criterion is the inability to account for any state involving shear stresses.

A table of important phenomenological yield functions that describe orthotropic

behavior taken from Barlat et al. (1991) is shown in Table 1-1. Some isotropic functions

are also included. The column labeled "'!., 1 indicates if shear terms appear in the

formulation. The "Doi, ii-. i column gives the number of stress components involved in

the formulation.

Barlat et al. (1991) extended the isotropic Hershey and Hosford model (see Equation

1-3) by applying a fourth order linear transformation operator on the Cauchy stress

tensor. The orthotropic criterion is


a = (Si C)rn + (y s)r + (s 1)r


(1 6)


where 1i, E2 and E3 are the principal values of the transformed Cauchy stress tensor


E La


(1-7)









Here, L is a 4th order tensor with orthotropic symmetry. Barlat proposed an improved

version of this criterion in 1997


a3(Si E2)m + aC( 2 :3)m + a2(3 1)m =- 2Y (1-8)

where a1, a2 and a3 are functions of the principal directions of E. Barlat has shown

that the linear transformation approach can be used to write any pressure independent

isotropic yield function in terms of the principal values of the Cauchy stress deviator.

Cazacu and Barlat (2003) and Cazacu et al. (2004) showed that one can extend

any isotropic criterion to anisotropy through generalized invariants using the theory of

representation of tensor functions. Using this approach they extended Drucker's isotropic

yield criterion to orthotropy as follows


(J)3 C(j2 = F (19)

where J2 and J3 are polynomials in terms of the Cauchy stress and independent of

pressure and respecting the orthotropic symmetries.

However, none of the approaches above can account for a strength differential between

tension and compression which is exibited by hcp materials. Hosford and Allen (1973)

used poly crystalline calculations to investigate the strength .-i-iiiii I ry in isotropic

materials and sI-:-, -. .1 that the .-i- ii. 1,1 ry was caused by twinning which is sensitive

the sign of the shear stress. Most recently, models have been proposed that allow for

an .-i-viiii.i I y between the strength in tension and in compression. Cazacu and Barlat

(2004) proposed an isotropic criterion (Equation 1-10) involving both the second and

third invariant of the stress deviator that can account for a strength .,-vi ii ., I ry and

they favorably compared this theory to the data given by Hosford and Allen (1973).

The proposed model extends the isotropic description of Cazacu and Barlat (2004) to

orthotropy using a linear transformation on the C, 1.r!: stress.


f = (J2)3/2 cJ3 = (1-10)









1.1.2.3 Flow rules

The evolution of the plastic deformation is given by a flow rule


S A ni (1-11)


The usual assumption is that direction of plastic deformation can be derived from a plastic

potential G(a) such that ni = a(. The flow rule is called associative if the plastic

potential function coincides with the yield function and non-associative otherwise.

Using the assumption of an additive decompostion of the total strain increment one

can write



e + CP C: + A fi

In the above equation, A, is a scalar and is non-zero only if plastic deformation occurs i.e.:

For plastic loading: A > 0, f = 0 and j > 0

For neutral loading: A = 0, f = 0 and j = 0

For elastic loading: A = 0, f < 0.

This is usually stated in a compact form as the Kuhn-Tucker conditions


Af 0, A >0, < 0 (112)


1.1.2.4 Hardening

As a metal is deformed plastically there is an increase in dislocation density resulting

in a strengthening or hardening of the material. One manifestation of this is in the

positive slope of the stress-strain curve although at a much lesser angle than in the

elastic region. For some materials twinning can pl i,- a significant role in hardening and

cause the yield surface not only to change in size but also distort significantly or harden

anisotropically. Of course the material can soften and show a decrease in the slope as well,

especially at higher temperatures or as the material accumulates internal damage and is

less capable of carrying a load.









Hardening can be described by one or several internal variables that may or may

not have physical meaning. An obvious and often used internal variable is the amount

of effective plastic strain that has been imparted to a material. A simple but very useful

empirical model for materials that have not undergone a lot of plastic deformation is the

Ramberg and Osgood (1943) model



S + K (113)

where is the effective plastic strain, a is the equivalent stress, E is Young's modulus and

K and m are material constants. Observing that the a/E is the elastic strain and with

K(a/E)m accounting for the plastic strain, the relationship can be rewritten in terms of a

yield stress ay and a new parameter a = K(ay/E)-1 as

C+ d y- (114)
E E (ay)

Ludwick (1903) presents a form that neglects elastic strains as


a = COr (11-5)


where C = ay(E/acry)", n is the work-hardening exponent related to m of Equation

(1-13) by n = 1/m. A\ i,: hardening rules that account for a particular material or

loading environment have been developed. For example, hardening at very high loading

rates can be described by the phenominological model of Johnson et al. (1997) that

incorporates Equation 1 15.



a = (y + a) (1 + b In*) (1 16)

where ay is the initial yield strength, i* is the dimensionless total strain rate j/ o. The

reference strain rate is taken as ,o = 1. Equation 1 15 is the second term in the first

bracket. The constants ay, a, b, and n are determined from experimental tests. The

full Johnson-Cook model includes terms not given here that account for the effects of









temperature and hydrostatic pressure. A more detailed description of the Johnson-Cook

model is given in Chapter 5 where it is used in the visco-plastic implementation of the

anisotropic model proposed in C'! lpter 4.

No simple relationship exists to describe anisotropic hardening. For materials with

only a slight anisotropy, the usual assumptions of isotrpoic or kinematic hardening

may be sufficient but for highly anisotropic materials some other approaches need to

be introduced. Plunkett et al. (2006) developed and demonstrated an interpolation

methodology that uses a reference hardening path and a series of yield surfaces established

at discrete levels of accumulated plastic strain. This is the approach used in the

implementation of the orthotropic model developed as part of this dissertation and is

described in detail in section 4.3.









CHAPTER 2
TITANIUM

2.1 Basic Properties

Although titanium is the ninth most abundant element on Earth, it was not

discovered until just before 1800 and has come into wide use only since the 1950's.

The 1i i, ri" producers of titanium are Australia, Canada, Norway, South Africa, Russia and

the United States. It provides high strength performance at a lower density than steels.

It has excellent corrosion resistance and performs well in high temperature environments.

Widely used in the aircraft industry it also has been extensively utilized in gas turbines,

for joint replacements and in the chemical industry. Titanium can be alloi, .1 with many

elements to tailor its performance to a wide variety of purposes. In the US about a

quarter of titanium production is pure titanium and more than half is Ti-6A1-4V according

to Lul! i -ii:. and Williams (2003). Mechanical and physical properties of titanium are

shown in Table 2-1 from Donachie (2000). Pure titanium melts around 16600 C. At room

temperature its crystal structure is hcp (known as a phase) but at 882 C there is an

allotropic phase transformation to a bcc structure (3 phase). When alloi-, .1 with elements

such as aluminum, oxygen and nitrogen, the a phase can be stabilized even at high

temperature. When alloi, .1 with other elements such as molybdenum, iron or vanadium, it

can be stabilized in 3 phase even at room temperatures. [Sergueeva et al. (2001)]

High purity titanium was used as the material of choice for this study for various

reasons. It exhibits a strong anisotropic behavior, is widely available and is used in many

applications of interest in both commercial and defense industries [Donachie (2000);

Lutjering and Williams (2003)]. As in the case of other hcp materials its behavior is

strongly influenced by its hcp cyrstalline structure. There is a competition between slip

and twinning that accounts for much of the anisotropic deformation. The hcp crystals are

much easier to deform in certain crystallographic directions than others. In particular,

titanium does not have enough easily activated slip systems to accommodate arbitrary









Table 2-1. Physical properties of Titanium
Property
Atomic number
Atomic weight
Atomic volume
Covalent radius
Ionization Potential
Thermal neutron absorption cross section
Crystal structure
Alpha (<882.5 o C, or 1620 o F)
Beta (>882.5 C, or 1620 F)
Color
Density
Melting point
Solidus/liquidus
Boiling point
Specific heat (at 25 0 C)
Thermal conductivity
Heat of fusion
Heat of vaporization
Specific gravity
Hardness
Tensile strength
Young's modulus
Poisson's ratio
Coefficient of friction
At 40 m/min (125 ft/min)
At 300 m/min (1000 ft/min)
Coefficient of linear thermal expansion
Electrical conductivity
Electrical resistivity (at 20 0 C)
Electrogativity
Temperature coefficient of electrical resistance
Magnetic susceptibility (volume at room temperature)


Description or value
22
47.90
10.6 W/D
1.32 A
6.8282 V
5.6 barns/atom

Close packed hexagonal
Body-centered cubic
Dark gray
4.51 g/cm3 (0.163 lb/in3)
1668 10 C ( 3035 F )
1725 o C ( 3135 F )
3260 C ( 5900 F )
0.5223 kJ/kg 0 K
11.4 W/m o K
440 kJ/kg (estimated)
9.83 MJ/kg
4.5
70 to 74 HRB
240 MPa ( 35 ksi) min
120 GPa (17 x 10 psi)
0.361

0.8
0.68
8.42 pi m/m o K
:, IACS
420 nQ- m
1.5 Pauling's
0.0026 /0 C
180(1.7)x10-6mks


deformation by slip [Salem et al. (2003); Nemat-Nasser et al. (1999)], therefore twinning

p1l ,i- an important role in the plastic deformation. This leads to s strength differential

effect since twinning is a directional shear deformation mechanism.

Several investigators have studied various aspects of the behavior of titanium and

its alloys. Gray (1997) studied the effects of strain rate and temperature in high purity

a-titanium but only for compressive loadings. Kalidindi and others [Kalidindi et al.










(2003); Li et al. (2004); Nemat-Nasser et al. (1999); Salem et al. (2002, 2003, 2004a,b)]

studied a high purity titanium plates similar to those used in this study but only for a few

loading paths and/or strain rates. One of the goals of this dissertation is to extend the

current knowledge by investigating a wide range of loading paths in order to more fully

characterize the behavior and to serve as a basis for development of improved material

models.

2.2 Single Crystal Properties

For pure Ti, the crystal lattice parameters correspond to c/a ratio of 1.587, which

is smaller than the ideal ratio of 1.633. The single i i--I I1 is highly anisotropic. The

elastic properties vary strongly with orientation. Figure 2-1 from Zarkades and Larson

(1970) shows the variation of the elastic modulus for various orientations at room

temperature. The modulus varies from 145 GPa along the c-axis to 100 GPa in the

direction perpendicular to the c-axis. There is a similar variation for the shear modulus.

The variations of these moduli in a pf li-,1 i --i 11iii:..i:-.-egate would of course also depend

on the variation of texture (Lutjering and Williams (2003)).



150


140 --


130


100
0 0 40 60 80
Declination angle


Figure 2-1. Variation of Ti single <- i--i I elastic modulus









2.3 Deformation Mechanisms

Figure 2-2 shows the three most densely packed types of planes, the basal plane

(0002), one of the three prismatic planes {1010 } and one of the six pyramidal planes.




2.3.1 Slip


(1011) ..





(0002)
a2

Figure 2-2. Titanium crystal structure


2.3.1 Slip

The primary slip directions are the three close packed (1120). The associated slip

planes are the three {1010} planes, the six {1011} planes and (0002) plane, for a total

of 12 slip systems. However, there are only 4 independent slip systems [Lutjering and

Williams (2003)]. The prism planes and basal (a) directions constitute the most favorable

slip while the basal planes and pyramidal planes in combination with appropriate

directions constitute the other probable slip systems. Since all of the slip systems have slip

directions that are restricted to the basal plane, they do not provide the five independent

slip systems necessary to accommodate arbitrary plastic strains [Gray (1997); AT, i-, rs

et al. (2001)]. This indicates that twinning can 1 'l a significant role in the deformation of
titanium.









2.3.2 Twinning

The primary twinning modes observed in pure a titanium are {1012}, {1121}, and

{1122}. Twinning modes are especially important for ductility at low temperatures if

the stress axis is parallel to the c axis and dislocation motion is inhibited. For this case

{1012} and {1121} twins are activated during tensile loading giving an extension along the

c-axis. The most frequently observed twins are of the {1012} type and the {1122} twins

activated under compression loading parallel to the c-axis which gives rise to contraction

along the c-axis (See Fig. 2-3). In comparison to other hcp materials, titanium is quite

ductile because it has more twinning and slip systems. Twinning can be suppressed

by alloying it to give higher strengths and can be retarded by interstitials found in

lower purity titanium. Because solute atoms suppress twinning, it is a in, ,ir pl .i, r in

deformation for pure titanium with low amounts of oxygen [Lutjering and Williams (2003)]














A B

Figure 2-3. Active twinning systems in Ti: A) Tensile {1012} B) Compression {1122}


2.3.3 Hardening

The deformation of hcp materials and titanium in particular is very complex. Strain

hardening is not only a function of dislocation movement, i.e. slip, but is strongly

influenced by twinning. The degree of twinning is dependent on the size of grains, the

texture of the material, temperature and rate of deformation and possibly other factors.









One effect of twinning is the reduction of effective slip distance which is effectively making

the grain size appear smaller, the Hall-Petch effect, and raises the hardening rate[Gray

(1997); A. i-, rs et al. (2001)]. Salem et al. (2006) observed evidence of the same Hall-Petch

effect and also showed evidence of two other effects on hardening resulting twnning. By

performing macro and micro- hardness tests, these authors showed that the twinned

regions are immediately harder than the bulk or matrix material. They attributed this

effect to sessile dislocations being trapped inside twinned regions (Basinski mechanism).

Thirdly, they found that there was softening from reorientation of the twinned region into

an orientation more aligned with easy slip.

Nemat-Nasser et al. (1999) -,i-.-, -1.I that an increased strain hardening rate is

associated with dynamic strain aging but other studies dispute this idea and indicate

that deformation twinning accounts for the change in strain hardening [Salem et al.

(2002)]. Gray (1997) states explicitly that the roles of slip and deformation twinning in

titannium are so intertwined that both effects must be accounted for in any physically

based constitutive model.

2.4 High Purity Titanium Plates

This research will be concerned primarily with high purity (99.9',' .) titanium whose

chemical analysis is shown in Table 2-2. Hardness tests were performed on this material

in plate form. The average hardness was of 43.1 HRB. This is a much softer material

than that reported in Table 2-1 which has a hardness of 70 to 74 HRB. The typical grain

structure for the material is shown in Figure 2-5. It shows somewhat equiaxed grains with

an average grain size of about 20 pm.

Two round plates of the material, 10 inches in diameter and 5/8 inch thick (see

Fig 2-4) were purchased from Alpha Aesar (A Johnson Matthey Company). The plates

were described as cross rolled, 99.9''1' pure, but no rolling direction was indicated. The

anisotropic texture was established via electron microscopy.









Table 2-2. C('!, I., II analysis of test material: Titanium metal disk 10 inch diameter,
x 0.625 inch thick, 0.010 inch, w 32RMS Surface o.b., cross rolled with 1 inch
square sample 99.9'1'1'.
Ag <0.05 Al 0.4 As <0.01 Au <0.05
B <0.01 Ba <0.005 Be <0.005 Bi <0.01
Br <0.05 C 10.5 Ca <0.2 Cd <0.05
Ce <0.005 Cl 0.105 Co 0.008 Cr 0.55
Cs <0.01 Cu 0.19 F <0.05 Fe 5.5
Ga <0.05 Ge <0.05 H 1 Hf <0.01
Hg <0.1 I <0.01 In <0.05 Ir <0.01
K <0.01 La <0.005 Li <0.005 Mg <0.05
Mn 0.0575 Mo <0.05 N <10 Na <0.01
Nb <0.2 Nd <0.005 Ni 0.11 O 156.5
Os <0.01 P <0.01 Pb <0.01 Pd <0.01
Pt <0.05 Rb* <5 Re <0.01 Rh <0.15
Ru <0.01 S <5 Sb <0.05 Sc <0.05
Se <0.05 Si 0.3 Sn <0.05 Sr* <3000
Ta** <5 Te <0.05 Th <0.0005 Tl <0.01
U <0.0005 V 0.135 W <0.01 Y* <200
Zn <0.005 Zr 0.6
Note: Values given in ppm unless otherwise noted. Carbon, hydrogen, nitrogen, oxygen
and sulfur determined by LECO, all other elements determined by GDMS
* Ion interference
** Instrument contamination

Figure 2-6 shows the through thickness (0002) pole figure which indicates no clear

anisotropy in the through thickness direction. A clear directionality is seen in the pole

figure for in-plane texture as shown in Figure 2-7. The rolling direction determined from

the texture measurements was marked on the plate shown on the right in Figure 2-4.

Test specimens were cut from the plate using Electrical Discharge Machining (EDM) for

different orientations relative to the established RD and the normal of the plate.

























Figure 2-4. Titanium plate A) as received B) With coupons cut and rolling direction
established


Figure 2-5. Micrograph of high purity Titanium plate material

































Figure 2-6. Plate 1 pole figure with center in TT direction


Figure 2-7. Plate 1 pole figure with center in RD









CHAPTER 3
EXPERIMENTS

An experimental investigation on the behavior of the high-purity, pc.li, i--i 1 11!ii

a-titanium plates described in C'!i pter 1 was carried out for both quasi-static and high

loading rates at room temperature. These tests were used to quantify the anisotropic

behavior, including the strength differential between tension and compression, for each

plate. It was observed that the response of Plate 1 is orthotropic and highly dependent on

the direction and sense of the applied load. Plate 2 is nearly isotropic in the plane of the

plate but has strong basal texture, which results in marked difference in response between

in-plane and through-thickness directions. Four-point bending tests were also performed

on beams cut from each plate in four configurations. A speckle pattern was deposited

on one profile of each beam and Digital Image Correlation (\!iguil-Touchal et al. (1997);

Hung and Voloshin (2003)) techniques were used to ,i, ,i-. .. the strain field. As a result of

the plate anisotropy and directionality of twinning, qualitative differences were observed

between the response of the upper and lower fibers of the different bent beams. The

beams were cut at the midpoint and the cross sections were observed and compared to

simulations for each loading orientation. The results indicate the need to use a constitutive

description for the material that accounts for the interplay between slip and twinning and

its effects on texture evolution and hardening response when simulating the behaviour of

Titanium.

Pre- and post-test textures of specimens were measured using neutron beam

techniques at the HIPPO facility at the Los Alamos National Laboratory (LANL).

Quasi-statically deformed samples from Plate 1 were also analyzed using Orientation

Imaging Microscopy (OIM). Significant texture evolution was observed only for compression

in the rolling direction. Both the OIM and neutron beam measurements revealed a high

volume fraction of twinned grains, the primary twin family being tensile twins.









3.1 Quasi-Static Tests


3.1.1 Characterization Tests

3.1.1.1 Test description

The quasi-static characterization tests for both plates consisted of uniaxial tension

and compression tests at a nominal strain rate of 0.001 per second. An Instron 1125

testing machine was used with an Instron 100 kN load cell for compression tests and

5 Kn load cell for tensile tests. An Instron extensometer model number G-51-17-A

with a gauge length of 12.7 mm was used for compression tests and an Instron model

number G-51-12-A extensometer with a gauge length of 25.4 mm was used for tensile

tests. To examine the effect of loading orientation on the mechanical response of these

two strongly basal-textured titanium plates, cylindrical compression specimens (0.3 x

0.3 in) were machined such that the axes of the cylinders are either in in-plane (IP) or

through-thickness (TT) plate directions (see Figure 3-2). For both plates, IP samples

were cut at 0 450 and 900 orientations to the rolling direction and labeled as in Figure

(3-4). Tensile tests in the IP directions were conducted using classical dog-bones shape

samples (Figure 3-3). A specialized miniature test specimen was used for the TT tests

(Figure 3-1). In order to examine the microstructural evolution at different levels of plastic

deformation as well as determine the Lankford coefficients, the tests were carried out

to approximately 1(' 21' 311' 4(0' strains respectively or until complete failure of

the specimen occurred. All IP specimens were labeled relative to the orientation with





SG a d o t -






Figure 3-1. Geometry and dimensions of the through-thickness tensile specimen











Lube Trap


S 32RMS Finish

+_ -----
.300


A B

Figure 3-2. Quasi-static compression specimens A) dimensions B) specimen with lube trap



-I ,""I F-, -- T,



S...______.__.__.- I






Figure 3-3. Geometry and dimensions of quasi-static in-plane specimens for tension


respect to the rolling direction (RD) as shown in Figure 3-4.


Figure 3-4. Definition of the specimen orientations relative to the established rolling
direction.










3.1.1.2 Plate 1 results

Quasi-static compression test results along the RD are shown in (Figure 3-5). For

Tests 301 and 401, the curves are not smooth at higher strains because the test specimens

for these tests included a small lube trap at one end (see Figure 3-2 B)). This was filled

with Moly grease in an effort to minimize friction at the platen faces. Subsequent tests

without the trap, using only Molykote lubricant, showed that friction was not a problem

and later tests did not include the trap. For all tests at lower strain levels, the lube trap

was not included. Note that strain-hardening is not linear. There is a distinct hump or

change in the slope of the stress-strain curves at about 1(' strain. This increase in the

strain-hardening rate may be associated with the onset of twinning. This hypothesis was

verified by subsequent OIM observations of the deformed specimens.


500


400


S 300 -------


200
Test Number
401
100 301
201
101

-0.4 -0.3 -0.2 -0.1 0
Strain

Figure 3-5. Results of quasi-static compression tests along the RD conducted at
0.001 sec-1 on Plate 1


The OIM map of the specimen deformed to 1(1' strain reveals that many grains have

twinned (twins appear red in Figure 3-6 ). The twin volume fraction was estimated to

be 17'. Figure 3-7 shows an OIM map corresponding to 211' strain, which indicates a

high volume fraction of twinned grains, about ,11' No other loading path produced this

level of twinning activity. These results are consistent with previous observations reported
























Figure 3-6. Orientation Imaging Microscopy map showing the evidence of twins (in red)
in sample from Plate 1 deformed to 10' strain in simple compression along
the rolling direction. The twin volume fraction is 17'.


Figure 3-7. Orientation Imaging Microscopy map showing significant twinning activity
(45' volume fraction) in sample from Plate 1 deformed to 21i i' strain in
simple compression along the rolling direction


by Salem et al. (2002, 2003, 2004b,a) on pcli ,i v-i i11iiw. a-titanium of similar purity.

Furthermore, as in the case of Plate 1 material, the maximum twin volume fraction was

observed in simple compression to 21i' strain, the reported volume fraction being of 45'

The stress-strain response under quasi-static tension in the same orientation (RD)

up to 10, 20, and 30 strain respectively, is shown in Figure 3-8 A). The initial yield

stress (at 0.'"-. offset)is about 175 MPa. The material gradually hardens until plastic











localization (necking) occurrs at strain levels of around :in'-. strains. A shear-type fracture

was observed. In contrast, tensile fracture of magnesium alloy AZ31B, which also has

hcp crystal structure is brittle (see data by Lou et al. (2006)). OIM observations for a

specimen deformed up to 30 strain show that most grains have less twins, although

some twinning is evident. Comparison between compression and tensile stress-strain curves

along the rolling direction (Figure 3-8 B)) shows a very large .,i-, iii I1 ry in hardening

evolution. Although, initially there is no significant difference in yielding behavior, at

about 7.5'. an especially sharp difference in response is observed. This striking strength

differential effect correlates with the onset of twinning in the compression sample.


Ca
vI


350

300 -------___'-

250 __ _

200

150

100 __ est Number
_- 101
50--- 201
50 ----------- -- 301 -
--- 301
C L .......L. I


rA


0 0.1 0.2 0.3 0.4 U 0.1 0.2 U.3 0.4
Strain Strain
A B


Figure 3-8. Results of quasi-static loading tests along the rolling direction conducted on
Plate 1 at 0.001sec-1. A) tensile tests B) Hardening in tension and
compression



Quasi-static test results in monotonic uniaxial tension and compression along the

transverse direction (TD) are shown in Figure 3-9. Notice that the stress-strain curves in

compression along the TD do not show the features present in the stress-strain response

in the RD compression. No significant change in strain-hardening is observed, which

correlates with minimal deformation twinning. Post test analysis using OIM confirms that

the tendency to twin is directional. There is little twining activity in compression (less

than 5' volume fraction) along TD as compared to the RD.















400


G, 300
200 0

S 15o0
200
100 Test Number Test Number
101 101l
201 10oo0 201
50 --301 -- 301
-- 401
0 1 ..0 I I .
0 0.05 0.1 0.15 0.2 5 -0.4 -0.3 -0.2 -0.1
Strain Strain

A B


Figure 3-9. Plate 1 results at 0.001 sec-1: A) Results of quasi-static tensile tests along the
TD B) Results of quasi-static compression tests along the TD



A comparison of tension versus compression response along the transverse direction


is shown in Figure 3-10. Again, there is little strength-differential effects in initial yielding


but strong ..- i :I I ry is observed after 1.5' strain.


,UU



400








200



100 Compression
I ension



o 1 .. .


0.1Strain
Strain


0.2 U.25 u.3


Figure 3-10. Hardening during uniaxial tension and compression in the TD for Plate 1



Quasi-static test results in monotonic uniaxial compression and tension along


the through-thickness direction are shown in Figure 3-11. There appears to be very


0.035












little variation in strain-hardening rate, which would indicate no significant amounts of

twinning.


350

300 0

250

200

150
Test Number
100 101
_- 201
50 301
S401


Test Number
SS 101
50- 201
--- 301
400-- 401
400 --------- __------


300 -


200


100


0 0.05 0.1 0.15 0.2 0.25 -0.4 -0.3 -0.2 -0.1
Strain Strain
A B


Figure 3-11. Results of quasi-static tests at 0.001 sec-1 along the through thickness
direction conducted on Plate 1 A) tensile B ) compression



Comparison between through-thickness uniaxial compression and tension stress-strain

curves (Figure 3-12) shows a strong tension/compression .-i-mmetry in initial yielding

and hardening behavior. This marked difference in response shows the strong dependency

between deformation mechanisms and loading conditions. Compressive loading is applied

essentially perpendicular to the basal plane, thus favors deformation twinning over

non-basal slip.

In conclusion, the various measured tensile yield stresses show in-plane anisotropy.

An anisotropy ratio for initial yield stress defined by the ratio of the yield stress in

the through-thickness direction (the largest) to that in the transverse direction (the

smallest value) is 1.27. The yield stress anisotropy in compression is 1.18, smaller than in

tension. This observed variation in compressive flow stresses is consistent with previously

observed results for po, li ii-I -11iii anisotropic hcp materials; Lou et al. (2006) on AZ31B

magnesium and Kaschner and Gray (2000) on Zirconium. The larger compressive flow

stress in the through-thickness direction as compared to the in-plane orientations is


)


(














500


400


300


200


100


n


0 0.1 0.2 0.3 0.4
Strain

Figure 3-12. Hardening in tension and compression in the TT direction for Plate 1


due to the strong basal texture of the material. For TT compression the load is applied

essentially perpendicular to the basal plane; thus plastic flow is achieved for higher stresses

than for in-plane samples which have more favorable conditions for activating prism,

pyramidal, and basal slip. Tensile deformation is slip-dominated hence the anisotropy

in yield stresses is stronger than twinning-dominated deformation, which is observed in

compression.

3.1.1.3 Plate 2 results

Measurements of the initial texture have shown that this plate di-pl~ 'i nearly

isotropically in-plane symmetry, while the c-axis of the grains are predominantly aligned at

150 -20 from the through thickness direction of the plate. To investigate the influence of

the loading orientation and thus of the texture on the response, compression samples were

cut in the through-thickness direction and at different in-plane directions. The in-plane

orientations were: 0 (rolling direction), 22.50 450 67.50 and 900 from the established

rolling direction. Based on the in-plane compression test results (see Figure 3-13 A)) it

can be concluded that indeed there is little in-plane anisotropy. For use in identifying the

parameters of the proposed model, all of the data for the in-plane quasi-static compression


-





Compression
Tension


I 'l lI I I I I I I











tests were averaged. The average in-plane response is depicted as solid black line in Figure

3-13 A). Note that this stress-strain curve indicates non-linear strain hardening which may

be indicative of twinning activity. Further, OIM investigations need to be performed in

order to verify this hypothesis.


500


400 -- ----- 300 ---- _

300 10
-200
In-planc
200 Direction 150
0
22 5
45 100 -- RD3
-- RD6
100 67.5 TDI
90 50 TD6
-- Avg -- HR Specimen

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.05 0.1 0.15 0.2
Strain Strain
A B

Figure 3-13. Plate 2 quasi-static in-plane data A) compression data B) tension data



Due to the in-plane isotropy established through texture measurements and

mechanical tests in compression, tensile tests were performed in only two in-plane

directions: at 0 and 900 from RD, respectively. The results of these tests are shown

in Figure 3-13 B). There is clearly more spread in the data than in compression. A

possible explanation of this is that the tensile specimens are relatively thin and are taken

from different locations along the thickness of the plate. If there is a gradient of texture

with the thickness, some specimens would be softer while some harder than others. Thus,

further texture measurements throughout the thickness of the plate need to be performed.

Such measurements have been performed for Plate 1 showing a noticeable texture gradient

though the thickness of this plate. One quasi-static test was made using the cylindrical

high rate specimen shown in Figure 3-34 B) and labeled "HR Specimen" in the figure.

This was done in an attempt to reduce the variation due to the position in the thickness

direction. The cross section of the high rate specimen (0.049 in2) was significantly larger










than for the flat tensile specimen (0.0078 in2). The results show that the high rate

specimen was stronger and more ductile. Results from this test were used in identifying

parameters for the proposed model.

The average uniaxial compression stress-strain curve versus the quasi-static tensile

data gathered using the high rate specimen are shown in Figure 3-14. The material

strength is similar in tension and compression up until about 1;:'. strain where the

compressive strength is larger. The strength differential becomes increasing larger at

higher strains.

600


500 -'_


400

300


200


100 Compression
S Tension

0 0.1 0.2 0.3 0.4 0.5
Strain

Figure 3-14. Plate 2 comparison of in-plane quasi-static tension versus compression data


Data for compression and tension tests from the TT direction of Plate 2 are shown in

Figure 3-15. As for Plate 1, the smaller tensile test specimen configuration was used due

to the limits of the thickness of the plate.

Figure 3-16 shows the comparison of the through thickness tension and compression

data. There is a significant strength differential from the beginning and both show a small

secondary yield point similar to that found in many steels.











700 400

600

300
500-

400 I
200
300

200 100

100


0 0.1 0.2 0.3 0.4 0.5 0 0.05 0.1
Strain Strain

A B

Figure 3-15. Plate 2 quasi-static data for TT: A) compression B) tension


Figure 3-16. Plate 2 comparison of TT quasi-static tension versus compression data



3.1.2 Four Point Beam Bend Tests

In order to validate the model developed to describe the material, four point beam

bend tests were carried out for each plate. Four beams were cut from each plate, two with

the long axis along the RD and two with the long axis along the TD. The beam length

was 57.15 mm with a square cross section of 6.35 mm as shown in Figure 3-17 A. For each











set, one beam was loaded in the through thickness direction and one in the plane of the

plate, normal to the beam axis. The four test configurations are shown in Figure 3-17 B.

END VIEW Case 1
TOP of PLATE x ( RD)
M 253216
Inplane Rolling Direclon s se y (TD) Loaing Direction

SideView I z (TT) Loading Direction
i63 Case 2
1 _L Hx ( RD)
BOTTOMofPLATE $15 77

57.15
END VIEW Case 3
TOP 01 PLATE 7 .............................................T D )

Transverse 90 to Rolling Direction a [ x ( RD) Loading Direction
z (TT)T
SideVew 9I 1 z Loading Direction
di rt ~ Case 4
,TD)
BOTTOMofPLATE


A B

Figure 3-17. Four point beam test specimens: A) Specimen dimensions B) Orientation
definitions: Case 1 and Case 2 have long axis aligned with the rolling
direction (x) Case 3 and Case 4 have the long axis aligned with the
transverse direction (y)


The testing jig is shown in Figure 3-18 including a test specimen. The two upper pins

were displacement controlled to approximately 5.5 mm. A typical load pin displacement

path is shown in Figure 3-19.

Along one side of the test beam, a speckle pattern was 'i .iv 1 and digital image

correlation or DIC (\ !i, m!-Touchal et al. (1997); Hung and Voloshin (2003)) was used

to determine the strain field after deformation. The image taken had 88 pixels along the

short direction of the beam. The beam dimension in that direction is 6.35 mm. therefore,

the physical distance between pixels is 6350 micron / 88 pixel = 72 micron/pixel. The

method can detect displacements of 0.01 pixel, therefore the error is less than 1 micron.

The strain field corresponds to the grid pattern set up on the undeformed speckle

field. The displaced field was used to map the strain field from the measurements from

the deformed speckle pattern using DIC. A typical undeformed and deformed grid are










.125"X 12" DOWEL PI


..... ...



125X 1-1 .125" DOWEL PI




"*'- 0 1625X 114" ROLL PINS =


A B

Figure 3-18. Four point beam test jig A) loaded with test specimen B) giving dimensions
and pin placements


2
3
2


Figure 3-19. Typical Load vs Displacement curve for bend tests: Left axis is pin
displacement in mm, right axis is load in kN plotted versus time on the
horizontal axis.


shown in Figure 3-20. Note that he grid and subsequent strain field does not cover the

entire speckle field. The deformed specimens were cut at the mid point along the axis to

examine the final deformed cross section. Measurements at this cross section were taken

for comparison to the FE simulations.



























Figure 3-20. Typical undeformed and deformed beam grid pattern used with DIC for
generating experimental strain field


3.1.2.1 Four point beam bend test results: Plate 1

For convienence in describing test results, a reference frame corresponding to

the three orthotropic axes was established with x = RD, y = TD, and z = TT. This

convention is emplo, 1 for all susequent beam bending tests results.

A contour plot of the experimental axial strain field for each of the cases (defined by

Figure 3-17) for Plate 1 are shown in Figures 3-21 to 3-24. The axial strain is defined as

the component relative to the long axis direction of the specimen. For Case 1 and 2 the

long axis is along the rolling direction so the axial strain component is Ex and for Case

3 and 4, the long axis corresponds to the transverse direction therefore the axial strain

is yE. These data are compared to simulation results in C!i lpter 5. For all cases, some

non-uniform deformation occurred in the direction normal to the plane for which the data

were reported. This would introduce a small error in the computation of the axial strains

using the DIC methodology.












10

5

0

-5

-10


-20


-10


0
x (mm)


Figure 3-21. Plate 1 experimental axial strain (F,) fields for Case 1: Long axis in x=RD,
loaded in y=TD.


-5

-10


-20 -10 0 10 20 30
x (mm)


Figure 3-22. Plate 1 experimental axial strain (ex) fields for Case 2: Long axis in x=RD,
loaded in z=TT.


x
-0.08 -0.04 0 0.04 0.08 0 12

- 1 1 1 I I ________









15

10

5

0

-5

-10


-30


/ 0
/ -,i1? 3
~-YU .


