2005_Efffect of Thermo Mechanical Treatment on Texture Evaluation of Polycrystalline

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THE FLORIDA STATE UNIVERSITY

COLLEGE OF ENGINEERING

EFFECT OF THERMO-MECHANICAL TREATMENT ON

TEXTURE EVOLUTION OF POLYCRYSTALLINE ALPHA

TITANIUM

By

GILBERTO ALEXANDRE CASTELLO BRANCO

A Dissertation submitted to the

Department of Mechanical Engineering

In partial fulfillment of therequirements for the degree of

Doctor of Philosophy

Degree Awarded

Summer Semester, 2005

Copyright 2005

Gilberto Alexandre Castello-Branco

All Rights Reserved


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The members of the Committee approve the dissertation of GILBERTO ALEXANDRE

CASTELLO BRANCO defended on May 16, 2005.

Hamid Garmestani

Professor Directing Dissertation

Chuk Zhang

Outside Committee Member

Justin Schwartz

Committee Member

Chiang ShihCommittee Member

Approved:

Chiang Shih, Chairman,Department of Mechanical Engineering

Ching-Jen Chen, Dean,

College of Engineering

The Office of Graduate Studies has verified and approved the above named committee members.


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Dedicated to my parents Gilberto and La, my sister Leila, my wife Cristiane

and my dear relatives Beatriz and Jorge Alberto.

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ACKNOWLEDGEMENTS

To God for giving me the strength to overcome all the obstacles that I have found in my

way.

I am very grateful and indebted to my advisor, Dr. Hamid Garmestani for his endless

support, encouragement and optimism during the course of this study. I would like to thank the

members of my committee.

I would like to thank my professors, Dr. Luiz Brando, Dr Said Ahzi and Dr Anthony

Rollett for their support in several occasions during my research program. I also would like to

tank Dr. Ayman Salem, Dr. Mike Glavicic for their help and suggestions, Dr. Scott Schoenfeld

and Dr. Lee Semiatin for providing funds and the material used in this research. This study was

partially funded under the AFOSR grant # F49620-03-1-0011 and Army Research Lab contract #

DAAD17-02-P-0398, DAAD17-02-P-0928.

I am grateful to the National High Magnetic Field Laboratory (NHMFL) and

MARTECH, Tallahassee, Florida for the facilities, the Department of Material Science and

Engineering of the Georgia Institute of Technology for allowing me to use the rolling facility,

and also to several members of the NHMFL, who in one way or another contributed to the

success of my work. Especial thanks go to: Mr. Robert Goddard for his guidance and assistance

in running the ESEM/OIM facility, the FSU staff personnel, especially Mr. George Green, my

friends in Tallahassee, especially Mr. Donald Hollett and family for their kindness, friendship

and support and my research colleagues at FSU and Georgia Tech.I would like to thank my friends in Brazil, who were always giving me support even

though the distance. A special thanks goes to my dear friend Bernardino.

Many thanks are due to my all colleagues at CEFET-RJ, for their support and

encouragement.

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I would like to express my profound gratitude to my parents, my sister, to Geracinda and

all my family, who have always given me their love, encouragement and endless support

throughout these years.

Finally I wish to express my heartful appreciation to my beloved wife, Cristiane, who has

always been walking by my side, sharing the good and bad moments, tirelessly helping and

encouraging me.

