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

Material Information

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

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Aerospace Engineering
Mechanical and Aerospace Engineering
Committee Chair:
Cazacu, Oana
Committee Members:
Sankar, Bhavani V.
Kumar, Ashok V.
Boginski, Vladimir L.
Graduation Date:
5/1/2008

Subjects

Subjects / Keywords:
Anisotropy ( jstor )
Axial strain ( jstor )
Cylinders ( jstor )
Deformation ( jstor )
Experimental data ( jstor )
Modeling ( jstor )
Simulations ( jstor )
Specimens ( jstor )
Strain rate ( jstor )
Titanium ( jstor )
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
anisotropic, material, strength, titanium, twinning
Genre:
Electronic Thesis or Dissertation
bibliography ( marcgt )
theses ( marcgt )
Aerospace Engineering thesis, Ph.D.

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. ( en )
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.
Statement of Responsibility:
by Michael E. Nixon

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Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright by Michael E. Nixon. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
7/11/2008
Classification:
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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.









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










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.










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











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









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









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









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

























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"













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.












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)









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.








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










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









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.




























(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









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














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









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










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















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










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.















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









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









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.











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--









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-









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.









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











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









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









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








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










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













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










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.









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.










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













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




















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.











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
















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









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

































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









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









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)









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










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









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.









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.









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.


























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 (%)








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



















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










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









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











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.


















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











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)


































2008 Michael E. Nixon









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

























(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.










(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









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.










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









Lemaitre, J., 2001. Deformation of Materials. Vol. 1 of Handbook of Materials Behavior
Models. Academic Press, San Diego, San Francisco, New York, Boston, London, Sydney,
Tokyo.

Li, Q., Xu, Y., Bassim, M., 2004. Dynamic mechanical behavior of pure titanium. Journal
of Materials Processing Technology 155-156, 1889-1892.

Logan, R. W., Hosford, W. F., 1980. Upper-bound anisotropic yield locus calculations
assuming <111>-pencil glide. International Journal of Mechanical Sciences 22 (7),
419-430.

Lou, X., Li, M., Boger, R., Agnew, S., Wagoner, R., 2006. Hardening evolution of az31b
mg sheet. International Journal of Plasticity.

Ludwick, P., 1903. Technische Blatter, 133-159.

Lutjering, G., Williams, J. C., 2003. Titanium. Springer, Berlin.

t. iv rs, M. A., Vohringer, O., Lubarda, V. A., 2001. The onset of twinning in metals: A
constitutive description. Acta Materialia 49, 4025-4039.

Miguil-Touchal, S., Morestin, F., Brunet, M., 1997. Various experimental applications of
digital image correlation method. In: Brebbia, C. A., Anagnostopoulos, P., Omagno,
G. C. (Eds.), Computational Methods in Experimental Measurements VIII. Vol.
Modeling and Simulation volume 17. Transactions of the Wessex Institute.

Nemat-Nasser, S., Guo, W. G., C'!, i.- J. Y., 1999. Mechanical properties and deformation
mechanisms of a commercially pure titanium. Acta Materialia 47, 3705-3720.

Plunkett, B. W., 2005. Plastic anisotropy of hexagonal closed packed metals. Ph.D. thesis,
University Of Florida.

Plunkett, B. W., Cazacu, O., Lebensohn, R. A., Barlat, F., 2006. Elastic-viscoplastic
anisotropic modeling of textured metals and validation using the taylor cylinder impact
test. International Journal of Plasticity.

Parametric Technology Corporation, 2007. Mathcad Version 14.

Ramberg, W., Osgood, W. R., 1943. Description of stress-strain curves by three
parameters. National Advisory Committee for Aeronautics (No. 902).

Salem, A., Kalidindi, S., Doherty, R., Glavicic, M., Semiatin, S., 2004a. Effect of texture
and deformation temperature on the strain hardening response of p li-vi il-1 i11!ii
a-titanium. Ti-2003 Science and Technology, 1429-1436.

Salem, A., Kalidindi, S., Doherty, R., Semiatin, S., 2004b. Strain hardening due to
deformation twinning in a-titanium: Part i mechanisms. Acta Materialia









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.









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














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










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










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









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





















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









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.










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



























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.











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.































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.










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











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
























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









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.

















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









Salem, A. A., Kalidindi, S. R., Doherty, R. D., 2002. Strain hardening regimes and
microstructure evolution during large strain compression of high purity titanium.
Scripta Materialia 46, 419423.