EI,


-20


-10


II II ii 4


0
y (mm)


" I


20


30


Figure 3-23. Plate 1 experimental axial strain (E,) fields for Case 3: Long axis in y=TD,
loaded in x RD.


* Zz:


-0.08


0 0.04 0.08 0.12


- ,I, I I I I


-20


-10


0
y (mm)


-30


Figure 3-24. Plate 1 experimental axial strain (yE) fields for Case 4: Long axis in y TD,
loaded in z=TT.


15

10

5

0

-5

-10


i


IIIIIIIIIIII I III I


II 1-71M









Each of the deformed beam specimens was sectioned at the midpoint to quantify

the deformed cross section at the middle of the beam. Figures 3-25 and 3-26 show the

cross sections for each case. Table 3-1 gives the dimensions (mm) measured at the three

locations shown in Figure 3-27 for each of the beams.


A B

Figure 3-25. Deformed cross section of beam from Plate 1 for Case 1 and 2


A B

Figure 3-26. Deformed cross section of beam from Plate 1 for Case 3 and 4










_____ >


C-


Figure 3-27. Measurement locations on deformed four point beam test specimens; values
given in Table 3-1 for Plate 1 and in Table 3-2 for Plate 2.

Table 3-1. Measurements of deformed beam bend specimens from Plate 1; locations
identified in Figure 3-27 (dimensions are mm)
Case A B C


6.4878
7.0212
6.6954
6.9367


6.3500
6.3348
6.4198
6.3786


6.1716
5.6890
6.0154
5.7988


3.1.2.2 Four point beam bend test results: Plate 2

A countour plot of the experimental axial strain field for each of the cases (defined

by Figure 3-17) for Plate 2 are shown in Figures 3-28 to 3-31. These are again compared

to simulations from FE simulations in C'! plter 5. The data from Case 1 and Case 3 are

very similar as is the data from Case 2 and Case 4. This is further evidence of the in-plane

isotropic texture of Plate 2 which means Case 1 and Case 3 are essentially the same test.

A similar argument holds for Case 2 and Case 4. As for Plate 1, the orthotropic axes

correspond to x = RD, y = TD, and z = TT.


L
F




















_In


-20


-10


20


0
x (mm)


Figure 3-28. Plate 2 experimental axial strain (Ex) fields for Case 1: Long axis is x RD,
loaded in y=TD.


-10,


-20


-10


0
x (mm)


Figure 3-29. Plate 2 experimental axial strain (Ex) fields for Case 2: Long axis is x RD,
loaded in z=TT.


-0.08 -0.04 0 0.04 0.08 0.12

I I I I I II I I I I I I I I


30


Ex
-0.08 -0.04 0 0.04 0.08 0.12
I I I I I I I I I I I I I I I I I I I


I A
-JU


30








15

10






-5 -0.08 -0.04 0 0.04 0.08 0.12

-30 -20 -10 0 10 20 31
y (mm)
Figure 3-30. Plate 2 experimental axial strain (Fy) fields for Case 3: Long axis is y=TD,
loaded in x RD.


15

10-------

5
0
-5 y

-S -0.08 -0.04 0 0.04 0,08 0.12

-10-20 -10 20 3
30 -20 -10 0 10 20 3


y (mm)


Figure 3-31. Plate 2 experimental
loaded in z=TT.


axial strain (Fy) fields for Case 4: Long axis is y=TD,


0


0









As for Plate 1, each of the deformed beam specimens for Plate 2 was sectioned at

the midpoint to quantify the deformed cross section at the middle of the beam. Figures

3-32 and 3-33 show the cross sections for each case. Table 3-2 gives the dimensions (mm)

measured at the three locations shown in Figure 3-27 for each of the beams.


A B

Figure 3-32. Deformed cross section of beam from Plate 2 for Case 1 and 2


A B

Figure 3-33. Deformed cross section of beam from Plate 3 for Case 3 and 4










Table 3-2. Measurements of deformed beam bend specimens from Plate 2; locations
identified in Figure 3-27(dimensions are mm)
Case A B C
1 6.6148 6.1081 6 -- ;
2 6.9469 6.3760 5.2902
3 6.5380 6.2738 6.2027
4 7.1945 6 ;,, .5.5467



3.2 High Rate Tests

3.2.1 Characterization Tests

High rate loading tests were carried out using a split Kolsky-Hopkinson pressure

bar. High rate compression specimens (Figures 3-34 A) ) are simple cylinders but

smaller (0.2 x 0.2 inches) than the specimens for the quasi-static tests. Tensile specimens

were cylinderical dogbones as shown in Figure 3-34 B) and were marked with an arrow

indicating the top of the plate and therefore the through thickness direction. The plate

thickness did not allow enough material to obtain high rate through thickness tension

specimens. All dimensions are in inches for both drawings. No post test metallography

was done on any of these specimens so texture evolution data are not directly available.






scLE 6 1

0 200 r tS.1SUNC IA
32RMSFnsh
20 0.
32 RMS FK----





A) B)

Figure 3-34. High rate test specimens: A) Compression cylinder B) Tension dogbone










All tensile tests were carried out to failure. A typical cross section of the failure

is shown in Figure 3-35. Note the elliptical shape with the harder, through thickness

direction in the direction of the major axis.

Side View

Lo 1,l
ri action














Axial View

Figure 3-35. Failed surface from high rate tension test specimen


3.2.1.1 Description of the split Hopkinson pressure bar

As described by Bacon and Lataillade (2001), the split Hopkinson pressure bar

(SHPB) (also referred to as the Kolsky-Hopkinson bar) is widely used to investigate the

dynamic response of a range of materials. The test allows the user to derive the applied

force and the load point displacement versus time by considering the propogating waves

in an instrumented elastic bar. Hopkinson (1914) described the technique initially in 1914

which involved a single long rod. The SHPB, introduced by Kolsky (1949), involved the

use of two instrumented bars and has become a standard setup for high rate material

deformation studies.

The SHPB consists of a striker, an incident or input bar, the specimen to be tested,

and a transmitter or output bar. A schematic of the SHPB setup is shown in Figure

3-36. The specimen is placed between the input and output bars. The striker impacts the

input bar and generates an elastic compressive wave moving towards the test specimen.









For testing tensile specimens, a cylindrical collar is placed around the specimen which

carries the compressive load without deforming the specimen. The compressive wave is

reflected from the end of the transmitter bar as a tensile wave and will eventually reach

the test specimen and load it in tension. For compression testing the specimen receives the

compressive wave directly.


Striker Input Bar Specimen Output Bar


Strain gage 1 Strain gage 2


Figure 3-36. Schematic of Split Hopkinson bar apparatus


The analysis of the waves depends on three primary assumptions: (1) the instrumented

bars remain linearly elastic throughout the test, (2) the diameter of the bars are small

relative to the smallest wavelength of the propagating wave along the bar, and (3) the

mechanical impedance of the bars is uniform. Equation 3-1 gives Kolsky's relation for

finding the stress in the specimen, a,(t) [Kolsky (1949)].


a,(t) = E (t) (3-1)
A,

where E is the elastic modulus of the output bar, A, is the cross sectional area of the

output bar, A, is the cross sectional area of the specimen, and ET(t) is the transmitted

strain as a function of time. The strain rate in the specimen is found from

d (t) 2CER(t) (3-2)
dt L

where ER(t) is the reflected strain history in the input bar, L is the initial length of the

specimen, and C, is the infinite wavelength velocity in the input bar. With p as the

density, this is calculated as:

C fE-/p (3 3)









Table 3-3. Strain rates acheived for tensile SHPB tests
Test Number 1 2 3 4 5 6 7 8 9 10
Strain Rate (sec-1) 522 517 643 665 652 627 662 653 555 543
Test Number 11 12 13 14 15 16 17 18 19 20
Strain Rate (sec-1) 585 534 547 563 573 528 552 557 529 516
Test Number 21 22 23
Strain Rate (sec-1) 517 616 531
Note: AVG = 567 and Standard Dev = 53
Table 3-4. Strain rates acheived for tensile SHPB tests
Test Number 1 2 3 4 5 6 7 8 9 10
Strain Rate (sec-1) 419 429 318 369 396 466 457 408 417 403
Note: AVG = 407 and Standard Dev = 42

The strain in the specimen can be found by integrating Equation 32 :


E (t)= t (tdt (3-4)


3.2.1.2 Test results

Both compressive and tensile tests were carried out at similar orientations as for

quasi-static tests at nominal strain rates from 400 to 600 per second (see Tables 3-3 and

3-4 ). The high rate compression tests for Plate 1 shows that it is also orthotropic at

high loading rates. Figure 3-37 shows stress-strain data for compressive loading along the

rolling direction, the transverse direction and the through thickness direction for Plate

1. As for the quasi-static results the plate is initially harder in the through thickness

direction but after 15'. strain, the rolling direction has hardening above even the though

thickness levels. This is not the case for the high rate results from Plate 2 as shown in

Figure 3-38 where both the transverse data and rolling direction data remain below the

through thickness data for all strain levels. The transverse direction curve remains below

the through thickness curve throughout, as in the quasi-static case for Plate 1 and 2.

Table 3-5 gives the yield values for several levels of strain as well as the anisotropy ratio

(defined as the ratio of highest to lowest yield at a given strain level) for both the high

loading rate dat and the quasi-static data for Plate 1.













600


500


400


300


200
-- TT
-- RD
100


0 0.1 0.2 0.3 0.4 0.5


Figure 3-37.


Experimental compression results for Plate 1 showing the anisotropy among
rolling direction, transverse direction and through thickness direction A) high
rate B) quasi-static


Strain


Strain


Figure 3-38. Experimental compression results for Plate 2 showing the isotropy between
rolling direction and transverse direction A) high rate B) quasi-static.





65


0.15 0.2
Strain









Table 3-5. Quasi-static and high rate compressive yield values for RD, TD, and TT
direction and anisotropy ratios for Plate 1
High Rate Quasi-static
Strain RD TD TT Max/\ I i RD TD TT Max/\ I,
0.05 411 398 294 1.398 225 276 330 1.467
0.1 498 467 399 1.248 271 307 361 1.332
0.15 537 498 492 1.091 323 331 389 1.204
0.2 548 521 573 1.099 379 352 419 1.190
0.25 589 548 626 1.142 426 388 448 1.155
0.3 620 575 654 1.137 456 418 477 1.141
0.35 NA NA NA NA 493 452 505 1.117
0.4 NA NA NA NA 522 483 537 1.112


3.2.1.3 Plate 1 HR characterization tests

The high rate tests on Plate 1 again show orthotropic behavior similar to the

quasi-static data. There is also an indication of significant twinning for the RD compression

loading based on the large changes in the slope of the stress-strain curve. The change in

slope is even more dramatic for the high rate tests than for the quasi-static. This is not

completely unexpected as it has been observed by Gray (1997) that titanium twins more

readily as loading rates increase In general the material is harder when loaded at the

higher rates.

Figure 3-39 A) shows the high rate results for uniaxial loading in the TD. The initial

yield points are very close but as more deformation occurs the strength in compression

becomes somewhat larger. This may indicate that some twinning is occurring in the

compression loading but this has not been confirmed by post test metallography.

Comparisons of high rate compression data with quasi-static data for the TD are shown

in Figure 3-39 B). A clear increase in strength is observed but the hardening rate remains

nearly unchanged.

The high rate results for compressive uniaxial loading in the TT direction compared

to data gathered at quasi-static loading rates is shown in Figure 3-40. Again a clear

increase in strength with loading rate is observed with very little change in hardening rate.



























0.05 0.1 0.15 0.2 025 0.3 0.
Strain


"0 0.05 0.1 0.15 0.2 0.25 0.3 0.
Strain


Figure 3-39. Plate 1 A) Experimental high rate data for the TD for tension and
compression and B) Comparison of experimental high rate data to
experimental quasi-static data


Due to geometry constraints, no through thickness tension data is available at high rates

of loading.


800

700

600

'5 500
-

400
c-l
-
- 300

200

100
-0
0


Figure 3-40.


0.15 0.2 0.25 0.3
Strain


Plate 1 Comparison of compressive high rate to quasi-static data for the TT
direction


Results for high rate uniaxial loading in the RD is shown in Figure 3-41 A). As for

the TD data, the initial yield points are similar but the difference increases with additional


G 500

400

S300











strain. This difference increases at a much higher rate than for the TD data. There is

also a clear change in hardening which indicates that significant twinning is occurring.

Figure 3-41 B) shows the comparison with quasi-static data where post test metallography

showed a significant amount of twinning. The data indicates that even higher levels of

twininng may be occurring at the higher loading rates. The slope of the curve is lower at

strains above 21i' which indicating that twinning has probably saturated.

800 800

700 700

600 600

2 500 500

400- 400 -

C 300 o 300
-- RDC
200 R-- RDT 200 -
--------- fl RDC HK
100 100 P RDC QS

0 0I I I I
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.05 0.1 0.15 0.2 0.25 0.3 0.3
Strain Strain

A B

Figure 3-41. Plate 1 A) Experimental high rate data for the RD for tension and
compression B) Comparison of high rate data to quasi-static data for the RD



3.2.1.4 Plate 2 HR characterization tests

As found from the quasi-static tests, the high rate data show that Plate 2 is nearly

isotropic in the plane of the plate but is stronger in the through thickness direction. As

expected Plate 2 is stronger in the through thickness direction since the texture indicates

that the c-axis is closely aligned with the through thickness direction. In all tests, the

material is harder under high rate loading than for quasi-static loading.

Figure 3-42 A) shows data from five in-plane directions. Although there is some

scatter in the data, it appears that the plate is nearly isotropic in the rolling plane of the

plate. Also shown in the figure (black line) an average of all in-plane data. This average

curve was used in all subsequent analysis and data identification.













500 o 500

400 400

300 300


200 -- RD 200
-TD
67 5
45
100 1 22 00 .. --- HR
-- Avg

0 05 0 05 01 02 0 25 0 3 05 0 1 0 15 02 025 3
Strain Strain
A B

Figure 3-42. Plate 2: A) high rate in-plane compression data B) high rate compression
data compared to quasi-static compression data


Figure 3-42 B) shows a comparison between the average in-plane compression data

at high rate loading compared to the average from quasi-static loading. There is a clear

strengthing effect from the high rate loading which r i ,-; fairly constant through out the

entire path, however the data do seem to show a slightly higher hardening rate for the

high rate loading. This may be an indication if twinning activity.

Data for high rate tensile tests are shown in Figure 3-43 A). The data shown are from

five directions relative to the RD within the plane of the plate. There is more scatter in

the tensile data than for the compressive data but there is no apparent trends indicating

that the tensile behavior is directional. As with the compressive data, an average of all

the data was made and is shown as the solid black line in Figure 3-43 A). Figure 3-43

B) shows a comparison of the average high rate in-plane tensile data to the quasi-static

tnesile data gathered using the round specimen test.

Figure 3-44 A) shows the high rate through thickness compression data for two tests

as well as an average (black line) of the two tests. Again, the average was used for all

analysis and parameter identification procedures. There is very little scatter between

the two tests. A comparison between the TT high rate compression and quasi-static TT















00 /- V vwJ0 V AJY0 300


200 200


-RDT
TDT QST
100 225 100 HRT
45
675
-Avg
0 I iI i I l i i I
0 005 01 015 0 0
005 01 015 02 0 005 01 015 02 02
Strain Strain
A B

Figure 3-43. Plate 2: A) Experimental high rate tension data B) High rate tension data
compared to quasi-static tension data



compression is shown in Figure 3-44 B). The increase in strength for the through thickness

compression at the higher loading rate remains quite constant to the strain levels shown.


Strain


Strain
B


Figure 3-44. Plate 2: A) Experimental high rate through thickness compression data B)
Experimental high rate data versus experimental quasi-static TT compression
data



3.2.2 Cylinder Impact Tests

High rate validation tests were carried out only for Plate 2 using cylinder impact

tests, also known as Taylor impact tests after G.I. Taylor. These tests examine the









very-high strain rate (104 to 105 s-1) response of materials during testing. A cylinderical

specimen is fired at high speeds onto a highly polished flat surface, in this case made

of high strength steel (see Figure 3-45). The velocity of the sample was measured using

a pair of pressure transducers mounted to the barrel and by a pair of lasers mounted

between the barrel exit and anvil impact surface. Sample profile is dynamically viewed

using high-speed photography with the laser closest to barrel (tri ---r laser) used to tri'--. r

the light source for the camera.



Polished surface






17mn reference





Figure 3-45. Taylor cylinder impact test setup


The axis of the barrel bore is aligned perpendicular to the impact face of the anvil.

Correct alignment of the barrel with respect to the anvil face is imperative so that the

leading edge of the sample is in perfect contact with the anvil face at initial impact. After

each test the anvil is rotated to ensure that the projected point of impact is free of defects

from previous experiments or other external sources. The ideal surface condition of the

anvil surface is flat and highly polished.

A Cordin 330A camera was loaded with 2 rolls of T-MAX P3200 film. The external

high intensity light source was positioned so that the test sample was directly between

the light and the camera lens and time of impact. The test sample is inserted into the

barrel opposite the anvil. One plastic obturator is inserted behind the sample prior to

cartridge insertion to limit gas discharge around the sample following propellant initiation.









Table 3-6. Impact velocities from high rate cylinder tests performed from Plate 2
Test Number Laser Velocity Transducer Velocity Angle
96 138 135 0
97 182 184 90
98 153 NR 0
99 182 184 90
100 191 188 45
101 163 NR 0
102 NR NR 90
103 199 200 22.5
104 188 189 67.5
105 185 188 0
106 NR 181 22.5
107 196 193 45
108 185 185 22.5


The appropriate cartridge, having been loaded with a predetermined amount of Red Dot

explosive, is last to be loaded into the bore before affixing the firing pin/cap assembly.

A total of 13 high rate cylinder impact tests were carried out for specimens from

Plate 2. Table 3-6 shows the velocities obtained during each test from both the lasers and

pressure transducers and the angle from the rolling direction associated with the specimen

axis. Table 3-7 gives the in ii' and minor axes of the deformed footprint (the surface

of the cylinder striking the anvil) and the ratio of 1n ii' ,r diameter to minor diameter.

In addition the initial cylinder length, the final cylinder length and the ratio of the two

are given. Note that the velocity from the pressure transducers for test number 101 was

not recorded, the velocity from the lasers for test number 106 was not recorded and and

neither velocity was recorded for test number 102. Figure 3-46 gives the ratio of 1n ii' 'r to

minor diameter and initial to final length as a function of the impact velocity. The ratio

of diameters are strongly influenced by frictional effects at the anvil interface and any

mis-alignment for the test.

The final profile of the deformed specimens were obtained using an optical comparator

model DIJ 415. The spatial measurements were made from enlarged images generated

from the comparator, accurate to within 0.0001 in. Due to the orthotropic texture of










Ratios of in i P" to minor final deformed diameters and ratios of final to initial
lengths from high rate cylinder tests performed on specimens from Plate 2


Table 3-7.

Test
Number
96
97
98
99
100
101
102
103
104
105
106
107
108


0.90

0.85

0o812


Major
Diameter
0.245
0.261
0.248
0.258
0.264
0.255
0.252
0.265
0.263
0.261
0.260
0.265
0.262


Minor
Diameter
0.231
0.243
0.228
0.241
0.246
0.234
0.238
0.246
0.244
0.243
0.237
0.245
0.244


Diameter
Ratio
1.061
1.074
1.088
1.071
1.073
1.090
1.059
1.077
1.078
1.074
1.097
1.082
1.074


X X
lxxx X
x


A


x
A
0 1


Diameter
Length


160 180
Impact velocity (m/s)


Initial
Length
2.097
2.100
2.101
2.100
2.099
2.100
2.100
2.100
2.097
2.099
2.099
2.100
2.097


Final
Length
1.900
1.790
1.879
1.804
1.788
1.851
1.840
1.752
1.778
1.797
1.816
1.770
1.789


Figure 3-46.


High rate cylinder test results giving the ratio of i i, i diameter to minor
diameter and the ratio of initial to final length plotted versus the impact
velocity


the specimen the initially circular cross section of the specimen deformed into an elliptical

shape. Both the i i, '.j and minor axis of the specimen were measured. As might be

expected, the data extraction is very time consuming and manpower intensive. Figure

3-47 shows the specimen dimensions and an undeformed compared to a deformed sample.


Length
Ratio
0.906
0.852
0.894
0.859
0.852
0.881
0.876
0.834
0.848
0.856
0.865
0.843
0.853


I I I























A B C
Figure 3-47. High rate cylinder impact test specimens A) dimensions of high rate
validation test specimen B) undeformed specimen compared to typical
deformed specimen C) High rate cylinder specimen showing arrow aligned
with the through thickness direction pointing to the top of the plate


All specimens are very similar with some variation due to the specimen machining

process. The specimen from test number 107 was judged to be definitive and used for

data extraction. Resources did not permit the detailed extraction of profile data from all

specimens. Both the 1i ii i and minor experimental profile data are shown in Figure 3-48.

During fabrication care was taken to identify the relation of the specimen with the TT

direction. A mark on the end of each specimen was made by the machine shop to indicate

the TT direction as shown in Figure 3-47 C). For all cases the deformation was less that

in the through thickness direction. This shows up in the elliptical foot print (initially

circular) of the deformed specimen. Figure 3-49 shows the experimental dimensions of the

final footprint from test number 107.

3.3 Texture

An investigation into the pre- and post- test textures for the quasi-static specimens

from Plate 1 was carried out to obtain data concerning the evolution of the initial texture

and to evaluate the level of twinning occurring for the various loading paths. An initial

texture for Plate 2 was determined in order to establish a reference direction for the rolling

direction. No post test texture measurements for Plate 2 specimens were carried out.















JVU



40



30












Minor


-10 -3 -2 -1 0 1 2 3
-4 -3 -2 -1 0 1 2 3 4


Diameter (mm)



Figure 3-48. Measured i1i i i" and minor profile data from
velocity 196 m/s)


test number 107 (impact


4


2 .



0


-2 -__


-4



-4 -3 -2 -1 0 1 2 3 4
Major Diameter (mm)



Figure 3-49. Measured deformed footprint from test number 107









Texture measurements were carried out using two approaches. Orientation Imaging

Microscopy (OIM) in a scanning electron microscope (SEM) was used to investigate

the amount of twinning occurring primarially in the compressive loading for the rolling

direction. The quasi static uniaxial tests drom Plate 1 indicated that this was the

only direction where a significant amount of twinning occurred. A single measurement

for a specimen loading uniaxially in the transverse direction was checked and only

a small amount ( 5' by volume) of twinning occurred for this case. All of the quasi

static compressive specimens were evaluating using the neutron time-of-flight (TOF)

diffractometer HIPPO (High-Pressure-Preferred Orientation) at LANSCE (Los Alamos

Neutron Science Center).

3.3.1 Grain Size

Figure 3-50 represents the arrangement of 40 pictures taken at a magnification of 50X

on an optical microscope. The top surface of the plate is on the left. As seen in the figure,

there are bands that are typical for a rolled material. The bands usually contain smaller

grains. The arrows show positions where higher magnification pictures, shown in Figures

3-51 to 3-54, were taken to measure the grain size. From these pictures, it is seen that the

grains are roughly equiaxed on the plane of the pictures but the grain size varies from one

position to another.


Plate Thickness 15.5 mm
4---







12 3 4 5 6 7 8


Figure 3-50. Locations along the thickness where micrographs of grain size data were
made for Plate 1.

























(1) (2)
Figure 3-51. Optical microscopy (50X) at locations 1 and 2 from Figure 3-50

















(3) (4)
Figure 3-52. Optical microscopy (50X) at locations 3 and 4 from Figure 3-50

The grain sizes at all positions except 1,2 and 7 appear uniform in size. For grains

at positions 1 and 2, there are big grains of 50 to 70 pm surrounded by grains similar to

those found at positions 3,4,5,6 and 8. At position 7 there are big grains of 40 to 50 pm

surrounded by smaller grains. Table 3-8 gives the average grain size at each position.




























(5) (6)


Figure 3-53. Optical microscopy (50X) at locations 5 and 6 from Figure 3-50


Figure 3-54. Optical microscopy (50X) at locations 7 and 8 from Figure 3-50


Table 3-8. Grain size averages at locations shown in Figure 3-50


Position
Grain size (pm)


1 2
26 26


3 4 5 6 7
16 15 16 15 20










3.3.2 Determination of Rolling Direction

The initial texture for both plates were established using OIM techniques. The plates

were initially 10 inches in diameter disk with a thickness of 5/8 inch. For Plate 1 twenty

metallurgical samples were removed via water jet from the perimeter of the titanium plate.

Each sample is approximately 0.75" X 0.75" X 0.75" and 11.320 from the neighboring

sample (See Figure 3-55). The samples were sequentially numbered counterclockwise

around the plate.







%-j




( .32)







Figure 3-55. Plate 1 with 20 coupons removed


Samples 20 and 11 were removed for metallurgical and textural a i --, Both were

sectioned at the midplane while parallel to the plane of the plate, then mounted in resin

(see Figure 3-56). Two views are available for each sample: "top-dc.vi: and "bottom-up."

The top-down view corresponds to the viewing direction necessary to read the sample

numbers. The bottom-up view is opposite this direction.

The plate exhibited strong fiber texture resulting from the rolling process. The basal

plane (0002) pole figures were examined to determine the rolling direction. Previous work

by Barrett and Massalski (1980) performed on rolled pure titanium states that the basal

planes align 350 from the plate normal during rolling. Figure 3-57 verifies this for Samples

























Figure 3-56. Definition of sample orientation from sectioned coupon used for initial texture
measurements


11 and 20. Notice that the top-down and bottom-up views for Sample 11 are essentially

mirror images.









0"/



(a) (b) (c)
Sample 20 Sample 11 Sample 11
Bottom-up view Top-down view Bottom-up view


Figure 3-57. Initial (0002) pole figures for Plate 1 from the two coupons used to identify
the rolling direction


The 12-o'clock position of each figure corresponds to the midpoint of the outer

edge of the sample. The rolling direction can be resolved in two dimensions since

it lies perpendicular to the basal texture). To translate the rolling direction to the

three-dimensional plate hardware, a texture map can be superimposed onto an image of

the plate itself. Figure 3-58 A) di-p-'v' the pole figures shown in Figure 3-57 with the


Sectioning Plane






f --
Analyzed Surface Analyzed Surface
"Top-down view" "Bottom-up view"










plate diagram. The pole figures were rotated such that the 12-o'clock position coincides

with their respective location on the plate hardware. It should also be noted that the

pole figure for Sample 20 (Figure 3-57 a ) is a mirror image of itself since Figure 3-58 A)

represents a top-down view rather than bottom-up.

A similar approach was taken to establish the rolling direction for Plate 2, two

coupons were cut from the outer edge of the plate and used to establish the rolling

direction. Since the plate was nearly isotropic in the plane of the plate this was somewhat

arbitrary. The established rolling direction was set relative to the texture as shown in

Figure 3-58 B).








RD







A B
"t-



^ ---1-

.. / ,I, /




-_ -- Sample 1 Sample 2
Rotated 55 Rotated -20
A B

Figure 3-58. Plate 1 and Plate 2 with pole figures superimposed to determine rolling
direction A) Plate 1 B) Plate 2


3.3.3 Variation of Texture in Through Thickness Direction

An investigation of the variation of the initial texture in the through thickness

direction was carried out on one of the initial coupons cut from the outer edge of Plate 1.

The specimens were taken from coupon number 18 which was approximately 340 from the

rolling direction.

A total of 17 scans were made in the through thickness direction as arranged in

Figure 3-59. Each scan covered an area of 200pm X 800 pm. The texture from all 17










Scan17 Scan2 Scan1




.................................................................. I il T |
g-




Figure 3-59. Position of scan locations for through thickness texture measurements


0001 1010





11
1001


RD o3m RD
mI n m O- 027

Figure 3-60. Bulk texture of Plate 1 found from averaging the 17 through thickness scans


scans were averaged to get a bulk texture as shown in Figure 3-60 which is similar to the

textures measured from the center of the coupons used to establish the rolling direction in

Figure 3-57.

Figures 3-61 to 3-69 show the pole figures for each of the 17 locations. Some

differences are apparent from these measurements. The scans taken near the center of

the plate has a similar texture to the bulk texture found from averaging all 17 scans.

Scans taken from the top and bottom 4 to 5 mm of the plate shows some non-symmetric

textures indicating the strong shear loading from the rolling process. This may account

for some of the variation in the uniaxial loading test results. The test specimens were

cut from various locations in the through thickness direction, some from softer or harder

regions of the plate.










1010 0001


Figure 3-61. Plate 1 pole figures from positions 1 and 2 in Figure 3-59


0001


1010


0001


1010


Figure 3-62. Plate 1 pole figures from positions 3 and 4 in Figure 3-59


0001


1010


0001


1010
eOM


Figure 3-63. Plate 1 pole figures from positions 5 and 6 in Figure 3-59


n n 1


1010











1010 0001


Figure 3-64. Plate 1 pole figures from positions 7 and 8 in Figure 3-59


o0n 0


1010


0001


1010


RD


(10)


Figure 3-65. Plate 1 pole figures from positions 9 and 10 in Figure 3-59


0001


1010


0001


1010


qTOC


(11)


(12)


Figure 3-66. Plate 1 pole figures from positions 11 and 12 in Figure 3-59


0001


1010











0001


RD


(13)


(14)


Figure 3-67. Plate 1 pole figures from positions 13 and 14 in Figure 3-59


1010


0001


(15)


(16)


Figure 3-68. Plate 1 pole figures from positions 15 and 16 in Figure 3-59


n (n i


1010
-w-


4T


Figure 3-69. Plate 1 pole figures from position 17 in Figure 3-59


1010


0001


1010


1010
or









3.3.4 Texture Evolution

Texture measurements were made for all of the compressive specimens from Plate

1 using neutron time-of-flight (TOF) diffractometer HIPPO (High-Pressure-Preferred

Orientation) at LANSCE (Los Alamos Neutron Science Center). This gives an indication

of the texture evolution for each loading path. Compressive loading in the transverse

and through thickness directions show less texture transition than for compression in the

rolling direction. This supports the indications from the uniaxial compression tests as well

as the OIM measurements.

Figure 3-70 shows the 0001 pole figure of the initial texture for plate 1 from three

different perspectives. Figure 3-70A has the transverse direction in the middle and the

through thickness direction from side to side, Figure 3-70B has the through thickness in

the center and the TD is side to side and Figure 3-70C has the rolling direction in the

center and transverse direction from side to side.




TT 4TD T



RD RD TT
A B C

Figure 3-70. Plate 1 (0001) PF of initial texture A) center is TD and TT side to side B)
center is TT and TD side to side C) center is RD and TD side to side


Figures 3-71 and 3-72 show 0001 pole figures for specimens loaded in compression

in the transverse direction at 10' 211' 311' and 40' strain. This shows a fairly strong

alignment of the c-axis with the through thickness direction when compared with the 0001

pole figure with the rolling direction in the center (Figure 3-70(c)). This is verified by the

uniaxial loading tests which show the plate is stronger in the transeverse direction than

the rolling direction.










There is only a small amount of evolution of the texture indicating that twinning is

not a dominant deformation mechanism for this loading path. Some variation occurs from

the variation with position of the initial texture. Again the transverse direction is in the

center and the through thickness direction from side to side.


001


mmn
-0.11


max
6.47


001


mm max
-0.02 5.85


10' ,


21'i


Figure 3-71. Plate 1 (0001) pole figure for specimens
Sin transverse direction, TD in center



001


loaded in compression to 10 and 20
and the TT from side to side


mmn max
0.06 4.99


max
4.19


Figure 3-72. Plate 1 (0001) pole figure for specimens loaded in compression to 30 and 40
Strain in transverse direction, TD in center and the TT from side to side










Figures 3-73 and 3-74 show results for the TT specimens loaded in compression. The

pole figures show only slight texture evolution indicating low levels of twinning for this

loading path. These show a strong orthogonal texture with a significant portion of the

grains aligned within 150 of the TT direction.


001


001


, -


max
4.89


minm max
-0 05 4.93

21 "


Figure 3-73. Plate 1 (0001) pole figure for specimens loaded in compression to 10 and 20
Sin through thickness direction, TT in center and the TD direction from
side to side




001 001












-0.04 4.28 -0.12 4.35

3i r., Ii' ,

Figure 3-74. Plate 1 (0001) pole figure for specimens loaded in compression to 30 and 40
strain in through thickness direction, TT in center and the TD direction
from side to side


mmi
-0.07


10' ,









Figures 3-75 and 3-76 show results for the RD specimens loaded in compression. The

pole figures show a significant texture evolution that would be expected from the high

levels of twinning shown both in the OIM measurements and the uniaxial stress-strain

curves. A significant amount of twinning has occurred by the point where 21i' strains have
been reached as shown by the OIM data.


001


Figure 3-75. Plate 1 (0001) pole figure for specimens loaded in compression to 10 and 20
Sin rolling direction, RD in center and the TD direction from side to side


001 001













Figure 3-76. Plate 1 (0001) pole figure for specimens loaded in compression to 30 and 40
strain in rolling direction, RD in center and the TD direction from side to
side










Figure 3-77, 3-78 and 3-79 show these same pole figures at the appropriate strain

levels on stress-strain curves for each loading condition. It is clear from Figure 3-77 that

the change in slope of the stress-strain curve coincides with the strong change in texture

from the pole pictures. The nearly linear hardening portion of the transverse and through

thickness directions correspond to the smaller texture changes observed for these loading

conditions.

The results of the OIM and texture measurements are consistent with the uniaxial

loading tests done on Plate 1. The stress-strain curve for compression in the rolling

direction shows a clear change in hardening that is indicative of twinning, the OIM

measurements show that there is significant twinning for the compression specimens

loaded in the rolling direction and The texture evolution for this case shows significant

changes arising from the large grains rotations occurring during twinning. Although

resources allowed only a limited investigation of the twinning for other loading paths, none

of the results indicate that twinning had a large role in the texture evolution for the other

loading conditions.


600


500


400


300

200


100 ----^ --


-0.1 0 0.1 02 0.3 0.4 0.5
Strain


Figure 3-77. Texture evolution for compressive loading in the rolling direction





















300 -. .



200


0001



0 01
DR01 0.2 0.3 0.4
Plastic Strain


Figure 3-78. Texture evolution for compressive loading in the transverse direction







600


5w0





30 ..
4a

cc 200

0001




0.1 0.2 0.3 0.4
Plastic Strain


Figure 3-79. Texture evolution for compressive loading in the through thickness direction









CHAPTER 4
MODELING

To model the behavior of high purity Titanium under quasi-static loading conditions

an elastic-plastic modeling approach is adopted. The onset of yielding will be described

using an anisotropic yield criterion that captures both strength differential effects and the

directionality in yielding. The parameter identification procedure based on experimental

results is given in section 4.2. The integration algorithm for the proposed model, and

its implementation in the explicit FE code EPIC follows. A comparison between model

predictions and the data is given in ('! Ilpter 5.