Gilberto Alexandre Castello Branco

Florida State University, TallahasseeMay, 2005

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TABLE OF CONTENTS

LIST OF TABLES........ ix

LIST OF FIGURES.. x

ABSTRACT... xiii

1. INTRODUCTION 1

2. BACKGROUND.. 4

2.1- Titanium and its Alloys. 4

2.1.1 - Physical metallurgy of Titanium and Titanium Alloys... 6

2.1.2 - Classification of Titanium Alloys . 7

2.1.2.1 - Alpha-Titanium Alloy .... 7

2.1.2.2 - Near-Alpha Titanium Alloys ...... 8

2.1.2.3 Alpha/Beta ( + ) Alloys.. 8

2.1.2.4 - Beta, Near-Beta and Metastable-Beta alloys.. 9

2.2 - Mechanical Behavior of Titanium and its Alloys... 11

2.2.1 Slip Modes in HCP Metals .... 11

2.3- Texture ... 17

2.3.1- Cold Rolling Texture 27

2.3.2- Hot and Warm Rolling Texture.. 28

2.4 X-ray Peak Profile Analysis 28

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2.4.1 - X-ray Peak Profile Analysis from MWP and Methodology for

Determining the Burgers Vector Populations 34

2.5 Self-Consistent Modeling of Deformation Texture... 38

2.5.1 The Single-Crystal Constitutive Law

40

2.5.2 Polycrystal Constitutive Law. 43

2.5.3 The Self-Consistent Approach... 47

3. EXPERIMENTAL PROCEDURE 50

3.1 - Material .... 50

3.2 - Thermo-Mechanical Processing...... 51

3.2.1- Cold Rolling... 54

3.2.2 - Hot Rolling........... 54

3.3 - Metallographic Sample Preparation... 55

3.3.1 - Mechanical Polishing... 56

3.4 - Characterization Techniques... 56

3.4.1- Texture Measurement.. 57

3.4.2 - Peak Profile Measurements 58

4. RESULTS......... 60

4.1 - Texture Evolution. 60

4.1.1 - As Received Sample .................... 60

4.1.2 Cold Rolled Sample......................................................................................... 61

4.1.3 - Warm Rolled Samples..................................................................................... 67

4.2 - X-ray Peak Profile Analysis. 72

4.3- Texture Simulation 78

5. DISCUSSION... 83

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5.1 - Deformation Texture.... 83

5.2 - X-Ray Peak Profile Analysis........... 86

5.3 - Self Consistent Simulation of the Deformation Texture... 87

6 - SUMMARY AND FUTURE WORK. 90

6.1 Summary... 90

6.2 Future Work. 91

REFFERENCES 93

BIBLIOGRAFICAL SKETCH.... 100

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LIST OF TABLES

Table 2.1 Some properties of titanium and its alloys..... 5

Table 2.2 Summary of commercial and semi-commercial grades and alloys of titanium... 10

Table 2.3- Number of grains showing a specific glide system for different samples. 14

Table 2.4 - The most important deformation systems in hcp metals and their influence on

the texture evolution .. 15

Table 2.5 The most typical correlations between diffraction peak aberrations and the

different elements of microstructure .. 30

Table 2.6 - The most common slip systems in hexagonal crystals: (a) Edge dislocations

and (b) Screw dislocations .. 33

Table 3.1 - Chemical composition (weight %) .. 50

Table 3.2 - Typical mechanical properties of the CP Ti Gr2.. 50

Table 3.3 Physical properties of the CP Ti Gr2 . 51

Table 3.4 - Nomenclature of the samples.... 54

Table 3.5 - Metallographic preparation procedure . 56

Table 4.1 - Dislocation densities and arrangement parameter, M, obtained from MWPevaluation for Ti samples deformed at different reduction levels ..

73

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LIST OF FIGURES

Figure 2 1 - Commercial production of Titanium ................................... 6

Figure 2.2 The hexagonal unit cell (a) and the first order slip and twinning planes for hcpmetals (b).................................................................

12

Figure 2.3 Glide systems in alpha titanium ..... 13

Figure 2.4 - Schematics of all investigations carried out and definition of sample shortnames. The starting texture of the different materials is given in the form of (0001) and{/1010/} X-ray pole figures. Sample short names are composed as follows: (1) chemical

composition; (2) sheet thickness in mm; (3) deformation mode; (4) angle between RD and

tension direction (0, 45, 90) or deformation degree (2%,4%) 16

Figure 2.5 Sheet textures in hcp materials as a function of c/a ratios (schematically). 18

Figure 2.6 Ideal cold rolling texture component for flat-cold rolled titanium: {2115}..