Salem, A. A., Kalidindi, S. R., Doherty, R. D., 2003. Strain hardening of titanium: role of
deformation twinning. Acta Materialia 51, 4225-4237.

Salem, A. A., Kalidindi, S. R., Doherty, R. D., Semiatin, S. L., 2006. Strain hardening
due to deformation twinning in a-titanium: Mechanisms. Metallurgical And Materials
Transactions A 37A, 259-268.

Sergueeva, A. V., Stolyarov, V., Valiev, R., Mukherjee, A., 2001. Advanced mechanical
properties of pure titanium with ultrafine grained structure. Scripta Materialia 45,
747-752.

Wi\-i. W. M., Sluys, L. J., De Borst, R., 1997. Viscoplasticity for instabilities due to
strain softening and strain-rate softening. International Journal for Numerical Methods
in Engineering 40, 3839-J;. 1.

Zarkades, A., Larson, F. R., 1970. The Science, Technology and Application of Titanium.
Pergamon Press, Oxford, UK.

Zyczkowski, M., 1981. Combined Loadings in the Theory of Plasticity. Polish Scientific
Publishers, Warsaw, Poland.









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

























,-.-


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









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)























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.









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..









Hill, R., 1950. The Mathematical Theory of Plasticity. Oxford University Press, London.

Hill, R., 1979. Theoretical plasticity of textured .,-.-regates. In: Mathematical Proceedings
of the Cambridge Philosophical Society, Cambridge University Press

Hopkinson, B., 1914. A method of measuring the pressure in the deformation of high
explosives by impact bullets. Philosophical Transactions of the Royal Society of London
A213, 437-452.

Hosford, W., 1972. A generalized isotropic yield criterion. Journal of Applied Mechanics
39, 607.

Hosford, W. F., 1966. Texture strengthening. Metals Engineering Quarterly 6 (4).

Hosford, W. F., 1993. The Mechanics of Crystals and Textured P.. i, -I ,-- .1- Oxford
Engineering Science Series. Oxford University Press, New York Oxford.

Hosford, W. F., Allen, T. J., 1973. Twinning and directional slip as a cause for a strength
differential effect. Metallurgical Transactions 4.

Hung, P.-C., Voloshin, A. S., 2003. In-plane strain measurement by digital image
correlation. Journal of the Brazilian Society of Mechanical Sciences and Engineering
XXV (3), 215-221.

Johnson, G., Beissel, S., Stryk, R., Gerlach, C., Holmquist, T., 2003. User instructions for
the 2003 version of the epic code. Tech. rep., Network Computing Services Inc.

Johnson, G., Cook, W., 1983. A constitutive model and data for metals subjected to large
strains, high strain rates, and high temperatures. In: Seventh International Symposium
on Ballistics. The Hague, The Netherlands.

Johnson, G., Stryk, R., Holmquist, T., Beissel, S., 1997. Numerical algorithms in a
lagrangian hydrocode. Tech. Rep. WL-TR-1997-7039.

Kalidindi, S. R., Salem, A. A., Doherty, R. D., 2003. Role of deformation twinning on
strain hardening in cubic and hexagonal p ..1, i--I 11iiw metals. Advanced Engineering
Materials 4, 229-232.

Kaschner, G. C., Gray, G. T., 2000. The influence of crystallographic texture and
interstitial impurities on the mechanical behavior of zirconium. Metallurgical and
Materials Transactions 31A.

Kelley, E. W., W. F. Hosford, J., 1968. The deformation characteristics of textured
magnesium. Transactions of the Metallurgical Society of AIME 242, 654-661.

Kolsky, H., 1949. An investigation of the mechanical properties of materials at very high
rates of strain. Proceedings of the Royal Physical Society B62, 676-700.









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











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.










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.









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









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.