4.1 Proposed Yield Criterion

One of the goals of this research is to advance the current state-of-the-art by

developing user-friendly, micro-structurally based and numerically robust macroscopic

constitutive models that can capture with accuracy the particularities of the plastic

response of hexagonal metals, in particular high puirity Titanium.

A full stress 3-dimensional anisotropic yield criterion is proposed. Key in this

development is the use of the isotropic yield function of Cazacu and Barlat (2004) that

captures the tension/compression .,i- ii ii i. First a brief overview of the criterion is

given. After reviewing the general aspects of a linear transformation operating on the

Cauchy stress tensor the anisotropic yield function is developed. The input data needed

for the calculation of the anisotropic yield function coefficients are discussed.

4.1.1 Isotropic Yield Function

If the internal shear mechanism of plastic deformation is sensitive to the sign of the

stress as is the case with twinning, the isotropic yield function ought to be represented by

an odd function in the principal values of the stress deviator. To describe the .i-viil. I1 ry

in yielding, due to twinning, Cazacu and Barlat (2004) proposed an isotropic yield

criterion of the form
3
f J cJ3 (41)









where ay is the yield stress in pure shear and c is a material parameter. J2 and J3 are

the second and third invariants of the stress deviator. The constant c can be expressed in

terms of the yield in uniaxial tension, oT, and yield in uniaxial compression -c, as

S ( ) (4 2)
2 (aTa)

When c = 0, i.e. UT = ac, the criterion (4-1) reduces to the von Mises criterion. To ensure

convexity, c is limited to: c E [-3-3 /2, 3/3/4].

In principal stress space, for plane stress conditions, the yield locus is

1 2 C
3(012 -- 1 2 27-2 2
3 (a1 12 + ) 27 [24 + 273 3- 3( + 2)l2] a (43)

When c / 0,this represents a triangle with rounded corners. Figure 4-1 shows Equation

4-3 plotted for different ratios of rT/ac. When this ratio equals 1, the curve corresponds

to the von Mises ellipse.

2,0
1.5 .- 23
S33 Von Mises ,"
1..-.. 4f3 '



0.0



S- Isotropic
-1.5 material

:-.O -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0



Figure 4-1. Plane stress yield locii for various rations of aT/ac


Figure 4-2 shows a comparison of the yield criterion described by Equation 4-1 to

data calculated by Hosford (1966) using a generalization of the Bishop and Hill (1951)

model. Assumptions for this approach include that deformation is accommodated by










dislocation glide only, deformation is uniform through the pi. -1 ,'i- iii.r11 material and the

material behaves as an elastic-plastic material. In this case, the only material parameter

is c which can be expressed in terms of the ratio aT/ac (see Equation4-2). The figure

shows the plane stress yield locus for a ratio of 0.78 (dashed curve) corresponding to an

fcc material as well as a ratio of 1.28 corresponding to a bcc material. The open and solid

circles are data as reported in Hosford (1966).



1.50 I
sotropic I I
material

a a I
O.0O _- ------- ---------- --------- :

0.50 ............

0.430






bcc
.4.5 ...--.--------------
-1.500
-1.50 -1.0 -O.1L 0.00 0.50 1.00 1.5 0
C2

Figure 4-2. Comparison with pol. i'-1 illii; simulations


Note that the yield locus generated with the proposed criterion coincides with

the yield locus obtained by p i1v iv I 11 i calculations. Also shown in Figure 4-2 is a

comparison between the yield locus predicted by the macroscopic model (rT/ac = 1.28)

and the p i.' li-,- ii -11 : model (full circles) for bcc p .vi-, i --I 1- Again, the yield loci

coincide. Next, in order to describe both the .i-vmmetry in yielding due to twinning and

anisotropy of rolled sheets, extensions to orthotropy of the isotropic criterion given by

Equation 4-1 are presented.

4.1.2 Extension to Orthotropy

A generalization to orthotropy can be obtained by using a linear transformation on

the Cauchy stress, c. Thus a is replaced by E = Lc, where L is a 4th order tensor, i.e.












t (y2 3/2 3r 3
f L: tr (3)ED y (4-4)

The tensor L satisfies (a) the symmetry conditions: Lijkl = Ljik = Ljik = Lklij ( i,j,k,l

= 1...3 ), (b) the requirement of invariance with respect to the symmetry group of the

material, and (c) Llk + L2k + L3k = 0 for k = 1, 2, and 3. This assures that E is traceless

and consequently yielding is independent of the hydrostatic pressure. Relative to the

orthotropy axes (x, y, z), L is represented by


(a2 +a3)
3

3

a2
3

0

0

0


a3
3
(al+a3)
3
al
3

0

0

0


a2
3
al1
3
(al +a2)
3

0

0

0


ai, i=1...6 are constants. In the (x, y, z) frame x represents the rolling direction, y the

transverse direction and z the thickness direction. This leads to: E =


'[(a2 + a3)(T a2


a47-xy

a57xz


1[-a3x + (al + a3)y al]Tz


a6Tyz 3 [


a57xz


a6Tyz


a2a1 alay + (a, + a2)0az


The proposed yield condition is






Where the deviatoric invariants are


J" 3/2 C_ J
^2 c'3


(4-5)











J = [(2 a+3 + 2) 32
+ (a2 + a + a23) 2
(2 aq 2 a3 )o

+ (a + 2 +a a1a) a2

+ (-2.,? + aia2 1j13 2a3) 3aaTy (4-6)

+ (-2a2 ala2 + ala3 a2a3) a

+ (-2a ala2 l3 + 23) 3yz,]
2 2 2 _22
+ a4xy + l:'z + '.z


jo [ (,2 + '.1 2) + (a3 + aa) + (2 + aa)

+ (-aia2 + ala 2. 2,,. ) 2,?

+ ( aia 2,,.) aTT

+ (-aa2 aa3 + 2aia ) (o2x

+ ( 2 2a a3 ia ,?,) o2

+ (-a2a2 a 2aa + 2, .) 2 (4-7)


2 a 2 2 2 2 2
+ (-2 2 + 2a a3 aia| ,2, .) oo

+ 2 (ai2 + i3 + aa + +2 + aja, + 2,.') xy ]

+ { ['-.'-4. + ai(oY (aia + a2a2) ( y

+ [, .,I. (aia + ,' ) ( + aia z] T-
2 2 (2 ] 2
+ [(- O-. + ,._]_ }

+ 2a4a5a67TyTxzTyz

For plane stress these reduce to

1 2 2
J = (-2a +2 a 2 -a23) 2a3) xy a4 1 y) c

+ (-2a + aa aa2 013 a3as) a + aT (48)








27 = [ (3 + 2) 73 + (a 2a3 + a1ai) 73+
+ (-ai + aa a a3 2a2a) on o

+ (- a2 a2 3 + 2a 2aia ) Cax ]
1 2 2
+ 3 ( a2a4cj + aQQacT) (4-9)

An expression for the stress in an arbitrary direction ao is useful in parameter identification
and can be found as follows. By definition (refer to Figure 4-3)
cx Cr cos 02 oy U0 sin 02 Txy 0 sin 0 cos 0











x





Figure 4-3. Arbitrary angle definition, x is rolling direction

Plugging in to 4-8 and 4-9 gives

S = { (a + ala3+ a) sin4 0 + (a + aa + a) cos 0

+ [ai(a3 2) + 2a + a2a3 a cos2 O sin2 0 }
1
27 { (ai + a3) aa3s sin6 0 + (a2 + a3)a2a3 cos6 0
+ [(a2 2ai)a (a2 3 + ) a + 9a ] sin4 6 cos2 0

+ [(ai 2a2)ta (ai +3) + 9a28a] cos4 0 sin2 O}1 (4 10)









In particular, with Rten and Rcomp defining the yield stress in tension and compression

along the rolling direction, then

3
Rtens a2 + a2 + a23 2 -c (a2a3 + 3 2) 3 (4-)

333
Rco7mp ( + a + a23)2 + c (a3 + az) 2 (4-12)

Similarly, with Tten and Tcorp being the yield stress in tension and compression along the

transverse direction, then

Ttens = TY {(a + a2 + aa32 c ( 3 + a2ai)} 3 (4 13)

Tcorap T{ {(a2 + a2 + a3) + c (a3 + aai) at) (4-14)
Tcomp 1 3 3 1 331 3

When ar =a2 ob and a3 = 0, yielding under equibiaxial occurs

aT r l b b3/) 2 a1a2(a1+ a2)] (415)
b = Ty (2ai 2b2 b3) / c 2-- (4-15)

and for equiaxial compression when al = -2 = ab

C F32 + a+2(a + a)2
ab T (2a 2b2 b)3/2 + c (4-16)


4.2 Identification of Material Parameters

The anisotropy coefficients involved in the proposed yield criteria can be found

through the minimization of cost functions. The experimental data in the cost functions

may consist of flow stresses in tension and compression corresponding to different

orientations in the plane of the plate and normal to the plane of the plate. If available,

Lankford coefficient data (or r-value data) can be used. If experimental data are not

available for a given strain path, they can be substituted with numerical data obtained

from pc-li-l i--i 11,ii. calculations as demonstrated by Plunkett (2005). All parameters were

determined using the built-in minimization function Minerr of the software Mathcad,

version 14. [PTC (2007)1









4.2.1 Cost function involving only yield stresses

When yield stresses measured at different orientations with respect to the rolling
direction are available, the cost function is

/7T 2 + 2
E(anisotropy coefficients, c) = wj 1) + -

where j is the number of experimental tensile yield stresses, k is the number of experimental
compressive yield stresses while wj and i,', are weights given to the respective experimental
values. The theoretical values are calculated according to the appropriate criteria.
4.2.2 Cost function involving Lankford coefficients

If experimental r-values (or Lankford coefficients) are available the cost function is
E T \ 2 / 7r 2
E(anisotropy coefficients, c) = j wy 1 + p 1


4.2.3 Cost function involving biaxial data

If biaxial data are available, the following procedure is followed: Let a1 and a2 be the
principal stress values. According to the proposed yield criterion, there is plastic flow if
and only if

F(ci, a2, anisotropy coefficients, c) = JL (91, a2, anisotropy coefficients)

c J3 (1, anisotropy coefficients)

The data are scaled with Xt the tensile yield value in the rolling direction. The cost
function to be minimized is then


F(tl, 0a, ai)
E(ai, c) Wj [F(,1l, 2, a/) -- ai kU2 I(7


where ai represents the anisotropy coefficients for the appropriate linear transformation
and










exp exp
1 2a2
XTV(e)2 + (XP)2 X (Utl ) + (P )2

4.3 Anisotropic Hardening

From the experimental data it appears that there is distortion of the yield surface

even for the simplest monotonic loading paths. This anisotropic hardening, which is due

to the evolving texture, cannot be described with the classical isotropic or kinematic

hardening laws. Recently, Plunkett et al. (2006) proposed a method for accounting for

the texture evolution. This method allows for the variation of the anisotropy coefficients

with accumulated plastic deformation, i.e. the linear transformation operator L is no

longer constant. However, obtaining analytic expressions for the evolution laws of all the

Lij components would be a formidable task. Rather, the anisotropy coefficients will be

calculated for a finite (descrete) set of values.

C'!....-i.g the effective plastic strain E as the hardening parameter, the current yield

stress ay and the current equivalent stress a are found by interpolation based on the

current level of the effective plastic strain. The procedure is as follows:

Prior to the FE simulation, a particular strain path is chosen. All of the quasi-static

simulations performed in this work used the uniaxial tensile curve in the rolling

direction as reference hardening curve. An alternative path could be have been

chosen if it were known a priori that a particular loading path would be used in a

given test.

Next, for a discrete set of strain levels, the yield strengths associated to this

reference loading path are determined. The anisotropic parameters for the yield

model are identified at the same set of decrete strain levels.

During the FE simulation the current effective plastic strain level is used an the

interpolation factor between two bounding levels of strain from the descrete set. The









interpolation factor is found as


J < E < E<
j+i


Using the interpolation parameter (, the equivalent stress is computed as


(O", ).current = & + (1 o&+l


The current yield stress is found similarly as


Y(-, )current = Yj + (1 ) Yj+l


The current yield function is then taken as


fj ( )current ) current j f) Y1current(, f) < 0

The model implemented for simulations of quasi-static loading for this work is based on

the transformed isotropic criterion given by Equation 1-10 assuming an associated flow

rule (Equation 1-11) and hardening as described by the interpolative procedure described

above.









CHAPTER 5
SIMULATIONS

Illustrative examples of the application of the proposed yield criteria to the

description of anisotropy of hexagonal materials based on experimental data available

in the literature is presented. The ability of the criteria to account for strength differential

is demonstrated by comparison to the widely known quadratic Hill (1950) yield surface

description. Finally, the proposed model is applied to experimental data gathered for the

high purity titanium material investigated in this research.

5.1 Application to Mg-Li Alloy

As an illustration of the identification procedure outlined, the criteria was applied to

Mg-Li alloy using the experimental data reported by Kelley and W. F. Hosford (1968).

The data consists of the results from plane-strain compression tests in six orientations

that correspond to the six combinations of the rolling direction, transverse direction, and

thickness direction ; uniaxial compression and uniaxial tension tests in the x, y, and z

directions respectively. Based on these data, the experimental yield loci corresponding

to several constant levels of the largest principal strain were reported. Due to the strong

basal pole alignment in the thickness direction, twinning is easily activated by compression

perpendicular to this direction, but is not active in tension within the plane. The effect

of twinning is clearly evident in the low compressive strengths at 1 At 10 strain, the

third quadrant strengths are comparable to the first quadrant owing to the exhaustion of

twinning. The parameters involved in the equations of the proposed model were calculated

using the procedure outlined in the previous chapter. The values of the anisotropy

coefficients are given in Table 5-1. Figure 5-1 shows the prediction of criterion given by

Table 5-1. Model parameters for the yield surface in Figure 5-1
a1 a2 a3 a4 C
0.8359 0.7564 1.1190 1.2170 -0.1532


Equation 4-5 in comparison with the experimental data at 10' strain. It is seen that the

criterion describes well the observed ..-i-i ii. I ry and anisotropy in yielding.















S20

10 ..




-10


-20


-30 -20 -10 0 10 20 30
a, (MPa)


Figure 5-1. Projection in the plane a3=0 for Mg-Li alloy sheet predicted with the criterion
and data (10'-. strain )


5.2 Application to High-purity Titanium

Next the proposed yield criteria is applied to the high-purity textured plates

experimentally characterized as part of this research (see C'! lpter 2).

5.2.1 Plate 1

Recall that Plate 1 is orthotropic. Based on the tensile flow data at 0 450 900 and

compressive flow data at 0 and 900 the anisotropy coefficients involved in the criteria

(Equation 4-5) were determined at fixed levels of accumulated plastic strain (up to 0.5).

For the tension data, this required an extrapolation above approxiamtely 21i'. strain

since the material began to have localized strains beyond this point. The corresponding

theoretical yield surfaces along with the experimental values (filled squares) are shown

in Figure 5-2. Note that the proposed criteria matches the data very well except for the

TD tension data. The optimization procedures used in determining the model parameters

were required to match the most significant data. Since the tensile data were extrapolated

beyond 211' it was allowed to have a larger error for these data in order to match the rest











of the data points. It was felt that this parameterization was very good for most of the

data and was sufficient to demonstrate the ability of the model to capture the anisotropic

behavior of the material.


I" I




IiI


____4M)__ ___


75
-10
-- 20
30
---- 30
40
-- 50


-800 -600 -400 -200 0 200 400 600 800
C1


Figure 5-2. Theoretical model (Equation 4-5) compared to experimental data for Plate 1
at various strain levels (data are represented by symbols)


5.2.2 Plate 2

The experimental data show that the material of Plate 2 is isotropic in the plane

of the sheet. Based on the average compressive flow stresses, a single tensile test at

22.50 and compressive and yield values from the TT direction, the anisotropy coefficients

involved in the anisotropic yield criterion were determined for different fixed levels of

accumulated plastic strain (up to 0.5). Again the data for tension are extrapolated beyond

211' strain. The tensile and compressive yield stress averages were found from averaging

the flow stresses from 0 22.50 450 67.50 and 900 directions in the plane of the plate.

It is illustrated in Figure 5-3, where the average values are represented by triangles. The

corresponding theoretical yield surfaces along with the experimental values (filled squares)


800-

600--










are shown in Figure 5-4. For this case, the theoretical yield surfaces have the largest error

in the biaxial data. Again, it was felt that this was sufficient to demonstrate the ability of

the model to capture the anisotropic behavior of the material.



500




400 ---




300

C)

200

0
22.5
45
100 45
1067.5
90
AVG


0 10 20 30 40 50
Strain (%)

Figure 5-3. Average experimental in-plane compression data for Plate 2


5.3 Comparison to Hill's Quadratic Model

The quadratic yield criterion of Hill (1948) is the most widely used orthotropic yield

criterion available and has proven to be accurate and robust for many materials, especially

steels. However, it can not account for the strength differential observed in hexagonal

materials. For comparison purposes, Hill's criterion is applied to the high purity Titanium

material used in this research. First, the identification procedure used to identify the

coefficients involved in Hill (1948) yield criterion is presented.













600


400


200


0
4-

-200 % Strain
-- 2.5

-400 -- 75
10
-600 40
-- 50

-80
-1800 -500 0 500 1000
Stress (MPa)

Figure 5-4. Theoretical model compared to experimental data for Plate 2 at various strain
levels (data are represented by symbols)


With respect to the orthotropy axes (x, y, z), the Hill (1948) orthotropic yield

criterion has the form


F(a a,)2 + G(, a,)2 + H(a ay) + 2LTZ + 2MrT + 2N' 1 (5-1)


where the coefficients F,G,H,L,M, and N are constants and x, y, and z are the orthotropic

axes. The coefficients can be found from mechanical tests. With X the yield strength in

the x direction, Y the yield strength in the y direction, and Z the yield strength in the z

direction, it can be shown that


F + x) (52)

1(1 1 1 (53)
-2 2 X2 y2
1 1 1 (54)
2 X2 Y2 Z2









Similarly with R is the (yz) shear yield, S is the (zx) shear yield and T is the (;,)

shear yield

1
L 2 (5-5)
2 R2
1
M = 2 (5-6)
2 S2
1
N = T (5-7)
2 T2

For the plane stress case i.e. uz = -rx = Tyz = 0, the criterion in Equation 5-1 reduces to


a (G + H) + a(F + H) 2Ha,ay + 2N2 1 (5-8)


By definition


Jc =2sx + sy (5-9)

O~y = s + 2Sy (5-10)


Substituting (5-9) and (5-10) into (5-8) gives


s(F + 4G + H) + (4F + G + H)+ sxs,(4F + 4G- 2H) + NT + x 1 (5-11)


In terms of the uniaxial yield at an arbitrary angle 0,


cr = c0 cos2 0 2 o = r sin2 0 Txy = or sin 0 cos 0


The deviator stresses become

2ax y 2 os 0-sin 3 cos2 0 1
S, 3 3 o0 (5-12)
3 3
20- 2sin2 0 8-cos2 2 3 COS0
sy = 3 3 o 3 o (5-13)


By definition the r-value in an arbitrary direction 0 is

sin2 0~ + cos2 0 sin 20 i
ro =f af (5-14)



107









Taking derivatives of (5-11)


2 1
2s,(F + 4G + H) + 2s,(4F + G + H)(- )
3 3


2
2H) + s,(4F + 4G
3


1
2H)(- )
3


2(2G + H)s, + 2(G- H)sy


2
2sx(F + 4G + H)(- fracl3) + 2sy(4F + G + H)
3


2
2H) + s,(4F + 4G
3


1
2H)(--
3


2(F H)sx + 2(2F + H)sy


2N-r,y


The numerator of (5-14) becomes


S i2 f
sin 20 O
OTXY


2sx [(2G + H) sin2 0 + (F


+ 2y [(G H) sin2 0 + (2F + H) cos2 0]

- sy 2N sin 20


The denominator of (5-14) becomes


2(F + 2G)sx + 2(2F + G)s,


Inserting 5-18 and 5-19 into 5-14 gives
[(2G + H) sin2 0 + (F H) cos2 0] sx + [(G H) sin O + (2F + H) cos2 0] s,
r (F + 2G)s, + (2F + G)s


Taking 0 = 90 so cos 0 = 0 and sin 0 = 1 then oy, = 90, cx = 0, ThXy


N -i 2- .

(5-20)

S0. Inserting


these into 5-8 gives


af
aj,


+ s,(4F + 4G


af
0j,

fa




af
9(7y





af
OTay
Qr~y


(5-15)


+ s,(4F + 4G


sin2 Of
(9car


(5-16)


(5-17)


+ os2 8
0ay


H) cos2 0]


af af
+Of Of


(5-18)


(5-19)











1
_2 y2
990 F+H


From (5-12) and (5-13)


and s = 90
and sy =0-90


Inserting these into (5-20) gives


(2G + H)(-90) + (G- H) 2
(F + 2G)(- 90) + (2F + G) 2

H


Now taking 0


0 so cos 8 = 1 and sin 0


0 then a,


00, ay 0,


Trx = 0. Inserting


these into (5-8) gives


Finally, taking 0


450, then a,


1
ay TX -45.


Inserting these into (5-8) gives


(5-24)


F+G+2N


Equations 5-21, 5-22, 5-23 and 5-24 gives four equations in the four unknowns H, G, H,

and N. These solve to


1
o90(1 + rgo)


1
G
G -
990


1 r90
0o 09o(1 + r90)


o90(1 + rgo)


N -2
2 j45


(5-25)


r90 1
90o(1 + r90o)


1
sx = 3 -90


(5-21)


(5-22)


G+H


(5-23)









The experimental data from the quasi-static tests on Plate 1 given in Table (5-2) and (5-3)

were used to determine the coefficients (given in Table 5-4) of the Hill(1948) criterion in

conjunction with relations given by Equation (5-25).

Table 5-2. Compressive yield data used to identify Hill48 parameter values
Direction x y z
Yield Strength (\iPa) 142.7 208.5 246.8


Table 5-3. Tensile yield data used to identify Hill48 parameter values
Direction x 450 y z
Yield Strength (\ Pa) 127.1 148.5 200.8 255.1


Table 5-4. Parameter values for Hill48 model using Plate 1 data
Hill Coeff F G H
Value 2.34E-05 -7.598E-06 3.15E-05


The theoretical Hill yield loci thus obtained are further compared to the theoretical

model and data in Figure 5-5. Note that Hill's yield surface cannot capture the observed

behavior while the proposed model describes very well the observed strength differential

effects.

5.4 FE Implementation of Proposed Model

Using the proposed orthotropic yield criteria, anisotropic elastic/viscoplastic models

are developed and implemented into the 2003 version [Johnson et al. (2003)] of the explicit

finite element code EPIC (Elastic Plastic Impact Calculations). The EPIC code has been

developed by Dr. Gordon Johnson under the primary sponsorship of the U.S. Air Force

and U.S. Army. The first documented (1977) version was 2D only but has evolved into a

1, 2 or 3D version with many additions and enhancements. All simulations were carried

out on a PC platform using Compag Visual Fortran Professional Edition 6.6a. The code

was compiled such that all real variables were double percision.


















400


200


0


-200


-400


-600 i i ii Ii
-600 -400 -200 0 200 400 600
Gx


Figure 5-5. Comparison of Hill's criterion to proposed criterion for Plate 1 data


5.4.1 Elastic-Plastic Model

In order to validate the model implementation against the four point beam test, the

proposed model was implemented into EPIC2003. The code has available the structure

to insert a user's subroutine to update the stress state given the current stress state, the

current strain rate and the current time step. The integration procedure implemented

solved for the updated stresses by enforcing the Kuhn-Tucker and consistency conditions.

An associated flow criteria is also assumed such that the stress potential G is the same as

the yield function. The derivative of the stress potential is used to determine the direction

of plastic flow. The flow rule becomes


S AO (5d26)
0o7









where 6Ep is the plastic strain increment, A is a scaler multiplier giving the magnitude of

the plastic strain increment, G is the stress potential, in this case the yield function f, and

a is the Cauchy stress tensor.

Within the stress update subroutine, an updated trial stress is computed assuming

an elastic increment, i.e. da = CedE, where C" is the elastic compliance tensor. The

updated trial stress is computed as updated = current + da. Next an equivalent stress

(a) is computed according to the proposed yield criterion and checked against the current

yield stress (Ys). If a < Ys the current stress state is elastic, the trial stresses are set to

the actual stresses. If a > Ys, the stress state is plastic. In order to update the stress

state for this case an implicit interactive scheme is used to return the stress state to the

yield surface. The initial guess for the iteration is the updated trial stresses. The total

accumulated plastic strain (E) is taken as the hardening parameter, the yield criterion

being

f(a,E) = (a,E) Y,(E) (5-27)

In the following j is used as the global counter, i.e. j is the current state and j + 1 is the

fully updated state and n is used as a local iteration counter, i.e. n is the current iteration

and n + 1 is the updated iteration count. Then the total increment of stress for a given

iteration step is

a Aa 1 + 6a\j4 (5-28)

where A indicates an increment over the entire time step and 6 indicates an increment for

each iteration. The correction to the stress due to plastic strains is

+1 +1 00 +l1
Jar$+l "ArA C07 n--I (5 29)

The derivatives of the equivalent stress are found by taking only the first term of a Taylor

series expansion about the current state

9aa j+1 9a J+1 2 j+1 2+} j+1
aa 1 a + 7 6aa + 7+ 6A (5-30)
a n+I a n aa2n 0a7 n









Keeping only the first term in (5-30)Equation (5-29) becomes


l6jl-- j cl (5 31)

Solving for the increment for a given iteration gives

( j+l c_ l,'-1^^j l 0c r j+l
6a+l- -C-16 ax+l a (5 32)

A Taylor series expansion of the yield criterion is used to obtain an approximation of the

increment of the effective plastic strain

f( J+ l .+ Il O jf ^l + f -j+ l j3
\ n+l n+l) cJ+n + r j+ 0 (5 33)


Evaluating derivatives at the previous step Equations (5-32) and (5-33) can be manipulated

to give
,(71 f l,j1 +1)
6+'1 a= C n a En a (5-34)
n+1 -asa ac aia s aS a
a 9a 1 0 dE
This can now be used in Equation (5-32) to find aj+ and finally the total stress

increment is found from Equation (5-28). The new stress is then evaluated to see if it

has converged within a specified tolerance. If not, this is used as the starting stress for

the next iteration. When the stress has converged, the global stress tensor is updated and

returned to the main program.

5.4.2 Elastic-viscoplastic Extension of the Proposed Model

The integration procedure used to implement the elastic-viscoplastic extension of the

model follows the consistency method described by Wang et al. (1997). Similarly to the

implementation of the rate-independent model, this method requires the stresses to ahv-w

lie on or in the interior of the rate-dependent yield surface f,

f a(a, Yp) Y(Ep, E,p) (5-35)









where the equivalent stress r(a, Ep) is computed using the proposed model given by

Equation (4-5); Y(-,p, Cp) is the current yield strength determined using an interpolation

hardening approach [Plunkett et al. (2006)] described in Section (5.4.5) and cr is the

current stress state passed in from the FE code; ,p and ,,p are the accumulated effective

visco-plastic strain and the effective visco-plastic strain rate, respectively.

Using an associated flow rule
8af
Ep = (5-36)

where A is a scalar multiplier since f is homogeneous of degree one in stresses. A is the

magnitude of the rate of change of the effective visco-plastic strain. It is assumed that

A = AA/At. The solution technique is to derive the stresses at time n + 1 based on the

state at time n and a known increment of total strain. In order to find this solution a

Taylor series expansion is made about the state at n

f f


+ 6An+l + f 6An+ (5-37)

Here, the subscripts refer to iteration steps. For the initial step, i.e. the state n = 0, is

the trial state of stress computed assuming elasticity. Denoted by A, a change in quantity

for a complete time step while 6 indicates the change during an integration step. So, for a

given time step
j=k
AA+1, -AAT + S JAj (5-38)
j=o
where k is the number of steps needed for convergence.

The stress variation is described by


6ar+l = -C6a,+l (5 39)
00,









Defining SA as 6A/At and using (5-39) in (5-37) an expression for the iterative variation

of A can be derived.

6A ti C = aa Y a' 1 9Y (5-40)
aT (T : t Evp
The stresses are then updated using


Ao-a+l Aao + 6ai+l Aj, C6A\+l (5 41)
Sn+1

The effective visco-plastic strain is updated as

-t+ALt t
P +At + AA (5-42)


Iterations continue until the yield criterion (5-35) is satisfied within a given tolerance. The

Johnson-Cook hardening law [Johnson et al. (1997)] described in C'! plter 1 was used for

simulations of the rate effects. This produced smoother derivatives than for the case of

piecewise linear hardening used for the elastic-plastic version of the model.

5.4.3 Effective Stress Calculation

The proposed criterion can be written as

a 1 [(Jl )3/2 -J ]C1/3 (543)


where 01 is a constant defined such as to assure that a reduces to the tensile yield stress

in the rolling direction. Thus,


S 2 )3/ -1/3
S= ( + a + a2a3)32 a2a3(a2 + a3) (5-44)


5.4.4 Derivatives of Yield Function

Derivatives of the yield function with respect to the stress are used in integrating the

increments of plastic strain in the FE implementation. The general form can be written as



OF OF OJ2 aO F OJ OE
00"r OJ2 OE ~cr O" 9. OE 9OC










OF OF aJ2 O9kl OF OJ3 O kl 4
+ (5-45)
0iaj aJ2 9 kl 01ij 9J83 9kl 90(ij

where
OF 0 11/2
29 (2 3/2 2/3
2 2(J2 C J3)

aF c
9J3 3/2 2/3
3 3(J2 c J3

The derivatives are

aJd2 1 aJ2 22
9S^ -11 2 22 S 33

=2 2E12 =', 2E13 j 2E23



9n11 1 8 22 1 O933 1
a- 1 (a2 + a3) a- -a3 -1 a2
01, 3 01, 3 07, 3

a311 1 al22 1 a3)33 1
a3 (a1 +a3) --3a1
0a7 3 0ay 3 0a7 3

8S11 1 8922 1 0833 1
a 2 1 (ai+ a
07-z 3 0jz 3 90z 3

az12 _9 13 23
Txy 0a4 z 05 7yz


All other derivatives are equal to zero.

5.4.5 Anisotropic Hardening

The FE implementation uses the procedure describing evolution of the yield surface

with texture changes proposed by Plunkett et al. (2006). It employs a linear interpolation

scheme between a discrete number of yield surfaces corresponding to fixed levels of

accumulated plastic strain. First, a given strain path on which to base the hardening

is chosen. For example, for all the quasi-static simulations performed in this study, the









Table 5-5. Plate 1 anisotropy coefficient values for discrete strain levels
Strain Yield al a2 a3 a4 a5 a6 c
0.0000 208 0.5454 0.5010 1.0900 0.7246 -0.8675 -0.8675 -0.2168
0.0250 245 0.5231 0.4745 0.9034 0.7309 0.7202 0.7202 -0.2198
0.0500 261 0.6694 0.5585 1.1030 0.9138 0.9381 0.9381 -0.2291
0.0750 273 0.6960 0.5969 1.1270 0.9838 0.9716 0.9716 -0.2607
0.1000 284 0.5356 0.4768 0.8603 0.7761 0.7714 0.7714 -0.2754
0.2000 317 0.0610 0.0576 0.0869 0.0870 0.0794 0.0794 -0.5908
0.4000 370 0.0632 0.0620 0.0788 0.0816 0.0801 0.0801 -1.0330
0.5000 389 0.9547 0.9570 1.2140 1.1810 1.1760 1.1760 -1.1480
Note: Yield Strength in MPa

hardening was based on the tensile strain path in the rolling direction. From this the Tc

stress versus strain curve, a discrete number of strain levels is chosen. A sufficient number

of points was used to ensure that the hardening curve was recreated with enough accuracy.

For each of these strain levels, ranging from 0 to 50' the corresponding yield surfaces

according to Equation (4-5) were determined. The anisotropiy coefficients shown in Tables

5-5, 5-6 were calculated following the procedure outlined in section 4.2.


5.4.6 Parameter Values

For Plate 1 iso-strain contours of the theoretical biaxial yield surfaces compared

to data are shown in Figure 5-6 A. The data used to identify the model parameters are

represented in Figure 5-6 B by symbols. A similar procedure for Plate 2, using the data

shown in Figure 5-7 B, give the yield loci shown in Figure 5-7 A. Recall that the tension

data required an extrpolation above approxiamtely 21' strain since the material began to

have localized strains beyond this strain level. Note that the model reproduces the data

quite well with the largest error for the biaxial data. Table 5-5 gives the uniaxial stresses

in the RD and anisotropy coefficient values determined at each strain level for Plate 1.

Table 5-6 gives the same information for Plate2.

5.5 FE Simulations

In order to verify that the model was implemented into the FE code correctly,

simulations were run and compared to experimental data from the high purity Titanium










































Figure 5-6.


2

C)
v^


2Strn
Strain (%)


41)0


Plate 1 yield data A) theoretical yield curves for fixed levels of accumulated
plastic strain B) Data used in identifying theoretical yield curves


(MPa) Strain

A B


Figure 5-7. Plate 2 yield data A) theoretical yield curves for fixed levels of accumulated
plastic strain B) Data used in identifying theoretical yield curves






118


600





400 ____________





200 ---M- IC

--A,- rr
9 rrr









Table 5-6. Plate 2 anisotropy coefficient values for discrete strain levels
Strain Yield al a2 a3 a4 a5 a6 c
0.0000 208 0.5454 0.5010 1.0900 0.7246 -0.8675 -0.8675 -0.2168
0.0250 245 0.5231 0.4745 0.9034 0.7309 0.7202 0.7202 -0.2198
0.0500 261 0.6694 0.5585 1.1030 0.9138 0.9381 0.9381 -0.2291
0.0750 273 0.6960 0.5969 1.1270 0.9838 0.9716 0.9716 -0.2607
0.1000 284 0.5356 0.4768 0.8603 0.7761 0.7714 0.7714 -0.2754
0.2000 317 0.0610 0.0576 0.0869 0.0870 0.0794 0.0794 -0.5908
0.4000 370 0.0632 0.0620 0.0788 0.0816 0.0801 0.0801 -1.0330
0.5000 389 0.9547 0.9570 1.2140 1.1810 1.1760 1.1760 -1.1480
Note: Yield Strength in MPa

obtained for the RD data used for the representation of hardening. Next simulations

of stress-strain response for other orientations were performed and compared to data.

Simulations involved a single computational cell with eight nodes with a single integration

point. The cell was streched in one direction along an axis and stress versus strain data

were collected and compared to the appropriate experimental data. The model was then

validated by simulating the four point bend tests described in C'!I pter 3. The comparison

was made in a qualitative way by juxtaposing contours of the experimental data against

the results from the simulation. This was done for all four orientations of the beam

specimens. Comparison of the experimental cross sections of the beams and simulated

ones using the model and an isotropic material with a von Mises yield surface were

performed. A more quantitative comparison was done by comparing axial strain versus

height at the centerline of the beam.