19

Figure 2.7 Typical textures 20

Figure 2.8 - Positioning and movement of the sample on the texture goniometer inside the

X-ray machine (a). The relation between crystallite coordinates (Xc, Yc, Zc) and samplecoordinates (Xs, Ys, Zs), (b), (c) and (d). 21

Figure 2.9 As received material: a) Pole figures and b) Inverse pole figures... 22

Figure 2.10 Pole figure representation of the cold rolling and the recrystalization texture

components... 23

Figure 2.11 - Three consecutives Euler rotations defining an orientation .. 24

Figure 2.12 Relationship between sample and crystal axis directions.. 25

Figure 2.13 Constant sections through the Eulerian space: a) 0, b) 20, c) 30, d) 40and e) 60..

26

Figure 2.14 Location of the cold rolling and recrystalization components on the constant

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phi sections of the Euler space using Roes definition. 27

Figure 2.15 The parabolas describing the average contrast factors for the eleven slip

systems, in the case of Titanium, as a function of x = (2/3)(l/ga)2

.. 35

Figure 2.16 - Slip systems in hexagonal crystal systems .... 36

Figure 3.1 As received material: OIM/SEM micrograph. 52

Figure 3.2 Rolling mill machine... 53

Figure 3.3 Schematic setup of the thermo-mechanical processing... 53

Figure 3.4 - X-ray machine Philips XPert MRD equipped with texture goniometer. 57

Figure 3.5 - Surfaces examined by X-ray diffraction: normal direction (ND); rolling

direction (RD); transverse direction (TD) 58

Figure 3.6 - Example of the instrumental broadening of the Alpha-1 PanalyticalDiffractometer measured using LaB6 660a NIST standard compared with the peak

broadening measured for deformed -Ti. The dashed line is the 220 reflection of LaB6 and

the continuous line is the 11.0 reflection of-Ti deformed at the 60% reduction rate... 59

Figure 4.1 - (0002) and (1010) pole figure for the as received sample... 61

Figure 4.2 - ODF sections of=0 and =30, Roe notation, for the as received sample. 61

Figure 4.3 (0002) and (1010) pole figures of the cold rolled samples.. 63

Figure 4.4- ODF sections of=0 and = 30, Roe notation, for the samples cold rolled at:a) 20%, b) 40%, c) 60%, d) 80% and e) 95%...................................................................

64

Figure 4.5 - Skeleton lines of the orientation distribution functions for the samples 20%,

40%, 60%, 80% and 95% cold rolled.. 65

Figure 4.6 - Development of the {0002}//ND fiber texture for the as received (AR) and

20%, 40%, 60%, 80% and 95% cold rolled (CR) samples.. 66

Figure 4.7 - Variation in volume fraction of the {0002}//ND fiber texture with degree ofcold rolling reduction. The as received material corresponds to the 0% cold rolling

reduction.................................................................................................................................. 67

Figure 4.8 (0002) and (1010) pole figures of the warm rolled samples... 68

Figure 4.9 - ODF sections of=0 and = 30, Roe notation, for the samples warm rolledat: a) 20%, b) 40%, c) 60%, d) 80% and e) 95%. 69

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Figure 4.10 - Skeleton lines of the orientation distribution functions for the samples 20%,

40%, 60%, 80% and 95% warm rolled ... 70

Figure 4.11 - Development of the {0002}//ND fiber texture for the as received (AR) and

20%, 40%, 60%, 80% and 95% warm rolled (WR) samples... 71

Figure 4.12 - Variation in volume fraction of the (0002)//ND fiber texture with degree of

warm rolling reduction. The as received material corresponds to the 0% cold rolling

reduction.. 72

Figure 4.13 - The hi fractions of the three fundamental Burgers vector types, , and