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


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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

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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










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-









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




















_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




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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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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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Li,Q.,Xu,Y.,Bassim,M.,2004.Dynamicmechanicalbehaviorofpuretitanium.JournalofMaterialsProcessingTechnology155-156,1889{1892. Logan,R.W.,Hosford,W.F.,1980.Upper-boundanisotropicyieldlocuscalculationsassuming<111>-pencilglide.InternationalJournalofMechanicalSciences22(7),419{430. Lou,X.,Li,M.,Boger,R.,Agnew,S.,Wagoner,R.,2006.Hardeningevolutionofaz31bmgsheet.InternationalJournalofPlasticity. Ludwick,P.,1903.TechnischeBlatter,133{159. Lutjering,G.,Williams,J.C.,2003.Titanium.Springer,Berlin. Meyers,M.A.,Vohringer,O.,Lubarda,V.A.,2001.Theonsetoftwinninginmetals:Aconstitutivedescription.ActaMaterialia49,4025{4039. Miguil-Touchal,S.,Morestin,F.,Brunet,M.,1997.Variousexperimentalapplicationsofdigitalimagecorrelationmethod.In:Brebbia,C.A.,Anagnostopoulos,P.,Omagno,G.C.(Eds.),ComputationalMethodsinExperimentalMeasurementsVIII.Vol.ModelingandSimulationvolume17.TransactionsoftheWessexInstitute. Nemat-Nasser,S.,Guo,W.G.,Cheng,J.Y.,1999.Mechanicalpropertiesanddeformationmechanismsofacommerciallypuretitanium.ActaMaterialia47,3705{3720. Plunkett,B.W.,2005.Plasticanisotropyofhexagonalclosedpackedmetals.Ph.D.thesis,UniversityOfFlorida. Plunkett,B.W.,Cazacu,O.,Lebensohn,R.A.,Barlat,F.,2006.Elastic-viscoplasticanisotropicmodelingoftexturedmetalsandvalidationusingthetaylorcylinderimpacttest.InternationalJournalofPlasticity. ParametricTechnologyCorporation,2007.MathcadVersion14. Ramberg,W.,Osgood,W.R.,1943.Descriptionofstress-straincurvesbythreeparameters.NationalAdvisoryCommitteeforAeronautics(No.902). Salem,A.,Kalidindi,S.,Doherty,R.,Glavicic,M.,Semiatin,S.,2004a.Eectoftextureanddeformationtemperatureonthestrainhardeningresponseofpolycrystallinea-titanium.Ti-2003ScienceandTechnology,1429{1436. Salem,A.,Kalidindi,S.,Doherty,R.,Semiatin,S.,2004b.Strainhardeningduetodeformationtwinningina-titanium:Parti-mechanisms.ActaMaterialia 166

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Salem,A.A.,Kalidindi,S.R.,Doherty,R.D.,2003.Strainhardeningoftitanium:roleofdeformationtwinning.ActaMaterialia51,4225{4237. Salem,A.A.,Kalidindi,S.R.,Doherty,R.D.,Semiatin,S.L.,2006.Strainhardeningduetodeformationtwinningin-titanium:Mechanisms.MetallurgicalAndMaterialsTransactionsA37A,259{268. Sergueeva,A.V.,Stolyarov,V.,Valiev,R.,Mukherjee,A.,2001.Advancedmechanicalpropertiesofpuretitaniumwithultranegrainedstructure.ScriptaMaterialia45,747{752. Wang,W.M.,Sluys,L.J.,DeBorst,R.,1997.Viscoplasticityforinstabilitiesduetostrainsofteningandstrain-ratesoftening.InternationalJournalforNumericalMethodsinEngineering40,3839-3864. Zarkades,A.,Larson,F.R.,1970.TheScience,TechnologyandApplicationofTitanium.PergamonPress,Oxford,UK. Zyczkowski,M.,1981.CombinedLoadingsintheTheoryofPlasticity.PolishScienticPublishers,Warsaw,Poland. 167

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MichaelEugeneNixonwasbornonJune5,1953inLafayetteIndiana,thethirdchildofRufusandIreneNixon.ThefamilymovedtonorthwestFloridawhileMichaelwasayoungchild.HegraduatedhighschoolinCrestviewFlorida.Michaelspent6yearsenlistedintheUnitedStatesAirForcebeforeearningadegreeinMechanicalEngineeringfromAuburnUniversityin1982.In1983hebeganworkattheAirForceArmamentTestLaboratory,nowtheAirForceResearchLabortory,atEglinAFB,FL.In1992heobtainedhisMaster'sDegreeinEngineeringMechanicsfromtheUniversityofFloridaandcompletedhisPh.D.workin2008attheUniversityofFloridaResearchandEngineeringEducationFacilityinShalimarFlorida.MichaeliscurrentlymarriedtoTammyNixonandresidesinCrestview,Florida. 168