5.5.1 Single Cell

In order to verify the implementation of the model in the FE code, single element

(cell) simulations were carried out for both plates under six different uniaxial stress

conditions. The single element, shown in Figure 5-8, was an eight noded constant strain

element with a single integration point. For each simulation, four nodes on one face of

the element were restrained and the four nodes on the opposite face were given a constant

velocity in either the tensile or compressive direction. Stress and strain data were collected









and compared to experimental data. For all plots, the stresses are the normal stresses in

the appropriate direction and the strains are the reported effective accumulated plastic

strains (- strains).


Figure 5-8. Single cell computational configuration


5.5.1.1 Plate 1 results

Figures 5-9 to 5-11 show the comparison of single cell simulations for Plate 1. Note

that simulations accurately reproduce the data for each condition. The largest error occurs

for the TD tension data, which is consistent with the discrepancy between experimental

and predicted flow stresses noted previously.







































10 20 30 40 50
Effective Strain (%)


A


10 20 30 40 50
Effective Strain (%)


Figure 5-9. Single cell simulation results for Plate 1 A) RD tension B) RD compression


400



300



200



100
-- 90T
A Data


0 10 20 30 40 50
Effective Strain (%)


Figure 5-10. Single cell simulation results for Plate 1 A) TD tension B) TD compression








121


























Effective Strain (%)


A B

Figure 5-11. Single cell simulation results for Plate 1 A) TT tension B) TT compression


5.5.1.2 Plate 2 results

Figures 5-12 and 5-13 show the comparison of single cell simulations for Plate 2. For

the in-plane plots, both the RD and TD data are shown on the same plot. It is again

clear that that Plate 2 in nearly isotropic in the plane of the plate. Very good agreement

is found for all cases in Plate 2. The largest errors occur in the biaxial data which

corresponds to the largest errors between the predicted flow stresses and experimental

data noted earlier.

Even though the RD direction tensile data was used for modeling isotropic hardening,

the simulations os all the other stress paths were in good agreement with the data.

Such good agreement can be achieved only by accounting for texture evolution i,e, the

anisotropy tensor is considered a function of the accumulated deformation.

5.5.2 Four Point Bend Tests

The beam bending tests were simulated using 21,600 four noded tetrahedral elements

with a single integration point. Only half of the beam was simulated with a plane

of symmetry along the centerline cross section. The elements were arranged into a


Effective Strain (%)

















400



300 -



200



100
RD
TD
Data
0 -- -- ----- I I


20 30
Effective Strain (%)


40 50


Effective Strain (%)


Figure 5-12. Single cell simulation results for Plate 2 A) In-plane tension B) In-plane

compression


20 30
Effective Strain (%)


40 50


600


500


400 A -





200 S


100
Data


0 10 20 30 40 50
Effective Strain (%)


Figure 5-13. Single cell simulation results for Plate 2 A) Through thickness tension B)

Through thickness compression



"symmetrical" brick arrangement. This arranges 24 tetrahedral elements into a hexagonal


or brick structure as shown in Figure 5-14 This is done to minimize the well known stiff


behavior of this type of element.


500



400 -



300



200 -



100 1 I
A DATA


0 10


2










SYMMETRIC BRICK ARRANGEMENT


SN4






NIS N15 CENTERR )






N2 0-- 8 N3

SN12




TOTAL OF
6 Ix. /24 EEVENTS
N1 N7



Figure 5-14. Symmetrical brick arrangement for tetrahedral elements


The loading profile was applied to the appropriate side of the beam at the same

distance as the center of the loading pin (10 mm from the centerline). A line of nodes

was restrained on the opposite face of the beam (at 20 mm from the plane of symetry) to

simulate the constraining pin. The constraining nodes were restrained in the direction of

loading but were free for the other two directions with no friction. A typical computational

mesh is shown in Figure 5-15 for loading as prescribed by Case 1 All other simulations

used the same mesh with loading and constraint directions appropriate for the particular

case. Figure 5-16 shows a typical deformed countoured mesh indicating the plane of

symmetry.


















STD


RestrinutE


RD




Figure 5-15. FE Computational mesh for beam bending tests


Y






E
S0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08


Figure 5-16. Typical deformed mesh showing plane of symmetry








5.5.2.1 Plate 1 results
Results from simulations of the four point beam bend tests clearly show that the
model captures in, i.i' features of the anisotropic behavior of the high purity titanium
tested. The cross sections for Plate 1 are shown compared in Figure 5-17. As expected
when the hard direction (TT) is perpendicular to the loading direction (Case 1 and 3) the
cross section remains nearly square. Case 2 and Case 4 are similar to each other with more
lateral strain shown by Case 4. This is consistent with Plate 1 being harder in the TD
than the RD as shown in the tests (see Figure 5-18). The data shows that for strain levels
below 15' the plate is stronger in both tension and compression at 900 from the rolling
direction (TD) as compared to RD.


axial
0.14
0.12
0.1
0.08
S0.06
0.04
0.02
0
S-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
-0.14


ading


/
TT

1bB.


Figure 5-17. Comparison of cross sectional area from simulations of the four beam
orientations from Plate 1

A simulation using an isotropic von Mises type model was used to simulate the
four point bend test for comparison to the simulations ran using the anisotropic model.
Figure 5-19 shows a comparison between the isotropic simulation against the four beam













350

300---------

250







50 0 Comp
------ 90Comp

-15 -10 -5 0 5 10 15
Strain (%)


Figure 5-18. Comparison of tension versus compression data for RD and TD in Plate 1


Case 1


Case 2


Case 4


Figure 5-19. Comparison of cross sections from Plate 1 beam simulations (red mesh is
from isotropic simulation)



test orientation. Note that in Case 1 and Case 3, the hard direction (TT) is the width

direction and the isotropic simulation shows more deformation than that using the

anisotropic model. For Case 2 and Case 4 there is less deformation in the height of the







127









beam and the width shows deformation similar to the isotropic simulation. Again this is

due to the softer direction being aligned with the width for these cases.

The beam orientation for Case 1 is shown in Figure 5-20 for reference. Figure 5-21

shows the comparison of the profile of axial strain contours for the simulation for Case

1 compared to the experimental data. Note that the data from the experiment does not

cover the entire profile area due to the DIC \ I '!uil-Touchal et al. (1997); Hung and

Voloshin (2003)] techniques used. Very good agreement is shown.

y=TD





Figure 5-20. Case : Long axis in RD loading in TD
Figure 5-20. Case 1: Long axis in RD, loading in TD


2

0

-2

-4

-6

-8_


0
x (mm)


Figure 5-21. Plate 1, Case 1: Comparison of axial strain countours (F,) from simulation
against experimental data: x RD, y=TD









A more qualitative comparison is shown in Figure 5-22 which shows a plot of the

axial strain versus the height of the beam at the center of the beam. This shows very good

agreement between the experiment and simulation and a clear upward shift of the neutral

axis of the beam.


-10 -5 0
6X N%


5 10 15


Figure 5-22. Plate 1, Case 1: Axial strains (Ex) versus height at centerline: x=RD, y=TD


As a final validation of the model, the beams were sectioned at the midpoint and an

image of the cross section was compared to the simulation. The comparison for Plate 1

for Case 1 is shown in Figure 5-23. There is very little deformation perpendicular to the

loading direction because this is the harder, TT direction.

The beam orientation for Case 2 is shown in Figure 5-24 for reference. Figure 5-25

shows the comparison of the profile of axial strain contours for the simulation for Case


-415
15














6-

5

4




2-

1

0-


-4 -2 0 2 4
z (mm)

Figure 5-23. Plate 1, Case 1: Comparison of cross sections from experiment (photo) and
simulation (symbols): y=TD, z=TT

2 compared to the experimental data. Again, very good agreement is shown between
experiment and simulation. The simulation does give somewhat less strain through the
thickness in the loading direction.

z-TT


y=TD ,-
'I---
Figure 5-24. Case 2 Long axis in RD loading in TT


Figure 5-24. Case 2: Long axis in RD, loading in TT































-0.12 -0.06 0


-6 -4 -2 0
x (mm)


1
0.06 0.12


2 4 6 8


Figure 5-25.


Plate 1, Case 2: Comparison of axial strain
against experimental data: = RD, z=TT


countours (Ex) from simulation


5


Figure 5-26. Plate 1, Case 2: Axial strains (Ex) versus height at centerline: x=RD, z=TT







131


2 --_ _:.2 __-

.. .. - -
Now
-2

-4


-8-
-8


4

3 -

2

I l:.i.of
neutral axis


-1 ---------__

-2 --- --_ _
Simulation
-3 Experiment

-4 0 :
-15 -10 -5 0 5 10 1
FIX N%









A plot of the axial strain versus the height of the beam at the center of the beam

is shown in Figure 5-26. This shows very good agreement between the experiment and

simulation and a clear upward shift of the neutral axis of the beam.

The comparison of cross sections for Plate 1 for Case 2 is shown in Figure 5-27 which

shows very good agreement. There is more deformation perpendicular to the loading

direction because this is now the softer transverse direction.


-3 -2 -1 0
y (mm)


1 2 3 4


Figure 5-27. Plate 1, Case 2: Comparison of cross sections from experiment (photo) and
simulation (symbols): y=TD, z=TT


The beam orientation for Case 3 is shown in Figure 5-28 for reference. Figure 5-29

shows the comparison of the profile of axial strain contours for the simulation for Case 3

compared to the experimental data. Again, very good agreement is shown.


i i i i i i i i i i i i i i i i i i i i i i i i i i i i i


,I,,,,I, ,,I,,,,I,,,,I, ,,I,,,,I,,,,I


I


\


-14
-4-










Sx=-RD




=y TD


Figure 5-28. Case 3: Long axis in TD, loading in RD




8

6
6 Simulation Experiment

4

2




-2

-4


-0.12 -0.06
, I . I ,


0.06 0.12
I I


-6 -4 -2 0
y (mm)


2 4 6 8


Figure 5-29. ]
Plate 1, Case


3: Comparison of axial strain countours (E ) from simulation against
experimental data: = RD, y=TD


A plot of the axial strain versus the height of the beam at the center of the beam

is shown in Figure 5-30. This shows very good agreement between the experiment and

simulation and a clear upward shift of the neutral axis. The comparison of cross sections

for Plate 1 for Case 3 is shown in Figure 5-31 which shows excellent agreement.


. . . .II/ I/ / /I/ /


-6-t


-8














3 --

2

1 Shift of
neutral axis
01

---2


-Simulation
-3 1 Experiment

-4
15 -10 -5 0 5 10 15



Figure 5-30. Plate 1, Case 3: Axial strains (Fy) versus height at centerline: x=RD, y=TD


-14
-4


I I I I i i i I
-2 0 2
z (mm)


Figure 5-31. Plate 1, Case 3: Comparison of cross sections from experiment (photo) and
simulation (symbols): x RD, z=TT





134


I I










The beam orientation for Case 4 is shown in Figure 5-32 for reference. Figure 5-33

shows the comparison of the profile of axial strain contours for the simulation for Case 4

compared to the experimental data. Again, very good agreement is shown.


iniTr


x RD
y= TI


Figure 5-32. Case 4: Long axis in TD, loading in TT


0

-2

-4

-6

-8_


-0.12 -0.08 -0.04 0 0.04 0.08 0.12
I I I I r I I I I I I i r I i l l l i ri l i i i


-6 -4 -2 0
y (mm)


2 4 6


Figure 5-33. ]
Plate 1, Case


4: Comparison of axial strain countours (Ey) from simulation against
experimental data: y=TD, z=TT


A plot of the axial strain versus the height of the beam at the center of the beam

is shown in Figure 5-34. This shows very good agreement between the experiment and


0-




















I ` O-- 'neutral axis
z 0
-1

-2
Simulation
-3 Experiment

-4
15 -10 -5 0 5 10 15



Figure 5-34. Plate 1, Case 4: Axial strains (Ey) versus height at centerline: y=TD, z=TT


-2 0 2
x (mm)


Figure 5-35. Plate 1, Case 4: Comparison of cross
simulation (symbols): x RD, z=TT


sections from experiment (photo) and


simulation and a clear upward shift of the neutral axis of the beam. The comparison of

cross sections for Plate 1 for Case 4 is shown in Figure 5-35 which shows very agreement.








5.5.2.2 Plate 2 results
The cross sectional area from simulations of the four point beam bend tests for
Plate 2 is shown in Figure 5-36. Again these is very good qualitative .'-:-:reement with
experimental data. As for Plate 1, for the Case 1 and Case 3, where the through thickness
direction (the harder direction) is normal to the loading direction, there is very little
variation from a rectangular cross section. There is a much greater deviation for Case 2
and Case 4 where the hardest direction is in the loading direction. It was also noted that
Case 1 and Case 3 as well as Case 2 and Case 4 are similar due to the in-plane isotropy of
Plate 2.

Case Case2




TD TT




TT TD

Case3 Case4





RD ---_ TT-



TT RD


Figure 5-36. Comparison of cross sectional area for Plate 2









For Plate 2, the variation in cross section versus a simulation using an isotropic, von

Mises material model is shown in Figures 5-37 and 5-38. Again, when the width of the

beam corresponds to the hard (through thickness) direction, very little distorsion of the

cross section is observed.


Figure 5-37. Plate 2 Isotropic simulation (black lines) versus model (blue and red lines)
Case 1 and 3


Figure 5-38. Plate 2 Isotropic simulation (black lines) versus model (blue and red lines)
Case 2 and 4


The beam orientations for Plate 2, Case 1 to Case 4 are the same as those for Plate 1.

Figure 5-25 shows the comparison of the profile of axial strain contours for the simulation

for Case 1 compared to the experimental data. Very good agreement is shown.
















6


4


2


0


-2


-4


-6


-8


2 4 6 8


Figure 5-39.


Plate 2, Case 1: Comparison of axial strain (
against experimental data: x RD, y=TD


IX) countours from simulation


4






Shift of
-neutral axis


-2

Simulation
--A--- Experiment

-4
-15 -10 -5 0 5 10 1:
Ex (%)


Figure 5-40. Plate 2, Case 1: Axial strains (Ex) versus height at centerline: x=RD, y=TD


A more qualitative comparison showing the plot of the axial strain versus the height

of the beam at the centerline is shown in Figure 5-40. This shows good agreement between

the experiment and simulation and a clear upward shift of the neutral axis of the beam.


139


-6 -4 -2 0
x (mm)


8


Simulation Experiment















-0.12 -0.08 -0.04 -0.00 0.04 0.08 0.12


5










The comparison of the experimental cross section for Plate 2, Case 1 is shown in

Figure 5-41 showing excellent agreement. Very little deformation occurs perpendicular to

the loading direction since this is the harder through thickness direction.


III Ii,, i, I I


-1 0
z (mm)


2 3 4


Figure 5-41. Plate 2, Case 1: Comparison of cross sections from experiment (photo) and
simulation (symbols): y=TD, z=TT


Figure 5-42 shows the comparison of the profile of axial strain contours for the

simulation for Case 2 compared to the experimental data. Again, very good agreement is

shown. A plot of the axial strain versus the height of the beam at the center of the beam

is shown in Figure 5-43. This shows very good agreement between the experiment and

simulation and a clear upward shift of the neutral axis of the beam.

The comparison of cross sections for Plate 2 for Case 2 is shown in Figure 5-44 which

shows very good agreement. There is more deformation perpendicular to the loading

direction because this is now the softer transverse direction.


0

-4
-4 -3 -2



















6


4


2


0


-2


-4


-6


-8


Figure 5-42.


Plate 2, Case 2: Comparison of axial strain countours
against experimental data: x RD, z=TT


(Fx) from simulation


2

Shift of
-- neutral axis




-2
Simulation
SExperiment

-4
20 -15 -10 -5 0 5 10 15 21
ex (%)


0


Figure 5-43. Plate 2, Case 2: Axial strains (Ex) versus height at centerline: x RD, z=TT





141


-6 -4 -2 0 2 4 6 8
x (mm)


8


Simulation Experiment
















-0.12 -0.08 -0.04 -0.00 0.04 0.08 0.12

































1 1 1 1 1 1 1 1 1 1 1 1 -


-3 -2 -1 0
y (mm)


1 2 3 4


Figure 5-44. Plate 2, Case 2: Comparison of cross sections from experiment (photo) and
simulation (symbols): y=TD, z=TT



Figure 5-45 shows the comparison of the profile of axial strain contours for the

simulation for Case 3 compared to the experimental data with excellent agreement.


8

6
Simulation I
4

2

0

-2

-4

-6 6Y
-0.12 -0.08 -0.04 -0.00

-8 -6 -4 -2 0
y (mm)


0.04 0.08 0.12

2 4 6 8


Figure 5-45. Plate 2, Case 3: Comparison of axial strain countours (Ey) from simulation
against experimental data: = RD, y=TD











A plot of the axial strain versus the height of the beam at the center of the beam

is shown in Figure 5-46 and the comparison of cross sections from simulation and

experiment for Plate 2 Case 3 is shown in Figure 5-47. Excellent agreement is shown

including a clear upward shift of the neutral axis.



4



2

Shift of
H neutral axis



-2
Simulation
Experiment

F4
15 -10 -5 0 5 10 15
igure 5-46. Plate 2, Case 3: Axial strains (%)


Figure 5-46. Plate 2, Case 3: Axial strains (y) versus height at centerline: x RD, y TD


ii Iii ii .. I 1.. 1. ii


-3 -2 -1 0
z (mm)


1 2 3 4


Figure 5-47. Plate 2, Case 3: Comparison of cross sections from experiment (photo) and
simulation (symbols): = RD, z=TT










Figure 5-48 shows the comparison of the profile of axial strain contours for the

simulation for Case 4 compared to the experimental data. Again, very good agreement is

shown.


Simulation Experiment












-0.12 -0.08 -0.04 -0.00 0.04 0.08 0.12

i1 j 1 ,7 1 i, I 'I 1....


-6 -4 -2 0
y (mm)


2 4 6


Figure 5-48.


Plate 2, Case 4: Comparison of axial strain countours (Fy)
against experimental data: y=TD, z=TT


from simulation


2

Shift of
0 -',"* -' neutral axis



-2
Simulation
------ Experiment

15 -10 -5 0 5 10 1
y (%)


Figure 5-49. Plate 2, Case 4: Axial strains (Ey) versus height at centerline: y=TD, z=TT




144


5









A plot of the axial strain versus the height of the beam at the center of the beam

is shown in Figure 5-49. This shows very good agreement between the experiment and

simulation and a clear upward shift of the neutral axis of the beam.

The comparison of cross sections for Plate 2 for Case 4 is shown in Figure 5-50 which

shows close agreement.


IlI I II I Ii I I I I I I iiliiiiliiii Ii I 11 1


-4 -3 -2 -1 0
x (mm)


1 2 3 4


Figure 5-50. Plate 2, Case 4: Comparison of cross sections from experiment (photo) and
simulation (symbols): x RD, z=TT


5.5.3 Cylinder Impact Tests

Cylinder impact validation tests were performed on specimens from Plate 2 only.

The geometry of the plate did not allow for specimens to be cut from the through

thickness direction thus specimens were cut such as the cylinder axis were ahv-- i along

the in-plane direction. The through thickness direction was marked on the end section









of each specimen during fabrication as shown in Figure 5-51A. Post test observations

confirmed the in-plane isotropy of the material. Also,the minor axis of the deformed

specimens were aligned to within 5 degrees of this mark. This is as expected since the

the specimens have basal texture i.e. the hard to deform c-axis direction lies in the cross

section. A photograph of the deformed footprint at the cylinder-anvil interface is shown

in Figure 5-51B and compared to a true circle clearly shows the anisotropic deformation

in this plane. The axis with less deformation, the minor axis, is nearly aligned with the

c-axis of the material.












Post test damage
A B
Figure 5-51. Deformed high rate specimen A) Test specimen with through thickness
direction identified by arrow. B) The deformed elliptical footprint from
experiment as compared to a circle.


5.5.3.1 Hardening

In the simulations of the high rate validation tests the phenominological Johnson-Cook

(J-C) hardening law was used which incorporates strain rate effects. The J-C law is given

by

Ys = (C'i + C2F) + 3 In*)(- T* ) (546)

where Ys is the yield strength, e is the accumulated effective plastic strain, and F* is

the dimensionless total strain rate such that >* = with ,o 1.Os-1. T* T is
an homologous temperature and T is the melting temperature. C C, lt and
an homologous temperature and Tmeit is the melting temperature. C(1, C2, C3, N and












6E+08


5E+08


4E+08


S3E+08


2E+08

HR Data
1E+08 Linear Fit


0 0.05 0.1 0.15 0.2 0.25 0.3
Strain


Figure 5-52. High rate compresive data with linear fit used in parameter identification


M are material constants. In all simulations the parameter M was set to zero, i.e. only

isothermal conditions were considered.

The parameters involved in Equation 5-46 were identified from in-plane compression

data rather than in-plane tensile data as for the four point beam beam simulations. This

is because the dominant loading during the Taylor tests is in-plane compression. The

quasi-staic data was used to obtain the parameters C1, C2 and N. A linear curve fit to the

high rate compression tests was used to identify the parameter C3 (see Figure 5-52).

The built in minerr function of MathCad was used to identify all parameters. The

cost function using only the quasi-static data is


Error =- [QSYeperiment Yj(C1, C2, C3 0, N)12


where QSYexperiment is the quasi-static experimental data at i descrete strain levels and

Yj-c(C1, C2, C3 0, N) is the yield strength computed from the J-C Model (Equation

5-46) at the same discrete strain levels.










The cost function including the high rate data is


Error E{ [QSYexperiment Y C1C, C2, C3 = 0, N)]

+ [HRYexperiment Yj-c(C1, C2, C3, N)1 }2


where HRYexperiment is the high rate experimental data at i discrete strain levels. The J-C

parameter values are given in Table 5-7.

Table 5-7. Johnson-Cook hardening law parameter values for Equation 5-46
C1 C2 C3 N
1.781e+007 6.477e+008 0.06375 0.4214


Figure 5-53 shows the comparison between the values obtained using the J-C model

for the set of values given in Table 5-7 and experimental data used in the identification of

the respective parameters.


700 = (C1+C2EN)(1+C3Edot)

600

500

2 400

S300"

200 QS Experiment
200 -- JC QS Fit
-- HR Experiment
100 Linear Fit HR Exp
-- JC HR Fit

0 0.1 0.2 0.3 0.4 0.5
Strain

Figure 5-53. Comparison of yield values obtained from J-C law to experimental data used
in parameter identification


5.5.3.2 Finite element mesh

Simulations were carried out with the EPIC2003 code using 34,560 four-node

tetrahedral elements with a single integration point. The initial computational mesh is

shown in Figure 5-54. Again, the "symmetrical" brick arrangement was used in order to









minimize the known overly stiff behavior of this type of element. The simulations were run

at an impact velocity of 196 m/s as in the experimental tests.


(c)


Figure 5-54. Initial FE mesh for Taylor impact simulations with 34,560 four-node
tetrahedral elements: specimen dimensions are: Height =2.1 inches,
Diameter = 0.21 inches. (a) 3-D view, (b) cross section, (c) initial profile


5.5.3.3 Simulations

First, simulation results are presented for the case where the anisotropy parameters

are set to unity isotropicc case) and with J-C hardening (Equation (5-46)) including rate

effects. Next results are given for a simulation using the proposed anisotropic visco-plastic

model with the parameters given in Table 5-6 and the J-C hardening law with C3 = 0 (i.e.

the rate term is not activated). Finally a simulation using the anisotropic visco-plastic

model and the J-C model with the rate term activated is given. For the isotropic case, Ys

in the hardening law is the effective von Mises stress, while for rhe anisotropy cases Ys is

the effective stress associated with the proposed yield function given by Equation 4-5.


Y +








A simulation was carried out with all six of the a, coefficients set to 1 and the
strength differential parameter, c, was set to 0 thus yielding is described by the von Mises
law. The results show that the deformed cross section remains circular, i.e. there is no
preferred direction. Figure 5-56 shows a comparison of profiles taken from 900 around the
deformed cylinder. Note that the two profiles lie on top of one another as expected for
an isotropic material. Figure 5-55 shows the deformed specimen and final cross section,
respectively.


(a)
(a)


x



(b)


L.


H o
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
S-0.35
-0.4
-0.45
-0.5


(c)


Figure 5-55. Cylinder impact simulation results using isotropic von Mises and J-C
hardening law with rate effects activated, (a) deformed profile, (b) 3D view
(c) deformed footprint





150

















30


S20


10 -
Profile 1
Profile 2
0-
2.7 2.8 2.9 3 3.1 3.2 3.3
Radius (mm)


Figure 5-56. Comparison of profiles from isotropic simulation taken at 900 around
circumference using J-C hardening law with rate effects activated



The simulation using the proposed anisotropic elastic/plastic model was carried out

where the anisotropic coefficients are functions of the plastic strain as discussed in the

interpolation procedure in section 5.4.5. Thus the texture evolution is accounted for.

Isotropic hardening is described by the J-C law without taking rate effects into account

(i.e. C3 = 0). The simulation results show an elliptical footprint, i.e. the surface of the

specimen impacting the rigid anvil. The minor axis is aligned with the through thickness

direction of the plate. The deformed mesh and footprint are shown in Figure 5-57. Profiles

from 900 along the circumference of the cylinder are compared in Figure 5-58 showing a

significant difference between the 1 ii, Pr and minor axes.

The cylinder impact tests were also simulated using the proposed anisotropic

viscoplastic model including the full J-C model with rate effects turned on. The results are

closer to experimental data and show less deformation than the case where no rate effects

were included. A comparison of the i1 i' j r and minor profiles is shown in Figure 5-59. The

deformed profiles and the final cross section are shown in Figure 5-60.

























,-.-


x-A
......,... ~


x-r


Figure 5-57. Cylinder impact simulation results using proposed anisotropic elastic/plastic
model and J-C hardening without rate effects (a) Major profile; aligned with
in-plane direction (b) Minor profile; aligned with through thickness direction
(c) 3D view of specimen with axial strain countours (d) Final cross section


r


















Major
S Minor
40


30 -


20


10



2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4
Radius


Figure 5-58. Comparison of deformed cylinder profile from two locations 900 apart for
simlation using proposed anisotropic model and isotropic hardening according
to J-C law with no rate effects included in the simulations)


50


40 -- Major_
-- Minor


30


20


10


'6 2 83


Radius (mm)


Figure 5-59. Comparison of deformed cylinder profile from two locations 900 apart
obtained using proposed anisotropic fit to proposed elastic/viscoplastic model
and J-C hardening with rate effects






153


J. J.J J.L J5


C






















NNA I\,_ A

N-,f


Nvl^
^ 4.1-^
''- -* 14'a
Ir-''-".E


X







:~c~ll
+,
ir





W


'-I
N


"-N-
17 ---1, [ I
Ls.! 11 -" :.LL1L~ L~'I' Ll"l'~' 'I" I I II'


Figure 5-60. Cylinder impact simulation results using anisotropic parameters for proposed
criteria using J-C hardening with rate effects (a) Major profile; aligned with
in-plane direction (b) Minor profile; aligned with through thickness direction
(c) 3D view of specimen with axial strain countours (d) Final cross section


~I''


nl-~lr' ~

.Ir~lXI~;GYI:;~:I;':I:r3;~'7:i~'T:.T r~T`n~T'rr:~ Ti7XTIITn'


i. '~:.I : -' .1.'1'-' ~. '-~~ '- .L '' L `-' I'' L'-YLLIL L _lYll llL IIII II_1II I_1 I I


t


o









Figures 5-61 to 5-62 show the comparison of deformed specimens obtained using

the isotropic model, anisotropic model with no rate effects and the elastic/viscoplastic

anisotropic model, respectively. Specifically, Figure 5-61 shows the comparison of 1n i.'

axes profiles, Figure 5-62 shows the comparison of minor profiles and Figure 5-63 compares

the foot print simulated in each case. Note that the total height of the deformed cylinder

with no rate effects is less than for both the isotropic simulation (using rate effects) and

the rate-dependent anisotropic model. Also, in the rate-independent simulations, there

is more radial deformation than for the other two rate-dependent cases. This clearly

demonstrates the need to model rate effects in order to capture the characteristics of the

deformation under high strain rates.

Initial


W/Rate Isotropic
No Rate

















(a) (b) (c) (d)

Figure 5-61. Comparison of 1 i' jr profiles obtained using the different models (a)
Undeformed mesh (b) anisotropic model with rate effects (c) isotropic von
Mises with rate effects (d) anisotropic model with no rate effects













W/R sato


IsotroDic


No Rate


Comparison of minor profiles obtained using the different models (a)
Undeformed mesh (b) anisotropic model with rate effects (c) isotropic von
Mises with rate effects (d) anisotropic model with no rate


Initial




(a)


Figure 5-63.


W/Rate
1JF-ff-


Isotropic

(c4 )


(c)


No Rate


Comparison of the predicted foot print obtained using the different models
(a) Undeformed mesh (b) anisotropic model with rate effects (c) isotropic von
Mises with rate effects (d) anisotropic model with no rate mesh


Test number RM107 was taken as typical from the 13 tests performed and was used

to compare to validation simulations. Profile data were taken from the 1n i, i and minor

axes as well as the final deformed footprint as described in section 3.2.2 of C!i ipter 3 using

using an optical comparator model DIJ 415.


Figure 5-62.










Figure 5-64 shows a comparison between the simulated and experimental data. The

simulation matches the ii, i, '.i axis very well while slightly underpredicting the minor

axis deformation. Figure 5-65 shows the comparison of axial strains along the ini i i.


-4 -3 -2 -1 0 1
In Plane (mm)


2 3 4 5


Figure 5-64. Comparison of deformed impact surface: FE simulations with viscoplastic
model to experimental data


axes versus height from experimental data to that obtained from simualtions including

rate effects. The strains from the simulation near the impact face are very similar to the

experimental data but show more deformation as the height increases. This probably

arises because the stress-strain behavior for only two different strain rates was available,

thus the uncertainty related to the determination of these parameters. Having data from

higher rates for the same material should provide a more accurate fit to the true hardening

behavior under dynamic conditions.


157


Model
























1 20
r



tO




0 0.1 0.2 0.3 0.4 0.5
ln[(D-D,)/D,]


Figure 5-65. Comparison of i i' 'r axis radial strain versus height predicted by the
anisotropic model and experimental data



Figure 5-66 shows axial strains along the minor axis versus height for the three cases.


Both simulations under predict the deformation along the minor axis. This is probably a


result of errors in the parameterization of the proposed model rather than entirely rate


effects.


50

45

40 Model
SExperiment
35 -|

30

E 25 ____ ________

.9 20
r
15

10

5 )



0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
ln[(D-Do)/Dol


Figure 5-66. Comparison of minor axis radial strain versus height predicted by the
anisotropic model and experimental data











A comparison between the measured and simulated eccentricity of the footprint, i.e.

the ratio of 1n i .i diameter to minor diameter, versus height is shown in Figure 5-67. Note

the very good agreement between experiment and simulation.


1 1.2
D ./D.
major minor


Figure 5-67. Comparison of ratio of ii i, Pi r to minor diameters versus heightpredicted by
the anisotropic model and experimental data


Experiment
Model


I I


I I









CHAPTER 6
CONCLUSIONS

6.1 Present Research

As so aptly stated by Lemaitre (2001), material modeling can be considered "a

science, a technique, and an art." In this dissertation all these facets of modeling have

been considered. The science aspect consists of the careful and systematic experimental

characterization of the behavior under loading and in the effort to include the main

features of the observed behavior in an anisotropic model. The technique is in identifying

model parameters and integrating the model to predict the behavior of the material under

loading conditions other than those used to build and parameterize the model. Equally

important is the engineering art of using the very nonlinear model to predict the nonlinear

behavior of a material that is evolving as the texture evolves.

This entails the incorporation of physics/phenomena at different length scales.

Classical plasticity accounts for plastic deformation, which at < i--I I1 scale occurs through

slip associated with dislocation motion. For hexagonal closed packed (hcp) materials,

at the single <( i I level, one has to include twinning as an additional mechanism of

plastic deformation. Twinning is responsible for drastic and abrupt lattice rotations

which in turn lead to significant texture evolution during even the simplest loading paths.

Twinning being a polar shear mechanism, induces a tension/compression .,-vmmmetry at

the macroscopic scale. C('! i:terization and modeling of the interplay between slip and

twinning and their effect on the mechanical response remains a great challenge. This

dissertation is an attempt to extend the current knowledge on hcp materials.

This work has consisted of three 1n i, r areas that somewhat correspond to the three

aspects described above. First, an experimental investigation into the behavior of high

purity titanium was conducted. Two high purity titanium plates were considered; one with

an orthotropic texture and one which was isotropic in the plane of the plate but differed in

the direction normal to the plane of the plate.









The experimental investigation of the two plates described in ('!i Ilter 3, which

included uniaxial tensile and compression tests at both quasi-static and high loading

rates. Validation experiments under quasi-static conditions consisted of a series of four

point bending tests on beams cut such that their long axes were aligned either with

the rolling direction or transverse direction; loading was applied either in the rolling,

transverse, or through thickness direction. Since the top beam fibers are in compression

while the bottom fibers are in tension, the bending tests results test the ability of models

to capture both the strength differential effects and anisotropy of a titanium. The classical

Taylor cylinder impact tests were carried out for validation at high loading rates. In

addition, investigations were made to establish the initial texture of both plates. For

Plate 1 the texture evolution with plastic deformation was investigated primarily under

compression. Furthermore, Orientation Image Microscopy (OIM) measurements were made

for specimens loaded in compression in the rolling direction since the stress-strain data

from these tests indicated a significant increase in hardening rate, hence the possibility of

deformation twinning. The texture measurements showed a clear rotation of the c-axes

of the grains associated with twinning. All of the texture measurements corraborated the

uniaxial stress-strain data which showed that twinning p1 i, d a significant role for this

loading condition.

A new anisotropic yield criterion was developed in order to model the observed

behavior (see C'!i pter 4). The model proposed is an extension to orthotropy of an

isotropic description from Cazacu and Barlat (2004) using a linear transformation

approach. For general (3D) conditions, the proposed anisotropic model involves seven

parameters: 6 anisotropy coefficients and 1 parameter associated to strength differential

effects. The approach to modeling the evolution of the yield surface to account for

the texture evolution occurring in the material was to use the linear interpolation

scheme developed by Plunkett et al. (2006). The methodology consists of computing

an equivalent stress according to the anisotropic criterion at discrete strain levels and









then use interpolation to determine the equivalent stress for effective strains lying between

these discrete levels. The versatility of the model was demonstrated through comparison

with data. Although, the piecewise linear hardening law was calibrated based on the

in-plane stress-strain curve in the rolling direction, the response of the material was well

described for all the other strain paths (i.e. tension and compression in the RD,TD, ND).

An elastic/viscoplastic extension of the model was also developed and used to describe the

dynamic plastic response of the material. Since the propensity for twinning is increasing

with increasing 1. lii:.- /strain rate, the in-plane stress-strain curve was used to calibrate

hardening.