, as a function of rolling reduction. Note that in the figure the solutions to equations.(2.8, 2.9 and 2.10), the hi fractions, were transformed in percentages... 74

Figure 4.14 - The line profiles of (0002) Bragg reflections for different deformations

levels. On the x-axes K is given by K=2sin/, where is the Bragg angle and is the

wave length of the used radiation 75

Figure 4.15 - The line profiles of (1120) Bragg reflections for different deformations

levels. On the x-axes K is given by K=2sin/, where is the Bragg angle and is thewavelength of the used radiation...... 76

Figure 4.16 - (2110), (0001) and (2113) pole figures of alpha titanium at a rolling reductionof (a) 0%, (b) 40% (c) 60%, (d) 80%, respectively.. 77

Figure 4.17 - Evolution of intensities of components with RD//2110, RD//0001 andRD//2113, respectively, during rolling reduction. 78

Figure 4.18 (0002) pole figures for the as-received material: (a) experimental and (b)discrete grains file. Axes convention: RD in the vertical direction and TD in the horizontal

direction... 79

Figure 4.19 Experimental and simulated results of the (0002) pole figures for the cold

rolled samples deformed (a) 20%, (b) 40%, (c) 60%, (d) 80% and (e) 95%........................... 81

Figure 4.20 Experimental and simulated results of the (0002) pole figures for the warm

rolled samples deformed (a) 20%, (b) 40%, (c) 60%, (d) 80% and (e) 95%........................... 82

Figure 5. 1 - Variation of: (a) twin volume fraction; (b) strain accommodated by twinningas a function of rolling temperature. 88

Figure 5.2 Optical micrographs of warm rolled 80% and 95% reduction 89

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ABSTRACT

The present work attempts to establish a unified path model for characterization as well

as prediction of microstructure evolution, in terms of texture, in commercially pure titanium that

have undergone thermo-mechanical processing. Two deformation temperatures, room

temperature (cold rolling) and 260C (warm rolling), and five different degrees of deformation,

20%, 40%, 60%, 80% and 95% were used in this investigation. X-ray measurements (texturemeasurements and peak profile analysis) have been used to characterize the texture and to

evaluate the relative activity of the various slips systems activated during the process.

Simulations of the resultant textures after each mode of deformation were performed using a

crystal plasticity self-consistent scheme, and comparisons, in the form of pole figures, between

the experimental results and the predicted deformation textures were performed in order to

validate the results obtained from peak profile analysis.

The experimental texture results show that except for the samples 95% deformed, the

warm rolling has shown to develop a deformed texture different from the cold rolling.

The results of peak profile analysis carried out for the 40%, 60% and 80% warm rolled

samples show that the type of dislocation was prevalent in all samples while the type of

dislocation was only marginal. The X-ray peak profile analysis, based on the dislocation model

of anisotropic peak broadening, show the dislocation densities, distributions and type during the

rolling process in good agreement with the texture evolution.

Even though twining was not taken into account during simulation of the cold rolled

samples, there was a reasonable agreement between the experimental and the predicted pole

figures with a small divergence on the distribution of in the TD-RD plane for the higher

deformed samples.

The results of simulated deformation texture of warm rolled CP-Ti are in good agreement

with the experimental results and with the peak profile analysis findings.

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CHAPTER 1

INTRODUTION

Despite being discovered as early as 1790, it was not until late 1940s that interest in

titanium and its alloys, as structural materials, began to accelerate, as their potential as high-

temperature, high-strength/density ratio and corrosion resistant materials with aeronautical

applications became apparent [Boyer et al., 1994; Froes, 1990] and in a relatively short time,

titanium has come to be used for many different and important purposes. Its greatest

disadvantage is the high cost compared to competing materials which frequently offsets

titaniums engineering advantages and restrings the market for titanium applications. Aiming to

change this perspective, just as other metals, such as aluminum, have had cost breakthroughs that

have dramatically expanded their use, a great deal of money and time has been put in basic

research to lower production costs improving both extraction and processing technologies.