Both the elastic/plastic and elastic/viscoplastic models were implented in an

explicit Finite Element code (see C'! lpter 5). The implementation was verified by

simulating the uniaxial loading tests using a single constant strain element. The four

point validation tests were simulated using the elastic plastic model developed for all four

beam configurations. The results show an excellent agreement between the simulation

results and the deformed specimens using various comparisons. For all four cases for each

plate, the model was able to closely match the cross sectional deformation. When the hard

to deform direction i.e. the through thickness direction, was perpendicular to the loading

direction the final (deformed) cross sections were nearly square while when the loading

direction was aligned with the through thickness direction, the deformed cross sections

were more wedge-shaped. A comparison was also made to the axial strain versus height at

the center of each specimen. Again the simulations showed excellent agreement with the

experiment for all cases including a clear shift of the neutral axis from the centerline of the

beams.

Finally, the rate sensitive version of the model was used in simulations of a cylinder

impact test for one of the plates. The simulated deformed profiles as well as the elliptical

footprint of the surface striking the anvil were compared to the profiles of the deformed

cylinders. For these simulations the Johnson-Cook [Johnson and Cook (1983)1 hardening









law was used to include the effects of strain rate. Good agreement was shown between

the simulations and the the test data. The simulation results using rate effects was

compared to the same simulation without including rate effects and with simulations

including rate effects but using an isotropic yield function. The results clearly show the

need to incorporate both rate effects and anisotropic behavior when simulating high rate

deformation of this material.

6.2 Future Research

Although the experimental portion of this research was quite extensive it did

not investigate all possible loading environments. Some of these are currently being

investigated and some are yet to be funded. All of the parameters of the proposed model

were identified based on monotonic uniaxial loading results. Further, validation of the

model is recommended for other strain paths There is a lack of data on a titanium when

subjected to simple shear or for complex loadings involving strain path changes. Further

experimental investigation is needed. Collaborative efforts with Univ Paris 13 (Dr Salima

Bouvier) are being done in order to provide such data.

6.3 Concluding Remarks

As stated in the introduction of the proposed model in section 4.1, a primary goal

of this research is to advance the current state-of-the-art by developing user-friendly,

micro-structurally based and numerically robust macroscopic constitutive models that

can capture with accuracy the particularities of the plastic response of hexagonal metals,

in particular high puirity titanium. It has been demonstrated that this goal has been

met to a large degree. Further research is needed to explore other loading environments

but this work has shown that the proposed model can be parameratrized by simple

uniaxial test data and used to simulate more complex loading. The ability to incorporate

data from other loading conditions is already in place. Although, this research was

concerned primarily with high purity titanium, it is felt that the proposed model and

implementation approach is quite valid for other HCP metals..









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Budianski, B., 1984. Anisotropic Plasticity of Plane-lsotropic Sheets. Mechanics of
Material Behavior. Elsevier, Amsterdam.

Cazacu, O., Barlat, F., 2003. Application of the theory of representation to describe
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Gray, G. T., 1997. Influence of strain rate and temperature on the structure/property
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Hosford, W., 1972. A generalized isotropic yield criterion. Journal of Applied Mechanics
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Hosford, W. F., 1966. Texture strengthening. Metals Engineering Quarterly 6 (4).

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BIOGRAPHICAL SKETCH

Michael Eugene Nixon was born on June 5, 1953 in Lafayette Indiana, the third

child of Rufus and Irene Nixon. The family moved to northwest Florida while Michael

was a young child. He graduated high school in Crestview Florida. Michael spent 6 years

enlisted in the United States Air Force before earning a degree in Mechanical Engineering

from Auburn University in 1982. In 1983 he began work at the Air Force Armament

Test Laboratory, now the Air Force Research Labortory, at Eglin AFB, FL. In 1992 he

obtained his Master's Degree in Engineering Mechanics from the University of Florida and

completed his Ph.D. work in 2008 at the University of Florida Research and Engineering

Education Facility in Shalimar Florida. Michael is currently married to T inirni, Nixon and

resides in Crestview, Florida.





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ThelistofpersonswhohaveinuencedmylifeandthisresearchistoolongtoproperlydocumentbutImustattempttothankanumberofpeople.FirstIwouldliketothankmyfamily,especiallymywifeandmother.IwouldalsoliketoacknowledgethestrongsupportofmyemployersattheAirForceResearchLabMunitionsDirectorate,inparticularDr.LarryLijewski.Icouldnothavebeguntoaccomplishthelevelofeortthatwentintothisresearchwithoutthecontributionsofmanyofmyprofessionalcollaborators.FirstmycolleaguesfromtheAirForceResearchLab.Dr.BrianPlunkettprovidedmanyinsightsintomaterialmodelingandwithhelpindistinguishingwhatwasimportantandwhatwaslessimportant.Havinghimnextdoortomyoceprovedtobeavaluableassetwhenmyinsightfailedmeormyenergieslagged.Dr.MartinSchmidtprovidedmewithinsightsinhowtogetthroughthemazesurroundingobtainingadegreeandwhentonallysayenoughisenough.ThankstoJoelStewartforhisdeepphilosophicalinsights.Dr.JoelHousehasprovidedmanyhoursofdiscussionsoverseveralyears.Technicaltopicsincludingdislocationmotionandtwinningandotherdiscussionsonhowtomaintainmysanitywhenitseemslikethewholeworldhasgonecrazy.IwouldalsoliketothankJoelandPhilipFlaterforprovidingmuchoftheexperimentaldata.IalsohadalotofhelpfrommycolleaguesattheLosAlamosNationalLabortorythatnotmanygraduatestudentsgettoenjoy.Dr.RicardoLebensohnnotonlyprovidedvaluableinsightintothemechanicsofdeformationinhcpmaterialsandthepolycrystallinecodeVPSCbutactedasmyliaisonformuchofthequasi-statictestingandtextureinvestigationsreportedhere.Hispatienceanddiligenceismuchappreciated.IwouldalsoliketoacknowledgethediscussionswithDr.CarlosTomeandDr.GeorgeKaschnerconcerningtheVPSCcode,multi-scalematerialbehavior,andexperimentaltechniques.SpecialthankstoManuelLovatoandDr.ChengLiuforperformingthequasi-static 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 9 LISTOFFIGURES .................................... 10 ABSTRACT ........................................ 17 CHAPTER 1GENERALINTRODUCTION ........................... 19 1.1MaterialModeling ............................... 20 1.1.1Elasticity ................................. 20 1.1.2Plasticity ................................. 21 1.1.2.1Isotropicyieldsurfaces .................... 22 1.1.2.2Anisotropicyieldsurfaces .................. 24 1.1.2.3Flowrules ........................... 27 1.1.2.4Hardening ........................... 27 2TITANIUM ...................................... 30 2.1BasicProperties ................................. 30 2.2SingleCrystalProperties ............................ 32 2.3DeformationMechanisms ............................ 33 2.3.1Slip .................................... 33 2.3.2Twinning ................................. 34 2.3.3Hardening ................................ 34 2.4HighPurityTitaniumPlates ......................... 35 3EXPERIMENTS ................................... 39 3.1Quasi-StaticTests ................................ 40 3.1.1CharacterizationTests ......................... 40 3.1.1.1Testdescription ........................ 40 3.1.1.2Plate1results ......................... 42 3.1.1.3Plate2results ......................... 47 3.1.2FourPointBeamBendTests ...................... 50 3.1.2.1Fourpointbeambendtestresults:Plate1 ......... 53 3.1.2.2Fourpointbeambendtestresults:Plate2 ......... 57 3.2HighRateTests ................................. 61 3.2.1CharacterizationTests ......................... 61 3.2.1.1DescriptionofthesplitHopkinsonpressurebar ...... 62 3.2.1.2Testresults .......................... 64 3.2.1.3Plate1HRcharacterizationtests .............. 66 6

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.............. 68 3.2.2CylinderImpactTests ......................... 70 3.3Texture ..................................... 74 3.3.1GrainSize ................................ 76 3.3.2DeterminationofRollingDirection .................. 79 3.3.3VariationofTextureinThroughThicknessDirection ........ 81 3.3.4TextureEvolution ............................ 86 4MODELING ..................................... 92 4.1ProposedYieldCriterion ............................ 92 4.1.1IsotropicYieldFunction ........................ 92 4.1.2ExtensiontoOrthotropy ........................ 94 4.2IdenticationofMaterialParameters ..................... 98 4.2.1Costfunctioninvolvingonlyyieldstresses .............. 99 4.2.2CostfunctioninvolvingLankfordcoecients ............. 99 4.2.3Costfunctioninvolvingbiaxialdata .................. 99 4.3AnisotropicHardening ............................. 100 5SIMULATIONS .................................... 102 5.1ApplicationtoMg-LiAlloy ........................... 102 5.2ApplicationtoHigh-purityTitanium ..................... 103 5.2.1Plate1 .................................. 103 5.2.2Plate2 .................................. 104 5.3ComparisontoHill'sQuadraticModel .................... 105 5.4FEImplementationofProposedModel .................... 110 5.4.1Elastic-PlasticModel .......................... 111 5.4.2Elastic-viscoplasticExtensionoftheProposedModel ........ 113 5.4.3EectiveStressCalculation ....................... 115 5.4.4DerivativesofYieldFunction ...................... 115 5.4.5AnisotropicHardening ......................... 116 5.4.6ParameterValues ............................ 117 5.5FESimulations ................................. 117 5.5.1SingleCell ................................ 119 5.5.1.1Plate1results ......................... 120 5.5.1.2Plate2results ......................... 122 5.5.2FourPointBendTests ......................... 122 5.5.2.1Plate1results ......................... 126 5.5.2.2Plate2results ......................... 137 5.5.3CylinderImpactTests ......................... 145 5.5.3.1Hardening ........................... 146 5.5.3.2Finiteelementmesh ..................... 148 5.5.3.3Simulations .......................... 149 7

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................................... 160 6.1PresentResearch ................................ 160 6.2FutureResearch ................................. 163 6.3ConcludingRemarks .............................. 163 REFERENCES ....................................... 164 BIOGRAPHICALSKETCH ................................ 168 8

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Table page 1-1Phenomenologicalyieldfunctions .......................... 25 2-1PhysicalpropertiesofTitanium ........................... 31 2-2Chemicalanalysisoftestmaterial .......................... 36 3-1MeasurementsofdeformedbeambendspecimensfromPlate1 ......... 57 3-2MeasurementsofdeformedbeambendspecimensfromPlate2 ......... 61 3-3StrainratesacheivedfortensileSHPBtests .................... 64 3-4StrainratesacheivedfortensileSHPBtests .................... 64 3-5Quasi-staticandhighratecompressiveyieldvaluesforPlate1 ......... 66 3-6Impactvelocitiesfromhighratecylindertests ................... 72 3-7Ratiosofmajortominornaldiametersandratiosofnaltoinitiallengthsfromhighratecylindertests ............................ 73 3-8GrainsizeaveragesatlocationsshowninFigure 3-50 ............... 78 5-1ModelparametersfortheyieldsurfaceinFigure 5-1 ............... 102 5-2CompressiveyielddatausedtoidentifyHill48parametervalues ......... 110 5-3TensileyielddatausedtoidentifyHill48parametervalues ............ 110 5-4ParametervaluesforHill48modelusingPlate1data ............... 110 5-5Plate1anisotropycoecientvaluesfordiscretestrainlevels ........... 117 5-6Plate2anisotropycoecientvaluesfordiscretestrainlevels ........... 119 5-7Johnson-Cookhardeninglawparametervalues ................... 148 9

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Figure page 1-1Elasticcoecientsrequiredforvariouscrystalsymmetries ............ 21 1-2ProjectionofTrescayieldsurface .......................... 23 2-1VariationofTisinglecrystalelasticmodulus .................... 32 2-2Titaniumcrystalstructure .............................. 33 2-3ActivetwinningsystemsinTi ............................ 34 2-4Titaniumplates ................................... 37 2-5MicrographofhighpurityTitaniumplatematerial ................ 37 2-6Plate1polegurewithcenterinTTdirection ................... 38 2-7Plate1polegurewithcenterinRD ........................ 38 3-1Geometryanddimensionsofthethrough-thicknesstensilespecimen ....... 40 3-2Quasi-staticcompressionspecimens ......................... 41 3-3Geometryanddimensionsofquasi-staticin-planespecimensfortension ..... 41 3-4Denitionofthespecimenorientations ....................... 41 3-5Resultsofquasi-staticcompressiontestsalongtheRDonPlate1 ........ 42 3-6OrientationImagingMicroscopymapat10%strain ................ 43 3-7OrientationImagingMicroscopymapat20%strain ................ 43 3-8Resultsofquasi-statictensiletestsalongtheRDconductedonPlate1 ..... 44 3-9Resultsofquasi-statictensiletestsalongtheTDconductedonPlate1 ..... 45 3-10HardeningduringuniaxialtensionandcompressionintheTDforPlate1 ... 45 3-11Resultsofquasi-statictensionandcompressiontestsinTTdirectionforPlate1 46 3-12HardeningintensionandcompressionintheTTdirectionforPlate1 ..... 47 3-13Plate2quasi-staticin-planedata .......................... 48 3-14Plate2comparisonofin-planequasi-statictensionversuscompressiondata .. 49 3-15Resultsofquasi-statictensionandcompressiontestsinTTdirectionforPlate2 50 3-16Plate2comparisonofTTquasi-statictensionversuscompressiondata ..... 50 10

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............ 51 3-18Fourpointbeamtestjig ............................... 52 3-19TypicalLoadvsDisplacementcurveforbendtests ................ 52 3-20BeamgridpatternusedinDICtocomputestraineld .............. 53 3-21Plate1experimentalaxialstrain("x)eldsforCase1 .............. 54 3-22Plate1experimentalaxialstrain("x)eldsforCase2 .............. 54 3-23Plate1experimentalaxialstrain("y)eldsforCase3 .............. 55 3-24Plate1experimentalaxialstrain("y)eldsforCase4 .............. 55 3-25DeformedcrosssectionofbeamfromPlate1forCase1and2 .......... 56 3-26DeformedcrosssectionofbeamfromPlate1forCase3and4 .......... 56 3-27Measurementlocationsondeformedfourpointbeamtestspecimens ...... 57 3-28Plate2experimentalaxialstrain("x)eldsforCase1 .............. 58 3-29Plate2experimentalaxialstrain("x)eldsforCase2 .............. 58 3-30Plate2experimentalaxialstrain("y)eldsforCase3 .............. 59 3-31Plate2experimentalaxialstrain("y)eldsforCase4 .............. 59 3-32DeformedcrosssectionofbeamfromPlate2forCase1and2 .......... 60 3-33DeformedcrosssectionofbeamfromPlate3forCase3and4 .......... 60 3-34Highratetestspecimens .............................. 61 3-35Failedsurfacefromhighratetensiontestspecimen ................ 62 3-36SchematicofSplitHopkinsonbarapparatus .................... 63 3-37ExperimentalcompressionresultsshowinganisotropyofPlate1 ......... 65 3-38ExperimentalcompressionresultsforPlate2 .................... 65 3-39Plate1HighrateTDdata .............................. 67 3-40Comparisonofcompressivehighratetoquasi-staticTTdataforPlate1 .... 67 3-41Plate1HighrateRDdata .............................. 68 3-42Plate2Highratein-planedata ........................... 69 3-43Plate2experimentalhighratetensiondata .................... 70 11

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....... 70 3-45Taylorcylinderimpacttestsetup .......................... 71 3-46Highratecylindertestresults ............................ 73 3-47Highratecylinderimpacttestspecimens ..................... 74 3-48Measuredmajorandminorproledatafromtestnumber107 .......... 75 3-49Measureddeformedfootprintfromtestnumber107 ................ 75 3-50MicrographlocationsforPlate1 .......................... 76 3-51Opticalmicroscopy(50X)atlocations1and2 ................... 77 3-52Opticalmicroscopy(50X)atlocations3and4 ................... 77 3-53Opticalmicroscopy(50X)atlocations5and6 ................... 78 3-54Opticalmicroscopy(50X)atlocations7and8 ................... 78 3-55Plate1with20couponsremoved .......................... 79 3-56Denitionofsampleorientationfromsectionedcoupon .............. 80 3-57Initial(0002)poleguresforPlate1 ........................ 80 3-58Plate1andPlate2withpoleguressuperimposedtodetermineRD ...... 81 3-59Positionofscanlocationsforthroughthicknesstexturemeasurements ..... 82 3-60BulktexturemeasurementofPlate1 ........................ 82 3-61Plate1poleguresfrompositions1and2 ..................... 83 3-62Plate1poleguresfrompositions3and4 ..................... 83 3-63Plate1poleguresfrompositions5and6 ..................... 83 3-64Plate1poleguresfrompositions7and8 ..................... 84 3-65Plate1poleguresfrompositions9and10 .................... 84 3-66Plate1poleguresfrompositions11and12 ................... 84 3-67Plate1poleguresfrompositions13and14 ................... 85 3-68Plate1poleguresfrompositions15and16 ................... 85 3-69Plate1poleguresfromposition17 ........................ 85 3-70Plate1Initialtexturefromthreeperspectives ................... 86 12

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................................ 87 3-72Plate1(0001)polegureforspecimensloadedincompressionto30and40%strainintransversedirection ............................ 87 3-73Plate1(0001)polegureforspecimensloadedincompressionto10and20%inthroughthicknessdirection ............................ 88 3-74Plate1(0001)polegureforspecimensloadedincompressionto30and40%straininthroughthicknessdirection ........................ 88 3-75Plate1(0001)polegureforspecimensloadedincompressionto10and20%inrollingdirection .................................. 89 3-76Plate1(0001)polegureforspecimensloadedincompressionto30and40%straininrollingdirection .............................. 89 3-77Textureevolutionforcompressiveloadingintherollingdirection ........ 90 3-78Textureevolutionforcompressiveloadinginthetransversedirection ...... 91 3-79Textureevolutionforcompressiveloadinginthethroughthicknessdirection .. 91 4-1PlanestressyieldlociiforvariousrationsofT=C 93 4-2Comparisonwithpolycrystillinesimulations .................... 94 4-3Arbitraryangledenition,xisrollingdirection .................. 97 5-1Projectionintheplane3=0forMg-Lialloysheet ................ 103 5-2TheoreticalmodelcomparedtoexperimentaldataforPlate1 .......... 104 5-3Averageexperimentalin-planecompressiondataforPlate2 ........... 105 5-4TheoreticalmodelcomparedtoexperimentaldataforPlate2 .......... 106 5-5ComparisonofHill'scriteriontoproposedcriterionforPlate1data ...... 111 5-6TheoreticalyieldcurvesforPlate1 ......................... 118 5-7TheoreticalyieldcurvesforPlate2 ......................... 118 5-8Singlecellcomputationalconguration ....................... 120 5-9SinglecellsimulationresultsforPlate1A)RDtensionB)RDcompression .. 121 5-10SinglecellsimulationresultsforPlate1A)TDtensionB)TDcompression .. 121 5-11SinglecellsimulationresultsforPlate1A)TTtensionB)TTcompression .. 122 13

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....................... 123 5-13Plate2TTsinglecellsimulations .......................... 123 5-14Symmetricalbrickarrangmentfortetrahedralelements .............. 124 5-15FEComputationalmeshforbeambendingtests .................. 125 5-16Typicaldeformedmeshshowingplaneofsymmetry ................ 125 5-17ComparisonofcrosssectionalareafromsimulationsofPlate1 .......... 126 5-18ComparisonoftensionversuscompressiondataforRDandTDinPlate1 ... 127 5-19ComparisonofcrosssectionsfromPlate1beamsimulations ........... 127 5-20Case1:LongaxisinRD,loadinginTD ...................... 128 5-21Plate1,Case1:Comparisonofaxialstraincountours ............... 128 5-22Plate1,Case1:Axialstrains("x)versusheightatcenterline:x=RD,y=TD 129 5-23Plate1,Case1:Comparisonofcrosssectionsfromexperimentandsimulation 130 5-24Case2:LongaxisinRD,loadinginTT ...................... 130 5-25Plate1,Case2:Comparisonofaxialstraincountours ............... 131 5-26Plate1,Case2:Axialstrains("x)versusheightatcenterline:x=RD,z=TT 131 5-27Plate1,Case2:Comparisonofcrosssectionsfromexperimentandsimulation 132 5-28Case3:LongaxisinTD,loadinginRD ...................... 133 5-29Plate1,Case3:Comparisonofaxialstraincountours ............... 133 5-30Plate1,Case3:Axialstrains("y)versusheightatcenterline:x=RD,y=TD 134 5-31Plate1,Case3:Comparisonofcrosssectionsfromexperimentandsimulation 134 5-32Case4:LongaxisinTD,loadinginTT ...................... 135 5-33Plate1,Case4:Comparisonofaxialstraincountours ............... 135 5-34Plate1,Case4:Axialstrains("y)versusheightatcenterline:y=TD,z=TT 136 5-35Plate1,Case4:Comparisonofcrosssectionsfromexperimentandsimulation 136 5-36ComparisonofcrosssectionalareaforPlate2 ................... 137 5-37Plate2IsotropicsimulationversusmodelforCase1and3 ............ 138 5-38Plate2IsotropicsimulationversusmodelforCase2and4 ............ 138 14

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............ 139 5-40Plate2,Case1:Axialstrains("x)versusheightatcenterline:x=RD,y=TD 139 5-41Plate2,Case1:Comparisonofcrosssectionsfromexperimentandsimulation 140 5-42Plate2,Case2:Comparisonofaxialstrain("x)countours ............ 141 5-43Plate2,Case2:Axialstrains("x)versusheightatcenterline:x=RD,z=TT 141 5-44Plate2,Case2:Comparisonofcrosssectionsfromexperimentandsimulation 142 5-45Plate2,Case3:Comparisonofaxialstrain("y)countours ............ 142 5-46Plate2,Case3:Axialstrains("y)versusheightatcenterline:x=RD,y=TD 143 5-47Plate2,Case3:Comparisonofcrosssectionsfromexperimentandsimulation 143 5-48Plate2,Case4:Comparisonofaxialstrain("y)countours ............ 144 5-49Plate2,Case4:Axialstrains("y)versusheightatcenterline:y=TD,z=TT .. 144 5-50Plate2,Case4:Comparisonofcrosssectionsfromexperimentandsimulation 145 5-51Deformedellipticalfootprintobtainedinhighratecylindertest. ......... 146 5-52Highratecompresivedatawithlineartusedinparameteridentication ... 147 5-53ComparisonofyieldvaluesobtainedfromJ-Clawtoexperimentaldata ..... 148 5-54InitialFEmeshforTaylorimpactsimulations ................... 149 5-55CylinderimpactsimulationresultsusingisotropicvonMisesandJ-Chardening 150 5-56Comparisonofprolesfromisotropicsimulation .................. 151 5-57Cylinderimpactsimulationresultsusinganisotropicelastic/plasticmodelandJ-Chardeningwithoutrateeects ......................... 152 5-58Comparisonofdeformedcylinderprolefromtwolocationsforsimlationusingproposedanisotropicmodelwithnorateeects .................. 153 5-59Comparisonofdeformedcylinderprolefromtwolocationsforsimlationusingproposedanisotropicviscoplasticmodel ....................... 153 5-60CylinderimpactsimulationresultsusinganisotropicparametersforproposedcriteriausingJ-Chardeningwithrateeects ................... 154 5-61Comparisonofmajorprolesobtainedusingthedierenctmodels ....... 155 5-62Comparisonofminorprolesobtainedusingthedierenctmodels ....... 156 5-63Comparisonofthepredictedfootprintobtainedusingthedierenctmodels .. 156 15

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................................. 157 5-65Comparisonofmajoraxisradialstrainversusheightpredictedbytheanisotropicmodelandexperimentaldata ............................ 158 5-66Comparisonofminoraxisradialstrainversusheightpredictedbytheanisotropicmodelandexperimentaldata ............................ 158 5-67Comparisonofratioofmajortominordiametersversusheightpredictedbytheanisotropicmodelandexperimentaldata ..................... 159 16

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Hosford ( 1993 ).AsubsetoftheseareshowninFigure 1-1 .Forexample,foranorthotropicmaterialthetensorChasnineindependentcoecients:C11;C22;C33;C12=C21;C13=C31;C23=C32;C44;C55;andC66. Figure1-1. Elasticcoecientsrequiredforvariouscrystalsymmetries 21

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22

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1-2 showsthebiaxialprojectionofthesurface. Figure1-2. ProjectionofTrescayieldsurfaceontheplane3=0 Saint-VenantbuiltontheworkofTrescaandLevy(1870)andlaidthefoundationsforthemathematicaltheoryofplasticity.Huber(1904)proposedarelationshipfortheconstantdistortionenergycriterionwhichwaslatterproposedbyvonMises(1913)asanapproximationofTresca.Thishasbeenthemostwidelyusedyieldsurfaceduetoitssimplicityandaccuracyformanymaterials.TheHubner-Misesyieldcriterioncanbewrittenas (yyzz)2+(zzxx)2+(xxyy)2+6(2yz+2xz+2xy)=22u(1{2)whereuistheyieldstressinuniaxialtension.ThiscriterioninvolvesonlythesecondinvariantofthedeviatoricstressJ2.Drucker(1949)proposedincludingeectsofthethird 23

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Hershey ( 1954 ). Zyczkowski ( 1981 )wheremorethan200yieldsurfacedescriptionsarediscussed.Onlyafewsignicantmodelswillbediscussedhere.Themostwidelyusedanisotropicyielddescriptionwaspresentedby Hill ( 1948 )andisgivenby ~=Fj23jm+Gj31jm+Hj12jm+Lj2123jm+Mj2231jm+Nj2312jm 24

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Phenomenologicalyieldfunctions YieldCriterionTypeShearDimension TrescaIsotropy--VonMisesIsotropy-Hill ( 1948 )PlanarAnisotropyyes6 Hershey ( 1954 )Isotropy-Hosford ( 1972 ) Gotoh ( 1977 )PlanarAnisotropyyes3 Bassani ( 1977 )Planarisotropy-Hill ( 1979 )PlanarAnisotropyno2 LoganandHosford ( 1980 )PlanarAnisotropyno Budianski ( 1984 )Planarisotropyno2 wherethecoecientsF,G,H,L,M,Nandmarematerialconstantsand1,2,and3aretheprinicipalstressescoincidentwiththeorthotropicaxes.Amajorlimitationofthiscriterionistheinabilitytoaccountforanystateinvolvingshearstresses.Atableofimportantphenomenologicalyieldfunctionsthatdescribeorthotropicbehaviortakenfrom Barlatetal. ( 1991 )isshowninTable 1-1 .Someisotropicfunctionsarealsoincluded.Thecolumnlabeled"Shear"indicatesifsheartermsappearintheformulation.The"Dimension"columngivesthenumberofstresscomponentsinvolvedintheformulation. Barlatetal. ( 1991 )extendedtheisotropicHersheyandHosfordmodel(seeEquation 1{3 )byapplyingafourthorderlineartransformationoperatorontheCauchystresstensor.Theorthotropiccriterionis ~=(12)m+(23)m+(31)m(1{6)where1;2and3aretheprincipalvaluesofthetransformedCauchystresstensor =L(1{7) 25

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CazacuandBarlat ( 2003 )and Cazacuetal. ( 2004 )showedthatonecanextendanyisotropiccriteriontoanisotropythroughgeneralizedinvariantsusingthetheoryofrepresentationoftensorfunctions.UsingthisapproachtheyextendedDrucker'sisotropicyieldcriteriontoorthotropyasfollows (Jo2)3c(Jo3)2=F(1{9)whereJo2andJo3arepolynomialsintermsoftheCauchystressandindependentofpressureandrespectingtheorthotropicsymmetries.However,noneoftheapproachesabovecanaccountforastrengthdierentialbetweentensionandcompressionwhichisexibitedbyhcpmaterials. HosfordandAllen ( 1973 )usedpolycrystallinecalculationstoinvestigatethestrengthasymmetryinisotropicmaterialsandsuggestedthattheasymmetrywascausedbytwinningwhichissensitivethesignoftheshearstress.Mostrecently,modelshavebeenproposedthatallowforanasymmetrybetweenthestrengthintensionandincompression. CazacuandBarlat ( 2004 )proposedanisotropiccriterion(Equation 1{10 )involvingboththesecondandthirdinvariantofthestressdeviatorthatcanaccountforastrengthasymmetryandtheyfavorablycomparedthistheorytothedatagivenby HosfordandAllen ( 1973 ).Theproposedmodelextendstheisotropicdescriptionof CazacuandBarlat ( 2004 )toorthotropyusingalineartransformationontheCauchystress. 26

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_"p=^n(1{11)TheusualassumptionisthatdirectionofplasticdeformationcanbederivedfromaplasticpotentialG()suchthat^n=@G() _"=_"e+_"p=C:_+^nIntheaboveequation,,isascalarandisnon-zeroonlyifplasticdeformationoccursi.e.: 27

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RambergandOsgood ( 1943 )model ~"=~ E+K Em(1{13)where~"istheeectiveplasticstrain,~istheequivalentstress,EisYoung'smodulusandKandmarematerialconstants.Observingthatthe~=EistheelasticstrainandwithK(~=E)maccountingfortheplasticstrain,therelationshipcanberewrittenintermsofayieldstressYandanewparameter=K(Y=E)m1as ~"=~ E+Y Ym(1{14) Ludwick ( 1903 )presentsaformthatneglectselasticstrainsas ~=C~"n(1{15)whereC=Y(E=Y)n,nisthework-hardeningexponentrelatedtomofEquation( 1{13 )byn=1=m.Manyhardeningrulesthataccountforaparticularmaterialorloadingenvironmenthavebeendeveloped.Forexample,hardeningatveryhighloadingratescanbedescribedbythephenominologicalmodelof Johnsonetal. ( 1997 )thatincorporatesEquation 1{15 ~=(Y+a~"n)(1+bln_")(1{16)whereYistheinitialyieldstrength,_"isthedimensionlesstotalstrainrate_"=_"o.Thereferencestrainrateistakenas_"o=1.Equation 1{15 isthesecondtermintherstbracket.TheconstantsY,a,b,andnaredeterminedfromexperimentaltests.ThefullJohnson-Cookmodelincludestermsnotgivenherethataccountfortheeectsof 28

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5 whereitisusedinthevisco-plasticimplementationoftheanisotropicmodelproposedinChapter 4 .Nosimplerelationshipexiststodescribeanisotropichardening.Formaterialswithonlyaslightanisotropy,theusualassumptionsofisotrpoicorkinematichardeningmaybesucientbutforhighlyanisotropicmaterialssomeotherapproachesneedtobeintroduced. Plunkettetal. ( 2006 )developedanddemonstratedaninterpolationmethodologythatusesareferencehardeningpathandaseriesofyieldsurfacesestablishedatdiscretelevelsofaccumulatedplasticstrain.Thisistheapproachusedintheimplementationoftheorthotropicmodeldevelopedaspartofthisdissertationandisdescribedindetailinsection 4.3 29

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LutjeringandWilliams ( 2003 ).MechanicalandphysicalpropertiesoftitaniumareshowninTable 2-1 from Donachie ( 2000 ).Puretitaniummeltsaround1660C.Atroomtemperatureitscrystalstructureishcp(knownasphase)butat882Cthereisanallotropicphasetransformationtoabccstructure(phase).Whenalloyedwithelementssuchasaluminum,oxygenandnitrogen,thephasecanbestabilizedevenathightemperature.Whenalloyedwithotherelementssuchasmolybdenum,ironorvanadium,itcanbestabilizedinphaseevenatroomtemperatures.[ Sergueevaetal. ( 2001 )]Highpuritytitaniumwasusedasthematerialofchoiceforthisstudyforvariousreasons.Itexhibitsastronganisotropicbehavior,iswidelyavailableandisusedinmanyapplicationsofinterestinbothcommercialanddefenseindustries[ Donachie ( 2000 ); LutjeringandWilliams ( 2003 )].Asinthecaseofotherhcpmaterialsitsbehaviorisstronglyinuencedbyitshcpcyrstallinestructure.Thereisacompetitionbetweenslipandtwinningthataccountsformuchoftheanisotropicdeformation.Thehcpcrystalsaremucheasiertodeformincertaincrystallographicdirectionsthanothers.Inparticular,titaniumdoesnothaveenougheasilyactivatedslipsystemstoaccommodatearbitrary 30

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PhysicalpropertiesofTitanium PropertyDescriptionorvalue Atomicnumber22Atomicweight47.90Atomicvolume10.6W/DCovalentradius1.32_AIonizationPotential6.8282VThermalneutronabsorptioncrosssection5.6barns/atomCrystalstructureAlpha(882.5C,or1620F)ClosepackedhexagonalBeta(882.5C,or1620F)Body-centeredcubicColorDarkgrayDensity4.51g/cm3(0.163lb/in3)Meltingpoint166810C(3035F)Solidus/liquidus1725C(3135F)Boilingpoint3260C(5900F)Specicheat(at25C)0.5223kJ/kgKThermalconductivity11.4W/mKHeatoffusion440kJ/kg(estimated)Heatofvaporization9.83MJ/kgSpecicgravity4.5Hardness70to74HRBTensilestrength240MPa(35ksi)minYoung'smodulus120GPa(17x106psi)Poisson'sratio0.361CoecientoffrictionAt40m/min(125ft/min)0.8At300m/min(1000ft/min)0.68Coecientoflinearthermalexpansion8.42m/mKElectricalconductivity3%IACSElectricalresistivity(at20C)420nmElectrogativity1.5Pauling'sTemperaturecoecientofelectricalresistance0.0026/CMagneticsusceptibility(volumeatroomtemperature)180(1:7)x106mks deformationbyslip[ Salemetal. ( 2003 ); Nemat-Nasseretal. ( 1999 )],thereforetwinningplaysanimportantroleintheplasticdeformation.Thisleadstosstrengthdierentialeectsincetwinningisadirectionalsheardeformationmechanism.Severalinvestigatorshavestudiedvariousaspectsofthebehavioroftitaniumanditsalloys. Gray ( 1997 )studiedtheeectsofstrainrateandtemperatureinhighpurity-titaniumbutonlyforcompressiveloadings.Kalidindiandothers[ Kalidindietal. 31

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2003 ); Lietal. ( 2004 ); Nemat-Nasseretal. ( 1999 ); Salemetal. ( 2002 2003 2004a b )]studiedahighpuritytitaniumplatessimilartothoseusedinthisstudybutonlyforafewloadingpathsand/orstrainrates.Oneofthegoalsofthisdissertationistoextendthecurrentknowledgebyinvestigatingawiderangeofloadingpathsinordertomorefullycharacterizethebehaviorandtoserveasabasisfordevelopmentofimprovedmaterialmodels. 2-1 from ZarkadesandLarson ( 1970 )showsthevariationoftheelasticmodulusforvariousorientationsatroomtemperature.Themodulusvariesfrom145GPaalongthec-axisto100GPainthedirectionperpendiculartothec-axis.Thereisasimilarvariationfortheshearmodulus.Thevariationsofthesemoduliinapolycrystallineaggregatewouldofcoursealsodependonthevariationoftexture( LutjeringandWilliams ( 2003 )). Figure2-1. VariationofTisinglecrystalelasticmodulus 32