Titanium and other metals with hexagonal crystal structure develop sharp deformation

textures that lead to a pronounced plastic anisotropy of the polycrystalline sample [Phillipe,

1995; Zaefferer, 2003]. Various factors can cause anisotropy in metals, among them are: grain

morphology [Kocks and Chandra, 1982], second phase precipitates [Mizera et al., 1996; Crosby

et al., 2000] and substitutional alloying elements [Phillipe, 1988]. As a consequence, the

deformation texture may vary with slight changes of the material composition [Zaefferer, 2003].

Researchers [Crosby et al., 2000; Fjeldly and Roven, 1996] agree that crystallographic textures

resulting from thermomecanical processing such as hot or cold rolling are most directly

responsible for anisotropy in metal alloys. Anisotropy of mechanical properties is a concern in

the forming of metals into shapes and parts; and the control of texture throughout the process can

provide beneficial use of the variety of available textures in , near- and other titanium alloys

[Zhu, 1997].

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In this scenario it becomes evident that an understanding on how the thermo-mechanical

processing affects the final properties of a semi-finished or finished material is of major

significance. Moreover, considering that the cost associated with the testing and development of

a product is somehow enormous and time consuming, availability of experimental

characterization techniques and computational tools capable of providing reliable data leading to

the prediction of optimal processing paths linking the commercially available raw material

to its semi-finished or finished forms, is of strategic importance.

In order to model deformation processes, it is fundamental the knowledge on the

evolution of parameters such as dislocation density and the relative activity of the various slips

systems activated during the process. The measurement of such parameters is normally executed

employing established techniques as transmission electron microscopy (TEM), electron back

scattering (EBSD) and trace analysis. However, the measurement of these parameters in

specimens that have undergone large strains and the consequent large number of dislocations

introduced (which are the actual characteristics of materials in industrial practices) is difficult

and time demanding with a considerable cost associated.

On the other hand, the characterization of material defects using X-ray and neutron-

diffraction techniques has received considerable attention during the past few decades and line

broadening analysis can be an attractive alternative, in substitution or associated with TEM, in

the study and evaluation of the substructure developed during thermo-mechanical processes. The

most attractive feature of these techniques is that they can be used to measure materials heavily

deformed. Other advantages are the easy sample preparation routine, relative short time required

by the state of the art equipments to render results and the fact that the data obtained is

statistically averaged over the area/volume irradiated.

The goal defined for this work is to establish a unified path model for characterization

and also prediction of microstructure evolution, in terms of texture, in materials that have

undergone thermo-mechanical processing. Once the behavior of a material is mapped by means

of a reliable characterization scheme, other possible routs of processing can be simulated

before any actual processing/testing is conducted in order to verify the accuracy of a chosen path

for processing.

In order to achieve this objective, it was decided to apply the following methodology to a

polycrystalline HCP material, which have undergone unidirectional rolling at different degrees of

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deformation (20% ~ 95% reduction in thickness or equivalent true strain of -0.22 ~ -3.0) and at

different isothermal conditions (25C and 260C). The choice of temperatures aimed to isolate

different mechanisms of deformation. The first step consisted in the characterization of

microstructure evolution (texture) by means of X-ray pole figure measurements and ODF

analysis. The second task was to employ X-ray peak profile analysis to study the microstructure

evolution evaluating the densities, distribution and type of dislocations. The third step consisted

in validating the results obtained from Peak Profile analysis by simulation of deformation texture

evolution using a crystal plasticity self-consistent scheme, and comparison of experimental

results with the predicted deformation texture in the form of pole figures.