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2-2 showsthethreemostdenselypackedtypesofplanes,thebasalplane(0002),oneofthethreeprismaticplanesf1010gandoneofthesixpyramidalplanes. Figure2-2. Titaniumcrystalstructure LutjeringandWilliams ( 2003 )].Theprismplanesandbasalhaidirectionsconstitutethemostfavorableslipwhilethebasalplanesandpyramidalplanesincombinationwithappropriatedirectionsconstitutetheotherprobableslipsystems.Sincealloftheslipsystemshaveslipdirectionsthatarerestrictedtothebasalplane,theydonotprovidetheveindependentslipsystemsnecessarytoaccommodatearbitraryplasticstrains[ Gray ( 1997 ); Meyersetal. ( 2001 )].Thisindicatesthattwinningcanplayasignicantroleinthedeformationoftitanium. 33

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2-3 ).Incomparisontootherhcpmaterials,titaniumisquiteductilebecauseithasmoretwinningandslipsystems.Twinningcanbesuppressedbyalloyingittogivehigherstrengthsandcanberetardedbyinterstitialsfoundinlowerpuritytitanium.Becausesoluteatomssuppresstwinning,itisamajorplayerindeformationforpuretitaniumwithlowamountsofoxygen[ LutjeringandWilliams ( 2003 )] ABFigure2-3. ActivetwinningsystemsinTi:A)Tensilef1012gB)Compressionf1122g 34

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Gray ( 1997 ); Meyersetal. ( 2001 )]. Salemetal. ( 2006 )observedevidenceofthesameHall-Petcheectandalsoshowedevidenceoftwoothereectsonhardeningresultingtwnning.Byperformingmacroandmicro-hardnesstests,theseauthorsshowedthatthetwinnedregionsareimmediatelyharderthanthebulkormatrixmaterial.Theyattributedthiseecttosessiledislocationsbeingtrappedinsidetwinnedregions(Basinskimechanism).Thirdly,theyfoundthattherewassofteningfromreorientationofthetwinnedregionintoanorientationmorealignedwitheasyslip. Nemat-Nasseretal. ( 1999 )suggestedthatanincreasedstrainhardeningrateisassociatedwithdynamicstrainagingbutotherstudiesdisputethisideaandindicatethatdeformationtwinningaccountsforthechangeinstrainhardening[ Salemetal. ( 2002 )]. Gray ( 1997 )statesexplicitlythattherolesofslipanddeformationtwinningintitanniumaresointertwinedthatbotheectsmustbeaccountedforinanyphysicallybasedconstitutivemodel. 2-2 .Hardnesstestswereperformedonthismaterialinplateform.Theaveragehardnesswasof43.1HRB.ThisisamuchsoftermaterialthanthatreportedinTable 2-1 whichhasahardnessof70to74HRB.ThetypicalgrainstructureforthematerialisshowninFigure 2-5 .Itshowssomewhatequiaxedgrainswithanaveragegrainsizeofabout20m.Tworoundplatesofthematerial,10inchesindiameterand5/8inchthick(seeFig 2-4 )werepurchasedfromAlphaAesar(AJohnsonMattheyCompany).Theplatesweredescribedascrossrolled,99.999%pure,butnorollingdirectionwasindicated.Theanisotropictexturewasestablishedviaelectronmicroscopy. 35

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Chemicalanalysisoftestmaterial:Titaniummetaldisk10inchdiameter,x0.625inchthick,0.010inch,w32RMSSurfaceo.b.,crossrolledwith1inchsquaresample-99.999% Ag<0.05Al0.4As<0.01Au<0.05B<0.01Ba<0.005Be<0.005Bi<0.01Br<0.05C10.5Ca<0.2Cd<0.05Ce<0.005Cl0.105Co0.008Cr0.55Cs<0.01Cu0.19F<0.05Fe5.5Ga<0.05Ge<0.05H1Hf<0.01Hg<0.1I<0.01In<0.05Ir<0.01K<0.01La<0.005Li<0.005Mg<0.05Mn0.0575Mo<0.05N<10Na<0.01Nb<0.2Nd<0.005Ni0.11O156.5Os<0.01P<0.01Pb<0.01Pd<0.01Pt<0.05Rb*<5Re<0.01Rh<0.15Ru<0.01S<5Sb<0.05Sc<0.05Se<0.05Si0.3Sn<0.05Sr*<3000Ta**<5Te<0.05Th<0.0005Tl<0.01U<0.0005V0.135W<0.01Y*<200Zn<0.005Zr0.6 Note:Valuesgiveninppmunlessotherwisenoted.Carbon,hydrogen,nitrogen,oxygenandsulfurdeterminedbyLECO,allotherelementsdeterminedbyGDMS*Ioninterference**Instrumentcontamination Figure 2-6 showsthethroughthickness(0002)polegurewhichindicatesnoclearanisotropyinthethroughthicknessdirection.Acleardirectionalityisseeninthepolegureforin-planetextureasshowninFigure 2-7 .TherollingdirectiondeterminedfromthetexturemeasurementswasmarkedontheplateshownontherightinFigure 2-4 .TestspecimenswerecutfromtheplateusingElectricalDischargeMachining(EDM)fordierentorientationsrelativetotheestablishedRDandthenormaloftheplate. 36

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TitaniumplateA)asreceivedB)Withcouponscutandrollingdirectionestablished Figure2-5. MicrographofhighpurityTitaniumplatematerial 37

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Plate1polegurewithcenterinTTdirection Figure2-7. Plate1polegurewithcenterinRD 38

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1 wascarriedoutforbothquasi-staticandhighloadingratesatroomtemperature.Thesetestswereusedtoquantifytheanisotropicbehavior,includingthestrengthdierentialbetweentensionandcompression,foreachplate.ItwasobservedthattheresponseofPlate1isorthotropicandhighlydependentonthedirectionandsenseoftheappliedload.Plate2isnearlyisotropicintheplaneoftheplatebuthasstrongbasaltexture,whichresultsinmarkeddierenceinresponsebetweenin-planeandthrough-thicknessdirections.Four-pointbendingtestswerealsoperformedonbeamscutfromeachplateinfourcongurations.AspecklepatternwasdepositedononeproleofeachbeamandDigitalImageCorrelation( Miguil-Touchaletal. ( 1997 ); HungandVoloshin ( 2003 ))techniqueswereusedtoanalyzethestraineld.Asaresultoftheplateanisotropyanddirectionalityoftwinning,qualitativedierenceswereobservedbetweentheresponseoftheupperandlowerbersofthedierentbentbeams.Thebeamswerecutatthemidpointandthecrosssectionswereobservedandcomparedtosimulationsforeachloadingorientation.TheresultsindicatetheneedtouseaconstitutivedescriptionforthematerialthataccountsfortheinterplaybetweenslipandtwinninganditseectsontextureevolutionandhardeningresponsewhensimulatingthebehaviourofTitanium.Pre-andpost-testtexturesofspecimensweremeasuredusingneutronbeamtechniquesattheHIPPOfacilityattheLosAlamosNationalLaboratory(LANL).Quasi-staticallydeformedsamplesfromPlate1werealsoanalyzedusingOrientationImagingMicroscopy(OIM).Signicanttextureevolutionwasobservedonlyforcompressionintherollingdirection.BoththeOIMandneutronbeammeasurementsrevealedahighvolumefractionoftwinnedgrains,theprimarytwinfamilybeingtensiletwins. 39

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3.1.1CharacterizationTests 3.1.1.1TestdescriptionThequasi-staticcharacterizationtestsforbothplatesconsistedofuniaxialtensionandcompressiontestsatanominalstrainrateof0.001persecond.AnInstron1125testingmachinewasusedwithanInstron100kNloadcellforcompressiontestsand5Knloadcellfortensiletests.AnInstronextensometermodelnumberG-51-17-Awithagaugelengthof12.7mmwasusedforcompressiontestsandanInstronmodelnumberG-51-12-Aextensometerwithagaugelengthof25.4mmwasusedfortensiletests.Toexaminetheeectofloadingorientationonthemechanicalresponseofthesetwostronglybasal-texturedtitaniumplates,cylindricalcompressionspecimens(0.3x0.3in)weremachinedsuchthattheaxesofthecylindersareeitherinin-plane(IP)orthrough-thickness(TT)platedirections(seeFigure 3-2 ).Forbothplates,IPsampleswerecutat0,45and90orientationstotherollingdirectionandlabeledasinFigure( 3-4 ).TensiletestsintheIPdirectionswereconductedusingclassicaldog-bonesshapesamples(Figure 3-3 ).AspecializedminiaturetestspecimenwasusedfortheTTtests(Figure 3-1 ).InordertoexaminethemicrostructuralevolutionatdierentlevelsofplasticdeformationaswellasdeterminetheLankfordcoecients,thetestswerecarriedouttoapproximately10%,20%,30%,40%strainsrespectivelyoruntilcompletefailureofthespecimenoccurred.AllIPspecimenswerelabeledrelativetotheorientationwith Figure3-1. Geometryanddimensionsofthethrough-thicknesstensilespecimen 40

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Quasi-staticcompressionspecimensA)dimensionsB)specimenwithlubetrap Figure3-3. Geometryanddimensionsofquasi-staticin-planespecimensfortension respecttotherollingdirection(RD)asshowninFigure 3-4 Figure3-4. Denitionofthespecimenorientationsrelativetotheestablishedrollingdirection. 41

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3-5 ).ForTests301and401,thecurvesarenotsmoothathigherstrainsbecausethetestspecimensforthesetestsincludedasmalllubetrapatoneend(seeFigure 3-2 B)).ThiswaslledwithMolygreaseinaneorttominimizefrictionattheplatenfaces.Subsequenttestswithoutthetrap,usingonlyMolykotelubricant,showedthatfrictionwasnotaproblemandlatertestsdidnotincludethetrap.Foralltestsatlowerstrainlevels,thelubetrapwasnotincluded.Notethatstrain-hardeningisnotlinear.Thereisadistincthumporchangeintheslopeofthestress-straincurvesatabout10%strain.Thisincreaseinthestrain-hardeningratemaybeassociatedwiththeonsetoftwinning.ThishypothesiswasveriedbysubsequentOIMobservationsofthedeformedspecimens. Figure3-5. Resultsofquasi-staticcompressiontestsalongtheRDconductedat0.001sec1onPlate1 TheOIMmapofthespecimendeformedto10%strainrevealsthatmanygrainshavetwinned(twinsappearredinFigure 3-6 ).Thetwinvolumefractionwasestimatedtobe17%.Figure 3-7 showsanOIMmapcorrespondingto20%strain,whichindicatesahighvolumefractionoftwinnedgrains,about40%.Nootherloadingpathproducedthisleveloftwinningactivity.Theseresultsareconsistentwithpreviousobservationsreported 42

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OrientationImagingMicroscopymapshowingtheevidenceoftwins(inred)insamplefromPlate1deformedto10%straininsimplecompressionalongtherollingdirection.Thetwinvolumefractionis17%. Figure3-7. OrientationImagingMicroscopymapshowingsignicanttwinningactivity(45%volumefraction)insamplefromPlate1deformedto20%straininsimplecompressionalongtherollingdirection by Salemetal. ( 2002 2003 2004b a )onpolycrystalline-titaniumofsimilarpurity.Furthermore,asinthecaseofPlate1material,themaximumtwinvolumefractionwasobservedinsimplecompressionto20%strain,thereportedvolumefractionbeingof45%.Thestress-strainresponseunderquasi-statictensioninthesameorientation(RD)upto10,20,and30%strainrespectively,isshowninFigure 3-8 A).Theinitialyieldstress(at0.2%oset)isabout175MPa.Thematerialgraduallyhardensuntilplastic 43

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Louetal. ( 2006 )).OIMobservationsforaspecimendeformedupto30%strainshowthatmostgrainshavelesstwins,althoughsometwinningisevident.Comparisonbetweencompressionandtensilestress-straincurvesalongtherollingdirection(Figure 3-8 B))showsaverylargeasymmetryinhardeningevolution.Although,initiallythereisnosignicantdierenceinyieldingbehavior,atabout7.5%anespeciallysharpdierenceinresponseisobserved.Thisstrikingstrengthdierentialeectcorrelateswiththeonsetoftwinninginthecompressionsample. ABFigure3-8. Resultsofquasi-staticloadingtestsalongtherollingdirectionconductedonPlate1at0.001sec1.A)tensiletestsB)Hardeningintensionandcompression Quasi-statictestresultsinmonotonicuniaxialtensionandcompressionalongthetransversedirection(TD)areshowninFigure 3-9 .Noticethatthestress-straincurvesincompressionalongtheTDdonotshowthefeaturespresentinthestress-strainresponseintheRDcompression.Nosignicantchangeinstrain-hardeningisobserved,whichcorrelateswithminimaldeformationtwinning.PosttestanalysisusingOIMconrmsthatthetendencytotwinisdirectional.Thereislittletwiningactivityincompression(lessthan5%volumefraction)alongTDascomparedtotheRD. 44

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Plate1resultsat0.001sec1:A)Resultsofquasi-statictensiletestsalongtheTDB)Resultsofquasi-staticcompressiontestsalongtheTD AcomparisonoftensionversuscompressionresponsealongthetransversedirectionisshowninFigure 3-10 .Again,thereislittlestrength-dierentialeectsininitialyieldingbutstrongasymmetryisobservedafter1.5%strain. Figure3-10. HardeningduringuniaxialtensionandcompressionintheTDforPlate1 Quasi-statictestresultsinmonotonicuniaxialcompressionandtensionalongthethrough-thicknessdirectionareshowninFigure 3-11 .Thereappearstobevery 45

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ABFigure3-11. Resultsofquasi-statictestsat0.001sec1alongthethroughthicknessdirectionconductedonPlate1A)tensileB)compression Comparisonbetweenthrough-thicknessuniaxialcompressionandtensionstress-straincurves(Figure 3-12 )showsastrongtension/compressionasymmetryininitialyieldingandhardeningbehavior.Thismarkeddierenceinresponseshowsthestrongdependencybetweendeformationmechanismsandloadingconditions.Compressiveloadingisappliedessentiallyperpendiculartothebasalplane,thusfavorsdeformationtwinningovernon-basalslip.Inconclusion,thevariousmeasuredtensileyieldstressesshowin-planeanisotropy.Ananisotropyratioforinitialyieldstressdenedbytheratiooftheyieldstressinthethrough-thicknessdirection(thelargest)tothatinthetransversedirection(thesmallestvalue)is1.27.Theyieldstressanisotropyincompressionis1.18,smallerthanintension.Thisobservedvariationincompressiveowstressesisconsistentwithpreviouslyobservedresultsforpolycrystallineanisotropichcpmaterials; Louetal. ( 2006 )onAZ31Bmagnesiumand KaschnerandGray ( 2000 )onZirconium.Thelargercompressiveowstressinthethrough-thicknessdirectionascomparedtothein-planeorientationsis 46

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HardeningintensionandcompressionintheTTdirectionforPlate1 duetothestrongbasaltextureofthematerial.ForTTcompressiontheloadisappliedessentiallyperpendiculartothebasalplane;thusplasticowisachievedforhigherstressesthanforin-planesampleswhichhavemorefavorableconditionsforactivatingprism,pyramidal,andbasalslip.Tensiledeformationisslip-dominatedhencetheanisotropyinyieldstressesisstrongerthantwinning-dominateddeformation,whichisobservedincompression. 3-13 A))itcanbeconcludedthatindeedthereislittlein-planeanisotropy.Foruseinidentifyingtheparametersoftheproposedmodel,allofthedataforthein-planequasi-staticcompression 47

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3-13 A).Notethatthisstress-straincurveindicatesnon-linearstrainhardeningwhichmaybeindicativeoftwinningactivity.Further,OIMinvestigationsneedtobeperformedinordertoverifythishypothesis. ABFigure3-13. Plate2quasi-staticin-planedataA)compressiondataB)tensiondata Duetothein-planeisotropyestablishedthroughtexturemeasurementsandmechanicaltestsincompression,tensiletestswereperformedinonlytwoin-planedirections:at0and90fromRD,respectively.TheresultsofthesetestsareshowninFigure 3-13 B).Thereisclearlymorespreadinthedatathanincompression.Apossibleexplanationofthisisthatthetensilespecimensarerelativelythinandaretakenfromdierentlocationsalongthethicknessoftheplate.Ifthereisagradientoftexturewiththethickness,somespecimenswouldbesofterwhilesomeharderthanothers.Thus,furthertexturemeasurementsthroughoutthethicknessoftheplateneedtobeperformed.SuchmeasurementshavebeenperformedforPlate1showinganoticeabletexturegradientthoughthethicknessofthisplate.Onequasi-statictestwasmadeusingthecylindricalhighratespecimenshowninFigure 3-34 B)andlabeled"HRSpecimen"inthegure.Thiswasdoneinanattempttoreducethevariationduetothepositioninthethicknessdirection.Thecrosssectionofthehighratespecimen(0.049in2)wassignicantlylarger 48

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3-14 .Thematerialstrengthissimilarintensionandcompressionupuntilabout13%strainwherethecompressivestrengthislarger.Thestrengthdierentialbecomesincreasinglargerathigherstrains. Figure3-14. Plate2comparisonofin-planequasi-statictensionversuscompressiondata DataforcompressionandtensiontestsfromtheTTdirectionofPlate2areshowninFigure 3-15 .AsforPlate1,thesmallertensiletestspecimencongurationwasusedduetothelimitsofthethicknessoftheplate.Figure 3-16 showsthecomparisonofthethroughthicknesstensionandcompressiondata.Thereisasignicantstrengthdierentialfromthebeginningandbothshowasmallsecondaryyieldpointsimilartothatfoundinmanysteels. 49

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Plate2quasi-staticdataforTT:A)compressionB)tension Figure3-16. Plate2comparisonofTTquasi-statictensionversuscompressiondata 3-17 A.Foreach 50

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3-17 B. ABFigure3-17. Fourpointbeamtestspecimens:A)SpecimendimensionsB)Orientationdenitions:Case1andCase2havelongaxisalignedwiththerollingdirection(x)Case3andCase4havethelongaxisalignedwiththetransversedirection(y) ThetestingjigisshowninFigure 3-18 includingatestspecimen.Thetwoupperpinsweredisplacementcontrolledtoapproximately5.5mm.AtypicalloadpindisplacementpathisshowninFigure 3-19 .Alongonesideofthetestbeam,aspecklepatternwassprayedanddigitalimagecorrelationorDIC( Miguil-Touchaletal. ( 1997 ); HungandVoloshin ( 2003 ))wasusedtodeterminethestraineldafterdeformation.Theimagetakenhad88pixelsalongtheshortdirectionofthebeam.Thebeamdimensioninthatdirectionis6.35mm.therefore,thephysicaldistancebetweenpixelsis6350micron/88pixel=72micron/pixel.Themethodcandetectdisplacementsof0.01pixel,thereforetheerrorislessthan1micron.Thestraineldcorrespondstothegridpatternsetupontheundeformedspeckleeld.ThedisplacedeldwasusedtomapthestraineldfromthemeasurementsfromthedeformedspecklepatternusingDIC.Atypicalundeformedanddeformedgridare 51

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FourpointbeamtestjigA)loadedwithtestspecimenB)givingdimensionsandpinplacements Figure3-19. TypicalLoadvsDisplacementcurveforbendtests:Leftaxisispindisplacementinmm,rightaxisisloadinkNplottedversustimeonthehorizontalaxis. showninFigure 3-20 .Notethathegridandsubsequentstrainelddoesnotcovertheentirespeckleeld.Thedeformedspecimenswerecutatthemidpointalongtheaxistoexaminethenaldeformedcrosssection.MeasurementsatthiscrosssectionweretakenforcomparisontotheFEsimulations. 52

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TypicalundeformedanddeformedbeamgridpatternusedwithDICforgeneratingexperimentalstraineld 3-17 )forPlate1areshowninFigures 3-21 to 3-24 .Theaxialstrainisdenedasthecomponentrelativetothelongaxisdirectionofthespecimen.ForCase1and2thelongaxisisalongtherollingdirectionsotheaxialstraincomponentis"xandforCase3and4,thelongaxiscorrespondstothetransversedirectionthereforetheaxialstrainis"y.ThesedataarecomparedtosimulationresultsinChapter 5 .Forallcases,somenon-uniformdeformationoccuredinthedirectionnormaltotheplaneforwhichthedatawerereported.ThiswouldintroduceasmallerrorinthecomputationoftheaxialstrainsusingtheDICmethodology. 53

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Plate1experimentalaxialstrain("x)eldsforCase1:Longaxisinx=RD,loadediny=TD. Figure3-22. Plate1experimentalaxialstrain("x)eldsforCase2:Longaxisinx=RD,loadedinz=TT. 54

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Plate1experimentalaxialstrain("y)eldsforCase3:Longaxisiny=TD,loadedinx=RD. Figure3-24. Plate1experimentalaxialstrain("y)eldsforCase4:Longaxisiny=TD,loadedinz=TT. 55

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3-25 and 3-26 showthecrosssectionsforeachcase.Table 3-1 givesthedimensions(mm)measuredatthethreelocationsshowninFigure 3-27 foreachofthebeams. ABFigure3-25. DeformedcrosssectionofbeamfromPlate1forCase1and2 ABFigure3-26. DeformedcrosssectionofbeamfromPlate1forCase3and4 56

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Measurementlocationsondeformedfourpointbeamtestspecimens;valuesgiveninTable 3-1 forPlate1andinTable 3-2 forPlate2. Table3-1. MeasurementsofdeformedbeambendspecimensfromPlate1;locationsidentiedinFigure 3-27 (dimensionsaremm) CaseABC 16.48786.35006.171627.02126.33485.689036.69546.41986.015446.93676.37865.7988 3-17 )forPlate2areshowninFigures 3-28 to 3-31 .TheseareagaincomparedtosimulationsfromFEsimulationsinChapter 5 .ThedatafromCase1andCase3areverysimilarasisthedatafromCase2andCase4.Thisisfurtherevidenceofthein-planeisotropictextureofPlate2whichmeansCase1andCase3areessentiallythesametest.AsimilarargumentholdsforCase2andCase4.AsforPlate1,theorthotropicaxescorrespondtox=RD,y=TD,andz=TT. 57

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Plate2experimentalaxialstrain("x)eldsforCase1:Longaxisisx=RD,loadediny=TD. Figure3-29. Plate2experimentalaxialstrain("x)eldsforCase2:Longaxisisx=RD,loadedinz=TT. 58

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Plate2experimentalaxialstrain("y)eldsforCase3:Longaxisisy=TD,loadedinx=RD. Figure3-31. Plate2experimentalaxialstrain("y)eldsforCase4:Longaxisisy=TD,loadedinz=TT. 59

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3-32 and 3-33 showthecrosssectionsforeachcase.Table 3-2 givesthedimensions(mm)measuredatthethreelocationsshowninFigure 3-27 foreachofthebeams. ABFigure3-32. DeformedcrosssectionofbeamfromPlate2forCase1and2 ABFigure3-33. DeformedcrosssectionofbeamfromPlate3forCase3and4 60

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MeasurementsofdeformedbeambendspecimensfromPlate2;locationsidentiedinFigure 3-27 (dimensionsaremm) CaseABC 16.61486.10816.283326.94696.37605.290236.53806.27386.202747.19456.38565.5467 3.2.1CharacterizationTestsHighrateloadingtestswerecarriedoutusingasplitKolsky-Hopkinsonpressurebar.Highratecompressionspecimens(Figures 3-34 A))aresimplecyclindersbutsmaller(0.2x0.2inches)thanthespecimensforthequasi-statictests.TensilespecimenswerecylindericaldogbonesasshowninFigure 3-34 B)andweremarkedwithanarrowindicatingthetopoftheplateandthereforethethroughthicknessdirection.Theplatethicknessdidnotallowenoughmaterialtoobtainhighratethroughthicknesstensionspecimens.Alldimensionsareininchesforbothdrawings.Noposttestmetallographywasdoneonanyofthesespecimenssotextureevolutiondataarenotdirectlyavailable. A)B)Figure3-34. Highratetestspecimens:A)CompressioncylinderB)Tensiondogbone 61

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3-35 .Notetheellipticalshapewiththeharder,throughthicknessdirectioninthedirectionofthemajoraxis. Figure3-35. Failedsurfacefromhighratetensiontestspecimen BaconandLataillade ( 2001 ),thesplitHopkinsonpressurebar(SHPB)(alsoreferredtoastheKolsky-Hopkinsonbar)iswidelyusedtoinvestigatethedynamicresponseofarangeofmaterials.Thetestallowstheusertoderivetheappliedforceandtheloadpointdisplacementversustimebyconsideringthepropogatingwavesinaninstrumentedelasticbar. Hopkinson ( 1914 )describedthetechniqueinitiallyin1914whichinvolvedasinglelongrod.TheSHPB,introducedby Kolsky ( 1949 ),involvedtheuseoftwoinstrumentedbarsandhasbecomeastandardsetupforhighratematerialdeformationstudies.TheSHPBconsistesofastriker,anincidentorinputbar,thespecimentobetested,andatransmitteroroutputbar.AschematicoftheSHPBsetupisshowninFigure 3-36 .Thespecimenisplacedbetweentheinputandoutputbars.Thestrikerimpactstheinputbarandgeneratesanelasticcompressivewavemovingtowardsthetestspecimen. 62

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Figure3-36. SchematicofSplitHopkinsonbarapparatus Theanalysisofthewavesdependsonthreeprimaryassumptions:(1)theinstrumentedbarsremainlinearlyelasticthroughoutthetest,(2)thediameterofthebarsaresmallrelativetothesmallestwavelengthofthepropagatingwavealongthebar,and(3)themechanicalimpedanceofthebarsisuniform.Equation 3{1 givesKolsky'srelationforndingthestressinthespecimen,s(t)[ Kolsky ( 1949 )]. 63

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StrainratesacheivedfortensileSHPBtests TestNumber12345678910StrainRate(sec1)522517643665652627662653555543 TestNumber11121314151617181920StrainRate(sec1)585534547563573528552557529516 TestNumber212223StrainRate(sec1)517616531 Note:AVG=567andStandardDev=53 Table3-4. StrainratesacheivedfortensileSHPBtests TestNumber12345678910StrainRate(sec1)419429318369396466457408417403 Note:AVG=407andStandardDev=42 ThestraininthespecimencanbefoundbyintegratingEquation 3{2 : 3-3 and 3-4 ).ThehighratecompressiontestsforPlate1showsthatitisalsoorthotropicathighloadingrates.Figure 3-37 showsstress-straindataforcompressiveloadingalongtherollingdiredtion,thetransversedirectionandthethroughthicknessdirectionforPlate1.Asforthequasi-staticresultstheplateisinitiallyharderinthethroughthicknessdirectionbutafter15%strain,therollingdirectionhashardeningaboveeventhethoughthicknesslevels.ThisisnotthecaseforthehighrateresultsfromPlate2asshowninFigure 3-38 whereboththetransversedataandrollingdirectiondataremainbelowthethroughthicknessdataforallstrainlevels.Thetransversedirectioncurveremainsbelowthethroughthicknesscurvethroughout,asinthequasi-staticcaseforPlate1and2.Table 3-5 givestheyieldvaluesforseverallevelsofstrainaswellastheanisotropyratio(denedastheratioofhighesttolowestyieldatagivenstrainlevel)forboththehighloadingratedatandthequasi-staticdataforPlate1. 64

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ExperimentalcompressionresultsforPlate1showingtheanisotropyamongrollingdirection,transversedirectionandthroughthicknessdirectionA)highrateB)quasi-static ABFigure3-38. ExperimentalcompressionresultsforPlate2showingtheisotropybetweenrollingdirectionandtransversedirectionA)highrateB)quasi-static. 65

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Quasi-staticandhighratecompressiveyieldvaluesforRD,TD,andTTdirectionandanisotropyratiosforPlate1 HighRate Quasi-static StrainRDTDTTMax/MinRDTDTTMax/Min 0.054113982941.3982252763301.4670.14984673991.2482713073611.3320.155374984921.0913233313891.2040.25485215731.0993793524191.1900.255895486261.1424263884481.1550.36205756541.1374564184771.1410.35NANANANA4934525051.1170.4NANANANA5224835371.112 Gray ( 1997 )thattitaniumtwinsmorereadilyasloadingratesincrease.Ingeneralthematerialisharderwhenloadedatthehigherrates.Figure 3-39 A)showsthehighrateresultsforuniaxialloadingintheTD.Theinitialyieldpointsareveryclosebutasmoredeformationoccursthestrengthincompressionbecomessomewhatlarger.Thismayindicatethatsometwinningisoccuringinthecompressionloadingbutthishasnotbeenconrmedbyposttestmetallography.Comparisonsofhighratecompressiondatawithquasi-staticdatafortheTDareshowninFigure 3-39 B).Aclearincreaseinstrengthisobservedbutthehardeningrateremainsnearlyunchanged.ThehighrateresultsforcompressiveuniaxialloadingintheTTdirectioncomparedtodatagatheredatquasi-staticloadingratesisshowninFigure 3-40 .Againaclearincreaseinstrengthwithloadingrateisobservedwithverylittlechangeinhardeningrate. 66

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Plate1A)ExperimentalhighratedatafortheTDfortensionandcompressionandB)Comparisonofexperimentalhighratedatatoexperimentalquasi-staticdata Duetogeometryconstraints,nothroughthicknesstensiondataisavailableathighratesofloading. Figure3-40. Plate1Comparisonofcompressivehighratetoquasi-staticdatafortheTTdirection ResultsforhighrateuniaxialloadingintheRDisshowninFigure 3-41 A).AsfortheTDdata,theinitialyieldpointsaresimilarbutthedierenceincreaseswithadditional 67

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3-41 B)showsthecomparisonwithquasi-staticdatawhereposttestmetallographyshowedasignicantamountoftwinning.Thedataindicatesthatevenhigherlevelsoftwininngmaybeoccuringatthehigherloadingrates.Theslopeofthecurveisloweratstrainsabove20%whichindicatingthattwinninghasprobablysaturated. ABFigure3-41. Plate1A)ExperimentalhighratedatafortheRDfortensionandcompressionB)Comparisonofhighratedatatoquasi-staticdatafortheRD 3-42 A)showsdatafromvein-planedirections.Althoughthereissomescatterinthedata,itappearsthattheplateisnearlyisotropicintherollingplaneoftheplate.Alsoshowninthegure(blackline)anaverageofallin-planedata.Thisaveragecurvewasusedinallsubsequentanalysisanddataidentication. 68

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Plate2:A)highratein-planecompressiondataB)highratecompressiondatacomparedtoquasi-staticcompressiondata Figure 3-42 B)showsacomparisonbetweentheaveragein-planecompressiondataathighrateloadingcomparedtotheaveragefromquasi-staticloading.Thereisaclearstrengthingeectfromthehighrateloadingwhichstaysfairlyconstantthroughouttheentirepath,howeverthedatadoseemtoshowaslightlyhigherhardeningrateforthehighrateloading.Thismaybeanindicationiftwinningactivity.DataforhighratetensiletestsareshowninFigure 3-43 A).ThedatashownarefromvedirectionsrelativetotheRDwithingtheplaneoftheplate.Thereismorescatterinthetensiledatathanforthecompressivedatabutthereisnoapparenttrendsindicatingthatthetensilebehaviorisdirectional.Aswiththecompressivedata,anaverageofallthedatawasmadeandisshownasthesolidblacklineinFigure 3-43 A).Figure 3-43 B)showsacomparisonoftheaveragehighratein-planetensiledatatothequasi-statictnesiledatagatheredusingtheroundspecimentest.Figure 3-44 A)showsthehighratethroughthicknesscompressiondatafortwotestsaswellasanaverage(blackline)ofthetwotests.Again,theaveragewasusedforallanalysisandparameteridenticationprocedures.Thereisverylittlescatterbetweenthetwotests.AcomparisonbetweentheTThighratecompressionandquasi-staticTT 69

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Plate2:A)ExperimentalhighratetensiondataB)Highratetensiondatacomparedtoquasi-statictensiondata compressionisshowninFigure 3-44 B).Theincreaseinstrengthforthethroughthicknesscompressionatthehigherloadingrateremainsquiteconstanttothestrainlevelsshown. ABFigure3-44. Plate2:A)ExperimentalhighratethroughthicknesscompressiondataB)Experimentalhighratedataversusexperimentalquasi-staticTTcompressiondata 70

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3-45 ).Thevelocityofthesamplewasmeasuredusingapairofpressuretransducersmountedtothebarrelandbyapairoflasersmountedbetweenthebarrelexitandanvilimpactsurface.Sampleproleisdynamicallyviewedusinghigh-speedphotographywiththelaserclosesttobarrel(triggerlaser)usedtotriggerthelightsourceforthecamera. Figure3-45. Taylorcylinderimpacttestsetup Theaxisofthebarrelboreisalignedperpendiculartotheimpactfaceoftheanvil.Correctalignmentofthebarrelwithrespecttotheanvilfaceisimperativesothattheleadingedgeofthesampleisinperfectcontactwiththeanvilfaceatinitialimpact.Aftereachtesttheanvilisrotatedtoensurethattheprojectedpointofimpactisfreeofdefectsfrompreviousexperimentsorotherexternalsources.Theidealsurfaceconditionoftheanvilsurfaceisatandhighlypolished.ACordin330Acamerawasloadedwith2rollsofT-MAXP3200lm.Theexternalhighintensitylightsourcewaspositionedsothatthetestsamplewasdirectlybetweenthelightandthecameralensandtimeofimpact.Thetestsampleisinsertedintothebarreloppositetheanvil.Oneplasticobturatorisinsertedbehindthesamplepriortocartridgeinsertiontolimitgasdischargearoundthesamplefollowingpropellantinitiation. 71

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ImpactvelocitiesfromhighratecylindertestsperformedfromPlate2 TestNumberLaserVelocityTransducerVelocityAngle 961381350971821849098153NR0991821849010019118845101163NR0102NRNR9010319920022.510418818967.51051851880106NR18122.51071961934510818518522.5 Theappropriatecartridge,havingbeenloadedwithapredeterminedamountofRedDotexplosive,islasttobeloadedintotheborebeforeaxingtheringpin/capassembly.Atotalof13highratecylinderimpacttestswerecarriedoutforspecimensfromPlate2.Table 3-6 showsthevelocitiesobtainedduringeachtestfromboththelasersandpressuretransducersandtheanglefromtherollingdirectionassociatedwiththespecimenaxis.Table 3-7 givesthemajorandminoraxesofthedeformedfootprint(thesurfaceofthecylinderstrikingtheanvil)andtheratioofmajordiametertominordiameter.Inadditiontheinitalcyclinderlength,thenalcylinderlengthandtheratioofthetwoaregiven.Notethatthevelocityfromthepressuretransducersfortestnumber101wasnotrecorded,thevelocityfromthelasersfortestnumber106wasnotrecordedandandneithervelocitywasrecordedfortestnumber102.Figure 3-46 givestheratioofmajortominordiameterandinitialtonallengthasafunctionoftheimpactvelocity.Theratioofdiametersarestronglyinuencedbyfrictionaleectsattheanvilinterfaceandanymis-alignmentforthetest.ThenalproleofthedeformedspecimenswereobtainedusinganopticalcomparatormodelDIJ415.Thespatialmeasurementsweremadefromenlargedimagesgeneratedfromthecomparator,accuratetowithin0.0001in.Duetotheorthotropictextureof 72

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RatiosofmajortominornaldeformeddiametersandratiosofnaltoinitiallengthsfromhighratecylindertestsperformedonspecimensfromPlate2 TestMajorMinorDiameterInitialFinalLengthNumberDiameterDiameterRatioLengthLengthRatio 960.2450.2311.0612.0971.9000.906970.2610.2431.0742.1001.7900.852980.2480.2281.0882.1011.8790.894990.2580.2411.0712.1001.8040.8591000.2640.2461.0732.0991.7880.8521010.2550.2341.0902.1001.8510.8811020.2520.2381.0592.1001.8400.8761030.2650.2461.0772.1001.7520.8341040.2630.2441.0782.0971.7780.8481050.2610.2431.0742.0991.7970.8561060.2600.2371.0972.0991.8160.8651070.2650.2451.0822.1001.7700.8431080.2620.2441.0742.0971.7890.853 Figure3-46. Highratecylindertestresultsgivingtheratioofmajordiametertominordiameterandtheratioofinitialtonallengthplottedversustheimpactvelocity thespecimentheinitiallycircularcrosssectionofthespecimendeformedintoanellipticalshape.Boththemajorandminoraxisofthespecimenweremeasured.Asmightbeexpected,thedataextractionisverytimeconsummingandmanpowerintensive.Figure 3-47 showsthespecimendimensionsandanundeformedcomparedtoadeformedsample. 73

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HighratecylinderimpacttestspecimensA)dimensionsofhighratevalidationtestspecimenB)undeformedspecimencomparedtotypicaldeformedspecimenC)Highratecylinderspecimenshowingarrowalignedwiththethroughthicknessdirectionpointingtothetopoftheplate Allspecimensareverysimilarwithsomevariationduetothespecimenmachiningprocess.Thespecimenfromtestnumber107wasjudgedtobedenitiveandusedfordataextraction.Resourcesdidnotpermitthedetailedextractionofproledatafromallspecimens.BoththemajorandminorexperimentalproledataareshowninFigure 3-48 .DuringfabricationcarewastakentoidentifytherelationofthespecimenwiththeTTdirection.AmarkontheendofeachspecimenwasmadebythemachineshoptoindicatetheTTdirectionasshowninFigure 3-47 C).Forallcasesthedeformationwaslessthatinthethroughthicknessdirection.Thisshowsupintheellipticalfootprint(initiallycircular)ofthedeformedspecimen.Figure 3-49 showstheexperimentaldimensionsofthenalfootprintfromtestnumber107. 74

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Measuredmajorandminorproledatafromtestnumber107(impactvelocity196m/s) Figure3-49. Measureddeformedfootprintfromtestnumber107 75

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3-50 representsthearrangementof40picturestakenatamagnicationof50Xonanopticalmicroscope.Thetopsurfaceoftheplateisontheleft.Asseeninthegure,therearebandsthataretypicalforarolledmaterial.Thebandsususallycontainsmallergrains.Thearrowsshowpositionswherehighermagnicationpictures,showninFigures 3-51 to 3-54 ,weretakentomeasurethegrainsize.Fromthesepictures,itisseenthatthegrainsareroughlyequiaxedontheplaneofthepicturesbutthegrainsizevariesfromonepositiontoanother. Figure3-50. LocationsalongthethicknesswheremicrographsofgrainsizedataweremadeforPlate1. 76

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Opticalmicroscopy(50X)atlocations1and2fromFigure 3-50 (3)(4)Figure3-52. Opticalmicroscopy(50X)atlocations3and4fromFigure 3-50 Thegrainsizesatallpositionsexcept1,2and7appearuniforminsize.Forgrainsatpositions1and2,therearebiggrainsof50to70msurroundedbygrainssimilartothosefoundatpositions3,4,5,6and8.Atposition7therearebiggrainsof40to50msurroundedbysmallergrains.Table 3-8 givestheaveragegrainsizeateachposition. 77

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Opticalmicroscopy(50X)atlocations5and6fromFigure 3-50 (7)(8)Figure3-54. Opticalmicroscopy(50X)atlocations7and8fromFigure 3-50 Table3-8. GrainsizeaveragesatlocationsshowninFigure 3-50 Position12345678Grainsize(m)2626161516152017 78

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3-55 ).Thesamplesweresequentiallynumberedcounterclockwisearoundtheplate. Figure3-55. Plate1with20couponsremoved Samples20and11wereremovedformetallurgicalandtexturalanalyses.Bothweresectionedatthemidplanewhileparalleltotheplaneoftheplate,thenmountedinresin(seeFigure 3-56 ).Twoviewsareavailableforeachsample:"top-down"and"bottom-up."Thetop-downviewcorrespondstotheviewingdirectionnecessarytoreadthesamplenumbers.Thebottom-upviewisoppositethisdirection.Theplateexhibitedstrongbertextureresultingfromtherollingprocess.Thebasalplane(0002)polegureswereexaminedtodeterminetherollingdirection.Previousworkby BarrettandMassalski ( 1980 )performedonrolledpuretitaniumstatesthatthebasalplanesalign35fromtheplatenormalduringrolling.Figure 3-57 veriesthisforSamples 79

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Denitionofsampleorientationfromsectionedcouponusedforinitialtexturemeasurements 11and20.Noticethatthetop-downandbottom-upviewsforSample11areessentiallymirrorimages. Figure3-57. Initial(0002)poleguresforPlate1fromthetwocouponsusedtoidentifytherollingdirection The12-o'clockpositionofeachgurecorrespondstothemidpointoftheouteredgeofthesample.Therollingdirectioncanberesolvedintwodimensionssinceitliesperpendiculartothebasaltexture).Totranslatetherollingdirectiontothethree-dimensionalplatehardware,atexturemapcanbesuperimposedontoanimageoftheplateitself.Figure 3-58 A)displaysthepoleguresshowninFigure 3-57 withthe 80

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3-57 a)isamirrorimageofitselfsinceFigure 3-58 A)representsatop-downviewratherthanbottom-up.AsimilarapproachwastakentoestablishtherollingdirectionforPlate2,twocouponswerecutfromtheouteredgeoftheplateandusedtoestablishtherollingdirection.Sincetheplatewasnearlyisotropicintheplaneoftheplatethiswassomewhatarbitrary.TheestablishedrollingdirectionwassetrelativetothetextureasshowninFigure 3-58 B). ABFigure3-58. Plate1andPlate2withpoleguressuperimposedtodeterminerollingdirectionA)Plate1B)Plate2 3-59 .Eachscancoveredanareaof200mX800m.Thetexturefromall17 81

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Positionofscanlocationsforthroughthicknesstexturemeasurements Figure3-60. BulktextureofPlate1foundfromaveragingthe17throughthicknessscans scanswereaveragedtogetabulktextureasshowninFigure 3-60 whichissimilartothetexturesmeasuredfromthecenterofthecouponsusedtoestablishtherollingdirectioninFigure 3-57 .Figures 3-61 to 3-69 showthepoleguresforeachofthe17locations.Somedierencesareapparentfromthesemeasurements.Thescanstakennearthecenteroftheplatehasasimilartexturetothebulktexturefoundfromaveragingall17scans.Scanstakenfromthetopandbottom4to5mmoftheplateshowssomenon-symmetrictexturesindicatingthestrongshearloadingfromtherollingprocess.Thismayaccountforsomeofthevariationintheuniaxialloadingtestresults.Thetestspecimenswerecutfromvariouslocationsinthethroughthicknessdirection,somefromsofterorharderregionsoftheplate. 82

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Plate1poleguresfrompositions1and2inFigure 3-59 (3)(4)Figure3-62. Plate1poleguresfrompositions3and4inFigure 3-59 (5)(6)Figure3-63. Plate1poleguresfrompositions5and6inFigure 3-59 83

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Plate1poleguresfrompositions7and8inFigure 3-59 (9)(10)Figure3-65. Plate1poleguresfrompositions9and10inFigure 3-59 (11)(12)Figure3-66. Plate1poleguresfrompositions11and12inFigure 3-59 84

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Plate1poleguresfrompositions13and14inFigure 3-59 (15)(16)Figure3-68. Plate1poleguresfrompositions15and16inFigure 3-59 Figure3-69. Plate1poleguresfromposition17inFigure 3-59 85

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3-70 showsthe0001polegureoftheintialtextureforplate1fromthreedierentperspectives.Figure 3-70 Ahasthetransversedirectioninthemiddleandthethroughthicknessdirectionfromsidetoside,Figure 3-70 BhasthethroughthicknessinthecenterandtheTDissidetosideandFigure 3-70 Chastherollingdirectioninthecenterandtransversedirectionfromsidetoside. ABCFigure3-70. Plate1(0001)PFofinitialtextureA)centerisTDandTTsidetosideB)centerisTTandTDsidetosideC)centerisRDandTDsidetoside Figures 3-71 and 3-72 show0001poleguresforspecimensloadedincompressioninthetransversedirectionat10%,20%,30%and40%strain.Thisshowsafairlystrongalignmentofthec-axiswiththethroughthicknessdirectionwhencomparedwiththe0001polegurewiththerollingdirectioninthecenter(Figure 3-70 (c)).Thisisveriedbytheuniaxialloadingtestswhichshowtheplateisstrongerinthetranseversedirectionthantherollingdirection. 86

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10%20%Figure3-71. Plate1(0001)polegureforspecimensloadedincompressionto10and20%intransversedirection,TDincenterandtheTTfromsidetoside 30%40%Figure3-72. Plate1(0001)polegureforspecimensloadedincompressionto30and40%strainintransversedirection,TDincenterandtheTTfromsidetoside 87

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3-73 and 3-74 showresultsfortheTTspecimensloadedincompression.Thepoleguresshowonlyslighttextureevolutionindicatinglowlevelsoftwinningforthisloadingpath.Theseshowastrongorthogonaltexturewithasignicantportionofthegrainsalignedwithin15oftheTTdirection. 10%20%Figure3-73. Plate1(0001)polegureforspecimensloadedincompressionto10and20%inthroughthicknessdirection,TTincenterandtheTDdirectionfromsidetoside 30%40%Figure3-74. Plate1(0001)polegureforspecimensloadedincompressionto30and40%straininthroughthicknessdirection,TTincenterandtheTDdirectionfromsidetoside 88

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3-75 and 3-76 showresultsfortheRDspecimensloadedincompression.ThepoleguresshowasignicanttextureevolutionthatwouldbeexpectedfromthehighlevelsoftwinningshownbothintheOIMmeasurementsandtheuniaxialstress-straincurves.Asignicantamountoftwinninghasoccuredbythepointwhere20%strainshavebeenreachedasshownbytheOIMdata. 10%20%Figure3-75. Plate1(0001)polegureforspecimensloadedincompressionto10and20%inrollingdirection,RDincenterandtheTDdirectionfromsidetoside 30%40%Figure3-76. Plate1(0001)polegureforspecimensloadedincompressionto30and40%straininrollingdirection,RDincenterandtheTDdirectionfromsidetoside 89

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3-77 3-78 and 3-79 showthesesamepoleguresattheappropriatestrainlevelsonstress-straincurvesforeachloadingcondition.ItisclearfromFigure 3-77 thatthechangeinslopeofthestress-straincurvecoincideswiththestrongchangeintexturefromthepolectures.Thenearlylinearhardeningportionofthetransverseandthroughthicknessdirectionscorrespondtothesmallertexturechangesobservedfortheseloadingconditions.TheresultsoftheOIMandtexturemeasurementsareconsistantwiththeuniaxialloadingtestsdoneonPlate1.Thestress-straincurveforcompressionintherollingdirectionshowsaclearchangeinhardeningthatisindicativeoftwinning,theOIMmeasurementsshowthatthereissignicanttwinningforthecompressionspecimensloadedintherollingdirectionandThetextureevolutionforthiscaseshowssignicantchangesarisingfromthelargegrainsrotationsoccuringduringtwinning.Althoughresourcesallowedonlyalimitedinvestigationofthetwinningforotherloadingpaths,noneoftheresultsindicatethattwinninghadalargeroleinthetextureevolutionfortheotherloadingconditions. Figure3-77. Textureevolutionforcompressiveloadingintherollingdirection 90

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Textureevolutionforcompressiveloadinginthetransversedirection Figure3-79. Textureevolutionforcompressiveloadinginthethroughthicknessdirection 91

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4.2 .Theintegrationalgorithmfortheproposedmodel,anditsimplementationintheexplicitFEcodeEPICfollows.AcomparisonbetweenmodelpredictionsandthedataisgiveninChapter 5 CazacuandBarlat ( 2004 )thatcapturesthetension/compressionasymmetries.Firstabriefoverviewofthecriterionisgiven.AfterreviewingthegeneralaspectsofalineartransformationoperatingontheCauchystresstensortheanisotropicyieldfunctionisdeveloped.Theinputdataneededforthecalculationoftheanisotropicyieldfunctioncoecientsarediscussed. CazacuandBarlat ( 2004 )proposedanisotropicyieldcriterionoftheform 22cJ3=3Y(4{1) 92

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2(3T+3C)(4{2)Whenc=0,i.e.T=C,thecriterion( 4{1 )reducestothevonMisescriterion.Toensureconvexity,cislimitedto:c23p 1 32112+223 2c 4-1 showsEquation 4{3 plottedfordierentratiosofT=C.Whenthisratioequals1,thecurvecorrespondstothevonMisesellipse. Figure4-1. PlanestressyieldlociiforvariousrationsofT=C 4-2 showsacomparisonoftheyieldcriteriondescribedbyEquation 4{1 todatacalculatedby Hosford ( 1966 )usingageneralizationofthe BishopandHill ( 1951 )model.Assumptionsforthisapproachincludethatdeformationisaccommodatedby 93

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4{2 ).Thegureshowstheplanestressyieldlocusforaratioof0.78(dashedcurve)correspondeingtoanfccmaterialaswellasaratioof1.28correspondingtoabccmaterial.Theopenandsolidcirclesaredataasreportedin Hosford ( 1966 ). Figure4-2. Comparisonwithpolycrystillinesimulations Notethattheyieldlocusgeneratedwiththeproposedcriterioncoincideswiththeyieldlocusobtainedbypolycrystallinecalculations.AlsoshowninFigure 4-2 isacomparisonbetweentheyieldlocuspredictedbythemacroscopicmodel(T=C=1.28)andthepolycrystallinemodel(fullcircles)forbccpolycrystals.Again,theyieldlocicoincide.Next,inordertodescribeboththeasymmetryinyieldingduetotwinningandanisotropyofrolledsheets,extensionstoorthotropyoftheisotropiccriteriongivenbyEquation 4{1 arepresented. 94

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2tr23=2c 3a3 3a1 3000000a4000000a5000000a63777777777777775ai,i=1...6areconstants.Inthe(x;y;z)framexrepresentstherollingdirection,ythetransversedirectionandzthethicknessdirection.Thisleadsto:=266666641 3[(a2+a3)xa3ya2z]a4xya5xza4xy1 3[a3x+(a1+a3)ya1z]a6yza5xza6yz1 3[a2xa1y+(a1+a2)z]37777775Theproposedyieldconditionis 95

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9a22+a23+a2a32x+a21+a23+a1a32y+a21+a22+a1a22z+2a23+a1a2a1a3a2a3xy +2a22a1a2+a1a3a2a3xz+2a21a1a2a1a3+a2a3yz+a242xy+a252xz+a262yz 27a22a3+a2a233x+a21a3+a23a13y+a21a2+a1a223z+a1a22+a1a23a22a32a2a232xy+a1a22a1a23a22a32a2a232xz+a21a2a21a3+a2a232a1a232yx+a21a22a21a3a1a23a2a232yz+a21a2a21a32a1a22+a22a32zx +2a21a2+a21a3a1a22a22a32zy+2a21a2+a21a3+a22a1+a22a3+a23a1+a23a2xyz+1 3fa2a24x+a1a24ya1a24+a2a24z2xy+a3a25xa1a25+a3a25y+a1a25z2xz+a2a26a3a26xa3a26y+a2a26z2yzg+2a4a5a6xyxzyzForplanestressthesereduceto 9a22+a23+a2a32x+a21+a23+a1a32y+2a23+a1a2a1a3a2a3xy+a242xy 96

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27a22a3+a2a233x+a21a3+a23a13y++a1a22+a1a23a22a32a2a232xy+a21a2a21a3+a2a232a1a232yx+1 3a2a24x2xy+a1a24y2xy Anexpressionforthestressinanarbitrarydirectionisusefulinparameteridenticationandcanbefoundasfollows.Bydenition(refertoFigure 4-3 )xcos2ysin2xysincos Arbitraryangledenition,xisrollingdirection Plugginginto 4{8 and 4{9 gives 27f(a1+a3)a1a3sin6+(a2+a3)a2a3cos6+(a22a1)a23(a2+a3)a21+9a1a24sin4cos2+(a12a2)a23(a1+a3)a22+9a2a24cos4sin2g3 97

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2ca22a3+a23a2o1 3(4{11) 2+ca22a3+a23a2o1 3(4{12)Similarly,withTtensandTcompbeingtheyieldstressintensionandcompressionalongthetransversedirection,then 2ca21a3+a23a1o1 3(4{13) 2+ca21a3+a23a1o1 3(4{14)When1=2=Tband3=0,yieldingunderequibiaxialoccurs 271 3(4{15)andforequiaxialcompressionwhen1=2=Cb 271 3(4{16) Plunkett ( 2005 ).Allparametersweredeterminedusingthebuilt-inminimizationfunctionMinerrofthesoftwareMathcad,version14.[ PTC ( 2007 )] 98

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99

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Plunkettetal. ( 2006 )proposedamethodforaccountingforthetextureevolution.Thismethodallowsforthevariationoftheanisotropycoecientswithaccumulatedplasticdeformation,i.e.thelineartransformationoperatorLisnolongerconstant.However,obtaininganalyticexpressionsfortheevolutionlawsofalltheLijcomponentswouldbeaformidabletask.Rather,theanisotropycoecientswillbecalculatedforanite(descrete)setofvalues.Choosingtheeectiveplasticstrain~"asthehardeningparameter,thecurrentyieldstressYandthecurrentequivalentstress~arefoundbyinterpolationbasedonthecurrentleveloftheeectiveplasticstrain.Theprocedureisasfollows: 100

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~"j~"~"j+1 ~(;~")current=~j+(1)~j+1 1{10 assuminganassociatedowrule(Equation 1{11 )andhardeningasdescribedbytheinterpolativeproceduredescribedabove. 101

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Hill ( 1950 )yieldsurfacedescription.Finally,theproposedmodelisappliedtoexperimentaldatagatheredforthehighpuritytitaniummaterialinvestigatedinthisresearch. KelleyandW.F.Hosford ( 1968 ).Thedataconsistsoftheresultsfromplane-straincompressiontestsinsixorientationsthatcorrespondtothesixcombinationsoftherollingdirection,transversedirection,andthicknessdirection;uniaxialcompressionanduniaxialtensiontestsinthex,y,andzdirectionsrespectively.Basedonthesedata,theexperimentalyieldlocicorrespondingtoseveralconstantlevelsofthelargestprincipalstrainwerereported.Duetothestrongbasalpolealignmentinthethicknessdirection,twinningiseasilyactivatedbycompressionperpendiculartothisdirection,butisnotactiveintensionwithintheplane.Theeectoftwinningisclearlyevidentinthelowcompressivestrengthsat1%.At10%strain,thethirdquadrantstrengthsarecomparabletotherstquadrantowingtotheexhaustionoftwinning.Theparametersinvolvedintheequationsoftheproposedmodelwerecalculatedusingtheprocedureoutlinedinthepreviouschapter.ThevaluesoftheanisotropycoecientsaregiveninTable 5-1 .Figure 5-1 showsthepredictionofcriteriongivenby Table5-1. ModelparametersfortheyieldsurfaceinFigure 5-1 Equation 4{5 incomparisonwiththeexperimentaldataat10%strain.Itisseenthatthecriteriondescribeswelltheobservedasymmetryandanisotropyinyielding. 102

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Projectionintheplane3=0forMg-Lialloysheetpredictedwiththecriterionanddata(10%strain) 2 ). 4{5 )weredeterminedatxedlevelsofaccumulatedplasticstrain(upto0.5).Forthetensiondata,thisrequiredanextrapolationaboveapproxiamtely20%strainsincethematerialbegantohavelocalizedstrainsbeyondthispoint.Thecorrespondingtheoreticalyieldsurfacesalongwiththeexperimentalvalues(lledsquares)areshowninFigure 5-2 .NotethattheproposedcriteriamatchesthedataverywellexceptfortheTDtensiondata.Theoptimizationproceduresusedindeterminingthemodelparameterswererequiredtomatchthemostsignicantdata.Sincethetensiledatawereextrapolatedbeyond20%,itwasallowedtohavealargererrorforthesedatainordertomatchtherest 103

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Theoreticalmodel(Equation 4{5 )comparedtoexperimentaldataforPlate1atvariousstrainlevels(dataarerepresentedbysymbols) 5-3 ,wheretheaveragevaluesarerepresentedbytriangles.Thecorrespondingtheoreticalyieldsurfacesalongwiththeexperimentalvalues(lledsquares) 104

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5-4 .Forthiscase,thetheoreticalyieldsurfaceshavethelargesterrorinthebiaxialdata.Again,itwasfeltthatthiswassucienttodemonstratetheabilityofthemodeltocapturetheanisotropicbehaviorofthematerial. Figure5-3. Averageexperimentalin-planecompressiondataforPlate2 Hill ( 1948 )isthemostwidelyusedorthotropicyieldcriterionavailableandhasproventobeaccurateandrobustformanymaterials,especiallysteels.However,itcannotaccountforthestrengthdierentialobservedinhexagonalmaterials.Forcomparisonpurposes,Hill'scriterionisappliedtothehighpurityTitaniummaterialusedinthisresearch.First,theidenticationprocedureusedtoidentifythecoecientsinvolvedin Hill ( 1948 )yieldcriterionispresented. 105

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TheoreticalmodelcomparedtoexperimentaldataforPlate2atvariousstrainlevels(dataarerepresentedbysymbols) Withrespecttotheorthotropyaxes(x;y;z),the Hill ( 1948 )orthotropicyieldcriterionhastheform 21 21 21 106

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2R2 2S2 2T2 Fortheplanestresscasei.e.z=zx=yz=0,thecriterioninEquation 5{1 reducesto Substituting( 5{9 )and( 5{10 )into( 5{8 )gives 3 Bydenitionther-valueinanarbitrarydirectionis @x+cos2@f @ysin2@f @xy @x+@f @y(5{14) 107

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5{11 ) @x=2sx(F+4G+H)2 3+2sy(4F+G+H)(1 3)+sy(4F+4G2H)2 3+sx(4F+4G2H)(1 3)@f @x=2(2G+H)sx+2(GH)sy @y=2sx(F+4G+H)(frac13)+2sy(4F+G+H)2 3+sx(4F+4G2H)2 3+sy(4F+4G2H)(1 3)@f @y=2(FH)sx+2(2F+H)sy @xy=2Nxy Thenumeratorof( 5{14 )becomes sin2@f @x+cos2@f @ysin2@f @xy=2sx(2G+H)sin2+(FH)cos2+2sy(GH)sin2+(2F+H)cos2sxy2Nsin2 Thedenominatorof( 5{14 )becomes @x+@f @y=2(F+2G)sx+2(2F+G)sy(5{19)Inserting 5{18 and 5{19 into 5{14 gives 5{8 gives 108

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5{12 )and( 5{13 ) 390andsy=2 390.Insertingtheseinto( 5{20 )gives F Nowtaking=0socos=1andsin=0thenx=0,y=0,xy=0.Insertingtheseinto( 5{8 )gives 245.Insertingtheseinto( 5{8 )gives 5{21 5{22 5{23 and 5{24 givesfourequationsinthefourunknownsH;G;H;andN.Thesesolveto 24 109

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5-2 )and( 5-3 )wereusedtodeterminethecoecients(giveninTable 5-4 )oftheHill(1948)criterioninconjunctionwithrelationsgivenbyEquation( 5{25 ). Table5-2. CompressiveyielddatausedtoidentifyHill48parametervalues DirectionxyzYieldStrength(MPa)142.7208.5246.8 Table5-3. TensileyielddatausedtoidentifyHill48parametervalues Directionx45oyzYieldStrength(MPa)127.1148.5200.8255.1 Table5-4. ParametervaluesforHill48modelusingPlate1data HillCoeFGHValue2.34E-05-7.598E-063.15E-05 ThetheoreticalHillyieldlocithusobtainedarefurthercomparedtothetheoreticalmodelanddatainFigure 5-5 .NotethatHill'syieldsurfacecannotcapturetheobservedbehaviorwhiletheproposedmodeldescribesverywelltheobservedstrengthdierentialeects. Johnsonetal. ( 2003 )]oftheexplicitniteelementcodeEPIC(E lasticP lasticI mpactC alculations).TheEPICcodehasbeendevelopedbyDr.GordonJohnsonundertheprimarysponsorshipoftheU.S.AirForceandU.S.Army.Therstdocumented(1977)versionwas2Donlybuthasevolvedintoa1,2or3Dversionwithmanyadditionsandenhancements.AllsimulationswerecarriedoutonaPCplatformusingCompagVisualFortranProfessionalEdition6.6a.Thecodewascompiledsuchthatallrealvariablesweredoublepercision. 110

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ComparisonofHill'scriteriontoproposedcriterionforPlate1data @(5{26) 111

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j+1n+1=j+1n+j+1n+1(5{28)whereindicatesanincrementovertheentiretimestepandindicatesanincrementforeachiteration.Thecorrectiontothestressduetoplasticstrainsis @j+1n+1(5{29)ThederivativesoftheequivalentstressarefoundbytakingonlythersttermofaTaylorseriesexpansionaboutthecurrentstate @j+1n+1@~ @j+1n+@2~ @2j+1n+@2~ @@"j+1n(5{30) 112

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5{30 )Equation( 5{29 )becomes @j+1n(5{31)Solvingfortheincrementforagiveniterationgives @j+1n(5{32)ATaylorseriesexpansionoftheyieldcriterionisusedtoobtainanapproximationoftheincrementoftheeectiveplasticstrain @"j+1nj+1n+1=0(5{33)EvaluatingderivativesatthepreviousstepEquations( 5{32 )and( 5{33 )canbemanipulatedtogive @C1@~ @+@YS @"(5{34)ThiscannowbeusedinEquation( 5{32 )tondj+1n+1andnallythetotalstressincrementisfoundfromEquation( 5{28 ).Thenewstressisthenevaluatedtoseeifithasconvergedwithinaspeciedtolerance.Ifnot,thisisusedasthestartingstressforthenextiteration.Whenthestresshasconverged,theglobalstresstensorisupdatedandreturnedtothemainprogram. Wangetal. ( 1997 ).Similarlytotheimplementationoftherate-independentmodel,thismethodrequiresthestressestoalwayslieonorintheinterioroftherate-dependentyieldsurfacef, 113

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4{5 );Y("vp;_"vp)isthecurrentyieldstrengthdeterminedusinganinterpolationhardeningapproach[ Plunkettetal. ( 2006 )]describedinSection( 5.4.5 )andisthecurrentstressstatepassedinfromtheFEcode;"vpand_"vparetheaccumulatedeectivevisco-plasticstrainandtheeectivevisco-plasticstrainrate,respectively.Usinganassociatedowrule _"vp=_@f @(5{36)where_isascalarmultipliersincefishomogeneousofdegreeoneinstresses._isthemagnitudeoftherateofchangeoftheeectivevisco-plasticstrain.Itisassumedthat_==t.Thesolutiontechniqueistoderivethestressesattimen+1basedonthestateattimenandaknownincrementoftotalstrain.InordertondthissolutionaTaylorseriesexpansionismadeaboutthestateatn f(;";_")n+1=f(;";_";T)n+@f @nn+1+@f @"vpnn+1+@f @_"vpnn+1 Here,thesubscriptsrefertoiterationsteps.Fortheintialstep,i.e.thestaten=0,isthetrialstateofstresscomputedassumingelasticity.Denotedby,achangeinquantityforacompletetimestepwhileindicatesthechangeduringaninterationstep.So,foragiventimestep n+1=n+j=kXj=0j(5{38)wherekisthenumberofstepsneededforconvergence.Thestressvariationisdescribedby @(5{39) 114

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5{39 )in( 5{37 )anexpressionfortheiterativevariationofcanbederived. @ C@ +@Y @"vp@ @"vp+1 t@Y @_"vp(5{40)Thestressesarethenupdatedusing n+1=n+n+1=nCn+1@ @n+1(5{41)Theeectivevisco-plasticstrainisupdatedas "t+tvp="tvp+(5{42)Iterationscontinueuntiltheyieldcriterion( 5{35 )issatisedwithinagiventolerance.TheJohnson-Cookhardeninglaw[ Johnsonetal. ( 1997 )]describedinChapter 1 wasusedforsimulationsoftherateeects.Thisproducedsmootherderivativesthanforthecaseofpiecewiselinearhardeningusedfortheelastic-plasticversionofthemodel. ~1h(J2)3=2cJ3i1=3(5{43)where1isaconstantdenedsuchastoassurethat~reducestothetensileyieldstressintherollingdirection.Thus, 1=1 3a22+a23+a2a33=2c @=@F @J2@J2 @J3@J3

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@ij=@F @J2@J2 @J3@J3 @J2=J1=22 @J3=c 3(a2+a3)@22 3a3@33 3a2@11 3a3@22 3(a1+a3)@33 3a1@11 3a2@22 3a1@33 3(a1+a2)@12 Plunkettetal. ( 2006 ).Itemploysalinearinterpolationschemebetweenadiscretenumberofyieldsurfacescorrespondingtoxedlevelsofaccumulatedplasticstrain.First,agivenstrainpathonwhichtobasethehardeningischosen.Forexample,forallthequasi-staticsimulationsperformedinthisstudy,the 116

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Plate1anisotropycoecientvaluesfordiscretestrainlevels StrainYielda1a2a3a4a5a6c 0.00002080.54540.50101.09000.7246-0.8675-0.8675-0.21680.02502450.52310.47450.90340.73090.72020.7202-0.21980.05002610.66940.55851.10300.91380.93810.9381-0.22910.07502730.69600.59691.12700.98380.97160.9716-0.26070.10002840.53560.47680.86030.77610.77140.7714-0.27540.20003170.06100.05760.08690.08700.07940.0794-0.59080.40003700.06320.06200.07880.08160.08010.0801-1.03300.50003890.95470.95701.21401.18101.17601.1760-1.1480 Note:YieldStrengthinMPa hardeningwasbasedonthetensilestrainpathintherollingdirection.FromthistheTCstressversusstraincurve,adiscretenumberofstrainlevelsischosen.Asucientnumberofpointswasusedtoensurethatthehardeningcurvewasrecreatedwithenoughaccuracy.Foreachofthesestrainlevels,rangingfrom0to50%,thecorrespondingyieldsurfacesaccordingtoEquation( 4{5 )weredetermined.TheanisotropiycoecientsshowninTables 5-5 5-6 werecalculatedfollowingtheprocedureoutlinedinsection 4.2 5-6 A.ThedatausedtoidentifythemodelparametersarerepresentedinFigure 5-6 Bbysymbols.AsimilarprocedureforPlate2,usingthedatashowninFigure 5-7 B,givetheyieldlocishowninFigure 5-7 A.Recallthatthetensiondatarequiredanextrpolationaboveapproxiamtely20%strainsincethematerialbegantohavelocalizedstrainsbeyondthisstrainlevel.Notethatthemodelreproducesthedataquitewellwiththelargesterrorforthebiaxialdata.Table 5-5 givestheuniaxialstressesintheRDandanisotropycoecientvaluesdeterminedateachstrainlevelforPlate1.Table 5-6 givesthesameinformationforPlate2. 117

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Plate1yielddataA)theoreticalyieldcurvesforxedlevelsofaccumulatedplasticstrainB)Datausedinidentifyingtheoreticalyieldcurves ABFigure5-7. Plate2yielddataA)theoreticalyieldcurvesforxedlevelsofaccumulatedplasticstrainB)Datausedinidentifyingtheoreticalyieldcurves 118

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Plate2anisotropycoecientvaluesfordiscretestrainlevels StrainYielda1a2a3a4a5a6c 0.00002080.54540.50101.09000.7246-0.8675-0.8675-0.21680.02502450.52310.47450.90340.73090.72020.7202-0.21980.05002610.66940.55851.10300.91380.93810.9381-0.22910.07502730.69600.59691.12700.98380.97160.9716-0.26070.10002840.53560.47680.86030.77610.77140.7714-0.27540.20003170.06100.05760.08690.08700.07940.0794-0.59080.40003700.06320.06200.07880.08160.08010.0801-1.03300.50003890.95470.95701.21401.18101.17601.1760-1.1480 Note:YieldStrengthinMPa obtainedfortheRDdatausedfortherepresentaionofhardening.Nextsimulationsofstress-strainresponseforotherorientationswereperformedandcomparedtodata.Simulationsinvolvedasinglecomputationalcellwitheightnodeswithasingleintegrationpoint.Thecellwasstrechedinonedirectionalonganaxisandstressversusstraindatawerecollectedandcomparedtotheappropriateexperimentaldata.ThemodelwasthenvalidatedbysimulatingthefourpointbendtestsdescribedinChapter 3 .Thecomparisonwasmadeinaqualitativewaybyjuxtaposingcontoursoftheexperimentaldataagainsttheresultsfromthesimulation.Thiswasdoneforallfourorientationsofthebeamspecimens.ComparisonoftheexperimentalcrosssectionsofthebeamsandsimulatedonesusingthemodelandanisotropicmaterialwithavonMisesyieldsurfacewereperformed.Amorequantitativecomparisonwasdonebycomparingaxialstrainversusheightatthecenterlineofthebeam. 5-8 ,wasaneightnodedconstantstrainelementwithasingleintegrationpoint.Foreachsimulation,fournodesononefaceoftheelementwererestrainedandthefournodesontheoppositefaceweregivenaconstantvelocityineitherthetensileorcompressivedirection.Stressandstraindatawerecollected 119

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Figure5-8. Singlecellcomputationalconguration 5-9 to 5-11 showthecomparisonofsinglecellsimulationsforPlate1.Notethatsimulationsaccuratelyreproducethedataforeachcondition.ThelargesterroroccursfortheTDtensiondata,whichisconsistentwiththediscrepancybetweenexperimentalandpredictedowstressesnotedpreviously. 120

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SinglecellsimulationresultsforPlate1A)RDtensionB)RDcompression ABFigure5-10. SinglecellsimulationresultsforPlate1A)TDtensionB)TDcompression 121

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SinglecellsimulationresultsforPlate1A)TTtensionB)TTcompression 5-12 and 5-13 showthecomparisonofsinglecellsimulationsforPlate2.Forthein-planeplots,boththeRDandTDdataareshownonthesameplot.ItisagainclearthatthatPlate2innearlyisotropicintheplaneoftheplate.VerygoodagreementisfoundforallcasesinPlate2.Thelargesterrorsoccurinthebiaxialdatawhichcorrespondstothelargesterrorsbetweenthepredictedowstressesandexperimentaldatanotedearlier.EventhoughtheRDdirectiontensiledatawasusedformodelingisotropichardening,thesimulationsosalltheotherstresspathswereingoodagreementwiththedata.Suchgoodagreementcanbeachievedonlybyaccountingfortextureevolutioni,e,theanisotropytensorisconsideredafunctionoftheaccumulateddeformation. 122

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SinglecellsimulationresultsforPlate2A)In-planetensionB)In-planecompression ABFigure5-13. SinglecellsimulationresultsforPlate2A)ThroughthicknesstensionB)Throughthicknesscompression "symmetrical"brickarrangement.Thisarranges24tetrahedralelementsintoahexagonalorbrickstructureasshowninFigure 5-14 .Thisisdonetominimizethewellknownstibehaviorofthistypeofelement. 123

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Symmetricalbrickarrangmentfortetrahedralelements Theloadingprolewasappliedtotheappropriatesideofthebeamatthesamedistanceasthecenteroftheloadingpin(10mmfromthecenterline).Alineofnodeswasrestrainedontheoppositefaceofthebeam(at20mmfromtheplaneofsymetry)tosimulatetheconstrainingpin.Theconstrainingnodeswererestrainedinthedirectionofloadingbutwerefreefortheothertwodirectionswithnofriction.AtypicalcomputationalmeshisshowninFigure 5-15 forloadingasprescribedbyCase1.Allothersimulationsusedthesamemeshwithloadingandconstraintdirectionsappropriatefortheparticularcase.Figure 5-16 showsatypicaldeformedcountouredmeshindicatingtheplaneofsymmetry. 124

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FEComputationalmeshforbeambendingtests Figure5-16. Typicaldeformedmeshshowingplaneofsymmetry 125

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5-17 .Asexpectedwhentheharddirection(TT)isperpendiculartotheloadingdirection(Case1and3)thecrosssectionremainsnearlysquare.Case2andCase4aresimilartoeachotherwithmorelateralstrainshownbyCase4.ThisisconsistentwithPlate1beingharderintheTDthantheRDasshowninthetests(seeFigure 5-18 ).Thedatashowsthatforstrainlevelsbelow15%,theplateisstrongerinbothtensionandcompressionat90fromtherollingdirection(TD)ascomparedtoRD. Figure5-17. ComparisonofcrosssectionalareafromsimulationsofthefourbeamorientationsfromPlate1 AsimulationusinganisotropicvonMisestypemodelwasusedtosimulatethefourpointbendtestforcamparisontothesimulationsranusingtheanisotropicmodel.Figure 5-19 showsacomparisonbetweentheisotropicsimulationagainstthefourbeam 126

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ComparisonoftensionversuscompressiondataforRDandTDinPlate1 Figure5-19. ComparisonofcrosssectionsfromPlate1beamsimulations(redmeshisfromisotropicsimulation) testorientation.NotethatinCase1andCase3,theharddirection(TT)isthewidthdirectionandtheisotropicsimulationshowsmoredeformationthanthatusingtheanisotropicmodel.ForCase2andCase4thereislessdeformationintheheightofthe 127

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5-20 forreference.Figure 5-21 showsthecomparisonoftheproleofaxialstraincontoursforthesimulationforCase1comparedtotheexperimentaldata.NotethatthedatafromtheexperimentdoesnotcovertheentireproleareaduetotheDIC[ Miguil-Touchaletal. ( 1997 ); HungandVoloshin ( 2003 )]techniquesused.Verygoodagreementisshown. Figure5-20. Case1:LongaxisinRD,loadinginTD Figure5-21. Plate1,Case1:Comparisonofaxialstraincountours("x)fromsimulationagainstexperimentaldata:x=RD,y=TD 128

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5-22 whichshowsaplotoftheaxialstrainversustheheightofthebeamatthecenterofthebeam.Thisshowsverygoodagreementbetweentheexperimentandsimulationandaclearupwardshiftoftheneutralaxisofthebeam. Figure5-22. Plate1,Case1:Axialstrains("x)versusheightatcenterline:x=RD,y=TD Asanalvalidationofthemodel,thebeamsweresectionedatthemidpointandanimageofthecrosssectionwascomparedtothesimulation.ThecomparisionforPlate1forCase1isshowninFigure 5-23 .Thereisverylittledeformationperpendiculartotheloadingdirectionbecausethisistheharder,TTdirection.ThebeamorientationforCase2isshowninFigure 5-24 forreference.Figure 5-25 showsthecomparisonoftheproleofaxialstraincontoursforthesimulationforCase 129

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Plate1,Case1:Comparisonofcrosssectionsfromexperiment(photo)andsimulation(symbols):y=TD,z=TT 2comparedtotheexperimentaldata.Again,verygoodagreementisshownbetweenexperimentandsimulation.Thesimulationdoesgivesomewhatlessstrainthroughthethicknessintheloadingdirection. Figure5-24. Case2:LongaxisinRD,loadinginTT 130

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Plate1,Case2:Comparisonofaxialstraincountours("x)fromsimulationagainstexperimentaldata:x=RD,z=TT Figure5-26. Plate1,Case2:Axialstrains("x)versusheightatcenterline:x=RD,z=TT 131

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5-26 .Thisshowsverygoodagreementbetweentheexperimentandsimulationandaclearupwardshiftoftheneutralaxisofthebeam.ThecomparisionofcrosssectionsforPlate1forCase2isshowninFigure 5-27 whichshowsverygoodagreement.Thereismoredeformationperpendiculartotheloadingdirectionbecausethisisnowthesoftertransversedirection. Figure5-27. Plate1,Case2:Comparisonofcrosssectionsfromexperiment(photo)andsimulation(symbols):y=TD,z=TT ThebeamorientationforCase3isshowninFigure 5-28 forreference.Figure 5-29 showsthecomparisonoftheproleofaxialstraincontoursforthesimulationforCase3comparedtotheexperimentaldata.Again,verygoodagreementisshown. 132

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Case3:LongaxisinTD,loadinginRD Figure5-29. ]Plate1,Case3:Comparisonofaxialstraincountours("y)fromsimulationagainstexperimentaldata:x=RD,y=TD AplotoftheaxialstrainversustheheightofthebeamatthecenterofthebeamisshowninFigure 5-30 .Thisshowsverygoodagreementbetweentheexperimentandsimulationandaclearupwardshiftoftheneutralaxis.ThecomparisionofcrosssectionsforPlate1forCase3isshowninFigure 5-31 whichshowsexcellentagreement. 133

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Plate1,Case3:Axialstrains("y)versusheightatcenterline:x=RD,y=TD Figure5-31. Plate1,Case3:Comparisonofcrosssectionsfromexperiment(photo)andsimulation(symbols):x=RD,z=TT 134

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5-32 forreference.Figure 5-33 showsthecomparisonoftheproleofaxialstraincontoursforthesimulationforCase4comparedtotheexperimentaldata.Again,verygoodagreementisshown. Figure5-32. Case4:LongaxisinTD,loadinginTT Figure5-33. ]Plate1,Case4:Comparisonofaxialstraincountours("y)fromsimulationagainstexperimentaldata:y=TD,z=TT AplotoftheaxialstrainversustheheightofthebeamatthecenterofthebeamisshowninFigure 5-34 .Thisshowsverygoodagreementbetweentheexperimentand 135

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Plate1,Case4:Axialstrains("y)versusheightatcenterline:y=TD,z=TT Figure5-35. Plate1,Case4:Comparisonofcrosssectionsfromexperiment(photo)andsimulation(symbols):x=RD,z=TT simulationandaclearupwardshiftoftheneutralaxisofthebeam.ThecomparisionofcrosssectionsforPlate1forCase4isshowninFigure 5-35 whichshowsveryagreement. 136

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5-36 .Againtheseisverygoodqualitativeaggreementwithexperimentaldata.AsforPlate1,fortheCase1andCase3,wherethethroughthicknessdirection(theharderdirection)isnormaltotheloadingdirection,thereisverylittlevariationfromarectanglularcrosssection.ThereisamuchgreaterdeviationforCase2andCase4wherethehardestdirectionisintheloadingdirection.ItwasalsonotedthatCase1andCase3aswellasCase2andCase4aresimilarduetothein-planeisotropyofPlate2. Figure5-36. ComparisonofcrosssectionalareaforPlate2 137

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5-37 and 5-38 .Again,whenthewidthofthebeamcorrespondstothehard(throughthickness)direction,verylittledistorsionofthecrosssectionisobserved. Figure5-37. Plate2Isotropicsimulation(blacklines)versusmodel(blueandredlines)Case1and3 Figure5-38. Plate2Isotropicsimulation(blacklines)versusmodel(blueandredlines)Case2and4 ThebeamorientationsforPlate2,Case1toCase4arethesameasthoseforPlate1.Figure 5-25 showsthecomparisonoftheproleofaxialstraincontoursforthesimulationforCase1comparedtotheexperimentaldata.Verygoodagreementisshown. 138

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Plate2,Case1:Comparisonofaxialstrain("x)countoursfromsimulationagainstexperimentaldata:x=RD,y=TD Figure5-40. Plate2,Case1:Axialstrains("x)versusheightatcenterline:x=RD,y=TD AmorequalitativecomparisonshowingtheplotoftheaxialstrainversustheheightofthebeamatthecenterlineisshowninFigure 5-40 .Thisshowsgoodagreementbetweentheexperimentandsimulationandaclearupwardshiftoftheneutralaxisofthebeam. 139

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5-41 showingexcellantagreement.Verylittledeformationoccursperpendiculartotheloadingdirectionsincethisistheharderthroughthicknessdirection. Figure5-41. Plate2,Case1:Comparisonofcrosssectionsfromexperiment(photo)andsimulation(symbols):y=TD,z=TT Figure 5-42 showsthecomparisonoftheproleofaxialstraincontoursforthesimulationforCase2comparedtotheexperimentaldata.Again,verygoodagreementisshown.AplotoftheaxialstrainversustheheightofthebeamatthecenterofthebeamisshowninFigure 5-43 .Thisshowsverygoodagreementbetweentheexperimentandsimulationandaclearupwardshiftoftheneutralaxisofthebeam.ThecomparisionofcrosssectionsforPlate2forCase2isshowninFigure 5-44 whichshowsverygoodagreement.Thereismoredeformationperpendiculartotheloadingdirectionbecausethisisnowthesoftertransversedirection. 140

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Plate2,Case2:Comparisonofaxialstraincountours("x)fromsimulationagainstexperimentaldata:x=RD,z=TT Figure5-43. Plate2,Case2:Axialstrains("x)versusheightatcenterline:x=RD,z=TT 141

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Plate2,Case2:Comparisonofcrosssectionsfromexperiment(photo)andsimulation(symbols):y=TD,z=TT Figure 5-45 showsthecomparisonoftheproleofaxialstraincontoursforthesimulationforCase3comparedtotheexperimentaldatawithexcellentagreement. Figure5-45. Plate2,Case3:Comparisonofaxialstraincountours("y)fromsimulationagainstexperimentaldata:x=RD,y=TD 142

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5-46 andthecomparisionofcrosssectionsfromsimulationandexperimentforPlate2Case3isshowninFigure 5-47 .Excellentagreementisshownincludingaclearupwardshiftoftheneutralaxis. Figure5-46. Plate2,Case3:Axialstrains("y)versusheightatcenterline:x=RD,y=TD Figure5-47. Plate2,Case3:Comparisonofcrosssectionsfromexperiment(photo)andsimulation(symbols):x=RD,z=TT 143

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5-48 showsthecomparisonoftheproleofaxialstraincontoursforthesimulationforCase4comparedtotheexperimentaldata.Again,verygoodagreementisshown. Figure5-48. Plate2,Case4:Comparisonofaxialstraincountours("y)fromsimulationagainstexperimentaldata:y=TD,z=TT Figure5-49. Plate2,Case4:Axialstrains("y)versusheightatcenterline:y=TD,z=TT 144

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5-49 .Thisshowsverygoodagreementbetweentheexperimentandsimulationandaclearupwardshiftoftheneutralaxisofthebeam.ThecomparisionofcrosssectionsforPlate2forCase4isshowninFigure 5-50 whichshowscloseagreement. Figure5-50. Plate2,Case4:Comparisonofcrosssectionsfromexperiment(photo)andsimulation(symbols):x=RD,z=TT 145

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5-51 A.Posttestobservationsconrmedthein-planeisotropyofthematerial.Also,theminoraxisofthedeformedspecimenswerealignedtowithin5degreesofthismark.Thisisasexpectedsincethethespecimenshavebasaltexturei.e.thehardtodeformc-axisdirectionliesinthecrosssection.Aphotographofthedeformedfootprintatthecylinder-anvilinterfaceisshowninFigure 5-51 Bandcomparedtoatruecircleclearlyshowstheanisotropicdeformationinthisplane.Theaxiswithlessdeformation,theminoraxis,isnearlyalignedwiththec-axisofthematerial. ABFigure5-51. DeformedhighratespecimenA)Testspecimenwiththroughthicknessdirectionidentiedbyarrow.B)Thedeformedellipticalfootprintfromexperimentascomparedtoacircle. TmeltisanhomologoustemperatureandTmeltisthemeltingtemperature.C1,C2,C3,Nand 146

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Highratecompresivedatawithlineartusedinparameteridentication 5{46 wereidentiedfromin-planecompressiondataratherthanin-planetensiledataasforthefourpointbeambeamsimulations.ThisisbecausethedominantloadingduringtheTaylortestsisin-planecompression.Thequasi-staicdatawasusedtoobtaintheparametersC1,C2andN.AlinearcurvettothehighratecompressiontestswasusedtoidentifytheparameterC3(seeFigure 5-52 ).ThebuiltinminerrfunctionofMathCadwasusedtoidentifyallparameters.Thecostfunctionusingonlythequasi-staticdatais Error=Xi[QSYexperimentYJC(C1;C2;C3=0;N)]2whereQSYexperimentisthequasi-staticexperimentaldataatidescretestrainlevelsandYJC(C1;C2;C3=0;N)istheyieldstrengthcomputedfromtheJ-CModel(Equation 5{46 )atthesamediscretestrainlevels. 147

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Error=Pif[QSYexperimentYJC(C1;C2;C3=0;N)]+[HRYexperimentYJC(C1;C2;C3;N)]g2whereHRYexperimentisthehighrateexperimentaldataatidiscretestrainlevels.TheJ-CparametervaluesaregiveninTable 5-7 Table5-7. Johnson-CookhardeninglawparametervaluesforEquation 5{46 Figure 5-53 showsthecomparisonbetweenthevaluesobtainedusingtheJ-CmodelforthesetofvaluesgiveninTable 5-7 andexperimentaldatausedintheidenticationoftherespectiveparameters. Figure5-53. ComparisonofyieldvaluesobtainedfromJ-Clawtoexperimentaldatausedinparameteridentication 5-54 .Again,the"symmetrical"brickarrangementwasusedinorderto 148

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Figure5-54. InitialFEmeshforTaylorimpactsimulationswith34,560four-nodetetrahedralelements:specimendimensionsare:Height=2.1inches,Diameter=0.21inches.(a)3-Dview,(b)crosssection,(c)initialprole 5{46 ))includingrateeects.Nextresultsaregivenforasimulationusingtheproposedanisotropicvisco-plasticmodelwiththeparametersgiveninTable 5-6 andtheJ-ChardeninglawwithC3=0(i.e.theratetermisnotactivated).Finallyasimulationusingtheanisotropicvisco-plasticmodelandtheJ-Cmodelwiththeratetermactivatedisgiven.Fortheisotropiccase,YSinthehardeninglawistheeectivevonMisesstress,whileforrheanisotropycasesYSistheeectivestressassociatedwiththeproposedyieldfunctiongivenbyEquation 4{5 149

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5-56 showsacomparisonofprolestakenfrom90aroundthedeformedcylinder.Notethatthetwoproleslieontopofoneanotherasexpectedforanisotropicmaterial.Figure 5-55 showsthedeformedspecimenandnalcrosssection,respectively. Figure5-55. CylinderimpactsimulationresultsusingisotropicvonMisesandJ-Chardeninglawwithrateeectsactivated,(a)deformedprole,(b)3Dview(c)deformedfootprint 150

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Comparisonofprolesfromisotropicsimulationtakenat90aroundcircumferenceusingJ-Chardeninglawwithrateeectsactivated Thesimulationusingtheproposedanisotropicelastic/plasticmodelwascarriedoutwheretheanisotropiccoecientsarefunctionsoftheplasticstrainasdiscussedintheinterpolationprocedureinsection 5.4.5 .Thusthetextureevolutionisaccountedfor.IsotropichardeningisdescribedbytheJ-Clawwithouttakingrateeectsintoaccount(i.e.c3=0).Thesimulationresultsshowanellipticalfootprint,i.e.thesurfaceofthespecimenimpactingtherigidanvil.Theminoraxisisalignedwiththethroughthicknessdirectionoftheplate.ThedeformedmeshandfootprintareshowninFigure 5-57 .Prolesfrom90alongthecircumferenceofthecylinderarecomparedinFigure 5-58 showingasignicantdierencebetweenthemajorandminoraxes.ThecylinderimpacttestswerealsosimulatedusingtheproposedanisotropicviscoplasticmodelincludingthefullJ-Cmodelwithrateeectsturnedon.Theresultsareclosertoexperimentaldataandshowlessdeformationthanthecasewherenorateeectswereincluded.AcomparisonofthemajorandminorprolesisshowninFigure 5-59 .ThedeformedprolesandthenalcrosssectionareshowninFigure 5-60 151

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Cylinderimpactsimulationresultsusingproposedanisotropicelastic/plasticmodelandJ-Chardeningwithoutrateeects(a)Majorprole;alignedwithin-planedirection(b)Minorprole;alignedwiththroughthicknessdirection(c)3Dviewofspecimenwithaxialstraincountours(d)Finalcrosssection 152

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Comparisonofdeformedcylinderprolefromtwolocations90apartforsimlationusingproposedanisotropicmodelandisotropichardeningaccordingtoJ-Clawwithnorateeectsincludedinthesimulations) Figure5-59. Comparisonofdeformedcylinderprolefromtwolocations90apartobtainedusingproposedanisotropicttoproposedelastic/viscoplasticmodelandJ-Chardeningwithrateeects 153

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CylinderimpactsimulationresultsusinganisotropicparametersforproposedcriteriausingJ-Chardeningwithrateeects(a)Majorprole;alignedwithin-planedirection(b)Minorprole;alignedwiththroughthicknessdirection(c)3Dviewofspecimenwithaxialstraincountours(d)Finalcrosssection 154

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5-61 to 5-62 showthecomparisonofdeformedspecimensobtainedusingtheisotropicmodel,anisotropicmodelwithnorateeectsandtheelastic/viscoplasticanisotropicmodel,respecively.Specically,Figure 5-61 showsthecomparisonofmajoraxesproles,Figure 5-62 showsthecomparisonofminorprolesandFigure 5-63 comparesthefootprintsimulatedineachcase.Notethatthetotalheightofthedeformedcylinderwithnorateeectsislessthanforboththeisotropicsimulation(usingrateeects)andtherate-dependentanisotropicmodel.Also,intherate-independentsimulations,thereismoreradialdeformationthanfortheothertworate-dependentcases.Thisclearlydemonstratestheneedtomodelrateeectsinordertocapturethecharacteristicsofthedeformationunderhighstrainrates. Figure5-61. Comparisonofmajorprolesobtainedusingthedierenctmodels(a)Undeformedmesh(b)anisotropicmodelwithrateeects(c)isotropicvonMiseswithrateeects(d)anisotropicmodelwithnorateeects 155

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Comparisonofminorprolesobtainedusingthedierenctmodels(a)Undeformedmesh(b)anisotropicmodelwithrateeects(c)isotropicvonMiseswithrateeects(d)anisotropicmodelwithnorate Figure5-63. Comparisonofthepredictedfootprintobtainedusingthedierenctmodels(a)Undeformedmesh(b)anisotropicmodelwithrateeects(c)isotropicvonMiseswithrateeects(d)anisotropicmodelwithnoratemesh TestnumberRM107wastakenastypicalfromthe13testsperformedandwasusedtocomparetovalidationsimulations.Proledataweretakenfromthemajorandminoraxesaswellasthenaldeformedfootprintasdescribedinsection 3.2.2 ofChapter 3 usingusinganopticalcomparatormodelDIJ415. 156

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5-64 showsacomparisonbetweenthesimulatedandexperimentaldata.Thesimulationmatchesthemajoraxisverywellwhileslightlyunderpredictingtheminoraxisdeformation.Figure 5-65 showsthecomparisonofaxialstrainsalongthemajor Figure5-64. Comparisonofdeformedimpactsurface:FEsimulationswithviscoplasticmodeltoexperimentaldata axesversusheightfromexperimentaldatatothatobtainedfromsimualtionsincludingrateeects.Thestrainsfromthesimulationneartheimpactfaceareverysimilartotheexperimentaldatabutshowmoredeformationastheheightincreases.Thisprobablyarisesbecausethestress-strainbehaviorforonlytwodierentstrainrateswasavailable,thustheuncertaintyrelatedtothedeterminationoftheseparameters.Havingdatafromhigherratesforthesamematerialshouldprovideamoreaccuratettothetruehardeningbehaviorunderdynamicconditions. 157

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Comparisonofmajoraxisradialstrainversusheightpredictedbytheanisotropicmodelandexperimentaldata Figure 5-66 showsaxialstrainsalongtheminoraxisversusheightforthethreecases.Bothsimulationsunderpredictthedeformationalongtheminoraxis.Thisisprobablyaresultoferrorsintheparameterizationoftheproposedmodelratherthanentirelyrateeects. Figure5-66. Comparisonofminoraxisradialstrainversusheightpredictedbytheanisotropicmodelandexperimentaldata 158

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5-67 .Notetheverygoodagreementbetweenexperimentandsimulation. Figure5-67. Comparisonofratioofmajortominordiametersversusheightpredictedbytheanisotropicmodelandexperimentaldata 159

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Lemaitre ( 2001 ),materialmodelingcanbeconsidered\ascience,atechnique,andanart."Inthisdissertationallthesefacetsofmodelinghavebeenconsidered.Thescienceaspectconsistsofthecarefulandsystematicexperimentalcharacterizationofthebehaviorunderloadingandintheeorttoincludethemainfeaturesoftheobservedbehaviorinananisotropicmodel.Thetechniqueisinidentingmodelparametersandintegratingthemodeltopredictthebehaviorofthematerialunderloadingconditionsotherthanthoseusedtobuildandparameterizethemodel.Equallyimportantistheengineeringartofusingtheverynonlinearmodeltopredictthenonlinearbehaviorofamaterialthatisevolvingasthetextureevolves.Thisentailstheincorporationofphysics/phenomenaatdierentlengthscales.Classicalplasticityaccountsforplasticdeformation,whichatcrystalscaleoccursthroughslipassociatedwithdislocationmotion.Forhexagonalclosedpacked(hcp)materials,atthesinglecrystallevel,onehastoincludetwinningasanadditionalmechanismofplasticdeformation.Twinningisresponsiblefordrasticandabruptlatticerotationswhichinturnleadtosignicanttextureevolutionduringeventhesimplestloadingpaths.Twinningbeingapolarshearmechanism,inducesatension/compressionasymmetryatthemacroscopicscale.Characterizationandmodelingoftheinterplaybetweenslipandtwinningandtheireectonthemechanicalresponseremainsagreatchallenge.Thisdissertationisanattempttoextendthecurrentknowledgeonhcpmaterials.Thisworkhasconsistedofthreemajorareasthatsomewhatcorrespondtothethreeaspectsdescribedabove.First,anexperimentalinvestigationintothebehaviorofhighpuritytitaniumwasconducted.Twohighpuritytitaniumplateswereconsidered;onewithanorthotropictextureandonewhichwasisotropicintheplaneoftheplatebutdieredinthedirectionnormaltotheplaneoftheplate. 160

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3 ,whichincludeduniaxialtensileandcompressiontestsatbothquasi-staticandhighloadingrates.Validationexperimentsunderquasi-staticconditionsconsistedofaseriesoffourpointbendingtestsonbeamscutsuchthattheirlongaxeswerealignedeitherwiththerollingdirectionortransversedirection;loadingwasappliedeitherintherolling,transverse,orthroughthicknessdirection.Sincethetopbeambersareincompressionwhilethebottombersareintension,thebendingtestsresultstesttheabilityofmodelstocaptureboththestrengthdierentialeectsandanisotropyoftitanium.TheclassicalTaylorcylinderimpacttestswerecarriedoutforvalidationathighloadingrates.Inaddition,investigationsweremadetoestablishtheinitialtextureofbothplates.ForPlate1thetextureevolutionwithplasticdeformationwasinvestigatedprimarilyundercompression.Furthermore,OrientationImageMicroscopy(OIM)measurementsweremadeforspecimensloadedincompressionintherollingdirectionsincethestress-straindatafromthesetestsindicatedasignicantincreaseinhardeningrate,hencethepossibilityofdeformationtwinning.Thetexturemeasurementsshowedaclearrotationofthec-axesofthegrainsassociatedwithtwinning.Allofthetexturemeasurementscorraboratedtheuniaxialstress-straindatawhichshowedthattwinningplayedasignicantroleforthisloadingcondition.Anewanisotropicyieldcriterionwasdevelopedinordertomodeltheobservedbehavior(seeChapter 4 ).Themodelproposedisanextensiontoorthotropyofanisotropicdescriptionfrom CazacuandBarlat ( 2004 )usingalineartransformationapproach.Forgeneral(3D)conditions,theproposedanisotropicmodelinvolvessevenparameters:6anisotropycoecientsand1parameterassociatedtostrengthdierentialeects.Theapproachtomodelingtheevolutionoftheyieldsurfacetoaccountforthetextureevolutionoccuringinthematerialwastousethelinearinterpolationschemedevelopedby Plunkettetal. ( 2006 ).Themethodologyconsistsofcomputinganequivalentstressaccordingtotheanisotropiccriterionatdiscretestrainlevelsand 161

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5 ).Theimplementationwasveriedbysimulatingtheuniaxialloadingtestsusingasingleconstantstrainelement.Thefourpointvalidationtestsweresimulatedusingtheelasticplasticmodeldevelopedforallfourbeamcongurations.Theresultsshowanexcellantagreementbetweenthesimulationresultsandthedeformedspecimensusingvariouscomparisons.Forallfourcasesforeachplate,themodelwasabletocloselymatchthecrosssectionaldeformation.Whenthehardtodeformdirectioni.e.thethroughthicknessdirection,wasperpendiculartotheloadingdirectionthenal(deformed)crosssectionswerenearlysquarewhilewhentheloadingdirectionwasalignedwiththethroughthicknessdirection,thedeformedcrosssectionsweremorewedge-shaped.Acomparisonwasalsomadetotheaxialstrainversusheightatthecenterofeachspecimen.Againthesimulationsshowedexcellentagreementwiththeexperimentforallcasesincludingaclearshiftoftheneutralaxisfromthecenterlineofthebeams.Finally,theratesensitiveversionofthemodelwasusedinsimulationsofacylinderimpacttestforoneoftheplates.Thesimulateddeformedprolesaswellastheellipticalfootprintofthesurfacestrikingtheanvilwerecomparedtotheprolesofthedeformedcylinders.ForthesesimulationstheJohnson-Cook[ JohnsonandCook ( 1983 )]hardening 162

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4.1 ,aprimarygoalofthisresearchistoadvancethecurrentstate-of-the-artbydevelopinguser-friendly,micro-structurallybasedandnumericallyrobustmacroscopicconstitutivemodelsthatcancapturewithaccuracytheparticularitiesoftheplasticresponseofhexagonalmetals,inparticularhighpuiritytitanium.Ithasbeendemonstratedthatthisgoalhasbeenmettoalargedegree.Furtherresearchisneededtoexploreotherloadingenvironmentsbutthisworkhasshownthattheproposedmodelcanbeparameratrizedbysimpleuniaxialtestdataandusedtosimulatemorecomplexloading.Theabilitytoincorporatedatafromotherloadingconditionsisalreadyinplace.Although,thisresearchwasconcernedprimarialywithhighpuritytitanium,itisfeltthattheproposedmodelandimplementationapproachisquitevalidforotherHCPmetals.. 163

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Bacon,C.,Lataillade,J.L.,2001.DevelopmentofKolsky-Hopkinsontechnicsandapplicationsfornon-conventionaltesting.Vol.3ofTrendsinMechanicsofMaterials.INBZTUREK,Warsaw,Poland. Barlat,F.,Lege,D.J.,Brem,J.C.,1991.Asix-componentyieldfunctionforanisotropicmaterials.InternationalJournalofPlasticity7,693{712. Barrett,C.S.,Massalski,T.B.,1980.StructureofMetals:CrystallographicMethods,Principles,andData,3rdEdition.Vol.35ofInternationalSeriesonMaterialsScienceandTechnology.PergamonPress,Oxford. Bassani,J.L.,1977.Yieldcharacterizationofmetalswithtransverselyisotropicplasticproperties.InternationalJournalofMechanicalSciences19,651{660. Bishop,J.,Hill,R.,1951.Atheoreticaldeviationoftheplasticpropertiesofapolycrystallineface-centeredmetal.PhilosophicalMagazine7(42),414{427. Budianski,B.,1984.AnisotropicPlasticityofPlane-lsotropicSheets.MechanicsofMaterialBehavior.Elsevier,Amsterdam. Cazacu,O.,Barlat,F.,2003.Applicationofthetheoryofrepresentationtodescribeyieldingofanisotropicaluminumalloys.InternationalJournalofEngineeringScience41,1367{1385. Cazacu,O.,Barlat,F.,2004.Acriterionfordescriptionofanisotropyandyielddierentialeectsinpressure-insensitivemetals.InternationalJournalofPlasticity20,2027{2045. Cazacu,O.,Barlat,F.,Nixon,M.E.,2004.Newanisotropicconstitutivemodelsforhcpsheetformingsimulations.In:The8thInternationalConferenceonNumericalMethodsinIndustrialFormingProcesses.TheOhioStateUniversity(OSU),Columbus,Ohio,U.S.A. Donachie,M.J.,2000.TitaniumAtechnicalGuideSecondEdition.TheMaterialsInformationSociety,MaterialsPark,Ohio. Gotoh,M.,1977.Theoryofplasticanisotropybasedonayieldfunctionoffourthorder(planestressstate).InternationalJournalofMechanicalSciences19(9),505. Gray,G.T.,1997.Inuenceofstrainrateandtemperatureonthestructure/propertybehaviorofhigh-puritytitanium.JournalDePhysique.IV:JP7(3),423{428. Hershey,A.V.,1954.Theplasticityofanisotropicaggregateofanisotropicfacecenteredcubiccrystals.JournalofAppliedMechanics21,241{249. Hill,R.,1948.Atheoryoftheyieldingandplasticowofanisotropicmetals.ProceedingsoftheRoyalSocietyofLondon.SeriesA,MathematicalandPhysicalSciences193,281{297. 164

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Hill,R.,1979.Theoreticalplasticityoftexturedaggregates.In:MathematicalProceedingsoftheCambridgePhilosophicalSociety,CambridgeUniversityPress Hopkinson,B.,1914.Amethodofmeasuringthepressureinthedeformationofhighexplosivesbyimpactbullets.PhilosophicalTransactionsoftheRoyalSocietyofLondonA213,437{452. Hosford,W.,1972.Ageneralizedisotropicyieldcriterion.JournalofAppliedMechanics39,607. Hosford,W.F.,1966.Texturestrengthening.MetalsEngineeringQuarterly6(4). Hosford,W.F.,1993.TheMechanicsofCrystalsandTexturedPolycrystals.OxfordEngineeringScienceSeries.OxfordUniversityPress,NewYorkOxford. Hosford,W.F.,Allen,T.J.,1973.Twinninganddirectionalslipasacauseforastrengthdierentialeect.MetallurgicalTransactions4. Hung,P.-C.,Voloshin,A.S.,2003.In-planestrainmeasurementbydigitalimagecorrelation.JournaloftheBrazilianSocietyofMechanicalSciencesandEngineeringXXV(3),215{221. Johnson,G.,Beissel,S.,Stryk,R.,Gerlach,C.,Holmquist,T.,2003.Userinstructionsforthe2003versionoftheepiccode.Tech.rep.,NetworkComputingServicesInc. Johnson,G.,Cook,W.,1983.Aconstitutivemodelanddataformetalssubjectedtolargestrains,highstrainrates,andhightemperatures.In:SeventhInternationalSymposiumonBallistics.TheHague,TheNetherlands. Johnson,G.,Stryk,R.,Holmquist,T.,Beissel,S.,1997.Numericalalgorithmsinalagrangianhydrocode.Tech.Rep.WL-TR-1997-7039. Kalidindi,S.R.,Salem,A.A.,Doherty,R.D.,2003.Roleofdeformationtwinningonstrainhardeningincubicandhexagonalpolycrystallinemetals.AdvancedEngineeringMaterials4,229{232. Kaschner,G.C.,Gray,G.T.,2000.Theinuenceofcrystallographictextureandinterstitialimpuritiesonthemechanicalbehaviorofzirconium.MetallurgicalandMaterialsTransactions31A. Kelley,E.W.,W.F.Hosford,J.,1968.Thedeformationcharacteristicsoftexturedmagnesium.TransactionsoftheMetallurgicalSocietyofAIME242,654{661. Kolsky,H.,1949.Aninvestigationofthemechanicalpropertiesofmaterialsatveryhighratesofstrain.ProceedingsoftheRoyalPhysicalSocietyB62,676{700. 165

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MichaelEugeneNixonwasbornonJune5,1953inLafayetteIndiana,thethirdchildofRufusandIreneNixon.ThefamilymovedtonorthwestFloridawhileMichaelwasayoungchild.HegraduatedhighschoolinCrestviewFlorida.Michaelspent6yearsenlistedintheUnitedStatesAirForcebeforeearningadegreeinMechanicalEngineeringfromAuburnUniversityin1982.In1983hebeganworkattheAirForceArmamentTestLaboratory,nowtheAirForceResearchLabortory,atEglinAFB,FL.In1992heobtainedhisMaster'sDegreeinEngineeringMechanicsfromtheUniversityofFloridaandcompletedhisPh.D.workin2008attheUniversityofFloridaResearchandEngineeringEducationFacilityinShalimarFlorida.MichaeliscurrentlymarriedtoTammyNixonandresidesinCrestview,Florida. 168