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CHAPTER 2

BACKGROUND

2.1 Titanium and its Alloys

Titanium is the fourth most abundant structural metal and the ninth most abundant

element, making up about 0.6% of the Earth's crust. It occurs in many mineral forms, but only

three present significant economical interest: leucoxene, rutile and ilmenite. Titanium was first

discovered (as rutile) by W. Gregor in 1791 and by M. H. Klaproth in 1795 [Boyer, Welsch and

Collings, 1994] who named the metal after the Greek mythological god Titan [Bomberger, Froes

and Morton, 1985; Ogdom and Gonser, 1956; McQuillan and McQuillan, 1956] and the first cost

effective non-vacuum process for titanium extraction from its ore was developed by W.J. Kroll

[Hunter, 1910]. Interest in titanium and its alloys, as structural materials, began to accelerate in

the late 1940s and early 1950s, as their potential as high-temperature, high-strength/density ratio

and corrosion resistant materials with aeronautical applications became apparent [Froes, 1990;Boyer, Welsch and Collings, 1994]. Due to its unique set of properties (see table 2.1), nowadays,

titanium and its alloys have been widely used throughout the aerospace industry for most types

of structural components, including airframes and engine components, as well as in many non-

aerospace applications. Just to mention a few, as a metal, cars, sports equipment such as racing

yacht parts, golf clubs, tennis racquets and bike frames, wrist watches, underwater craft, and

general industrial equipment. Its non-toxicity also makes it useful for surgical implants such as

pacemakers, artificial joints and bone pins. Titanium is also used to manufacture chlorine. As

Titanium dioxide, it is used in paints (replacing the use of lead), lacquers, paper, plastics, ink,

rubber, textiles, cosmetics, sunscreens, leather, food coloring, and ceramics. It is also used as a

coating on welding rods. Titanium dioxide is one of the whitest and brightest substances known.

Due to its reflective properties, Titanium dioxide, add richness/brightness to colors and provides

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UV protection. Finally, as a compound known as titanium tetrachloride, it is used for

smokescreens and skywriting.

Titanium usage is, however, strongly limited by its higher extraction and production cost

relative to competing materials such as aluminum, grades of stainless steel and other steels just

to mention.

Table 2.1 Some properties of titanium and its alloysHigh strength-to-weight ratio

Corrosion-resistant

High melting point (1660C).Non-toxic

Titanium dioxide is one of the whitest and brightest substances

Provides protection from UV rays

The basic commercial production process of titanium (figure 2.1) starts with theextraction, which involves treatment of the ore (leucoxene, rutile or ilmenite) with chlorine gas

to produce titanium tetra-chloride, which is purified and reduced to what is known as titanium

sponge. The sponge is then blended with alloying elements and vacuum melted giving origin to

an ingot.

After obtaining a homogeneous ingot, it is processed into suitable shapes and sizes,

typically, by forging followed by rolling. Forging and rolling are not the only forming processes

employed to transform the material from its raw form into the final desired product. There are

many other processing routes for the thermo-mechanical processing of titanium products

described elsewhere. The present investigation is focused in the texture and microstructure

evolution resultant from rolling, therefore this discussion will be limited to cold and warm

rolling.

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Figure 2 1- Commercial production of Titanium

2.1.1 - Physical Metallurgy of Titanium and Titanium Alloys

Titanium is an allotropic element, which means that it exists in more than one

crystallographic form. At room temperature, titanium exhibits a hexagonal close-packed (hcp)

crystal structure, referred to as "alpha" phase. However, as the temperature is raised through

882.5C (1621 F) [Collings, 1994], pure titanium undergoes an allotropic transformation from

the -phase (hcp) to a body-centered cubic (bcc) crystal structure, called "beta" () phase. Thetemperature at which this transformation occurs is known asbeta transus and it is defined as the

lowest equilibrium temperature at which the material is 100% (bcc) phase. Alloying element

addition to pure titanium can either cause the transformation temperature to increase decrease or

remain unaffected. These elements are generally classified as alpha or beta stabilizers. The group

of alloying elements that favor the -phase and stabilize it by raising the beta-transus

temperature include aluminum, gallium, germanium, carbon, oxygen and nitrogen. The phase

is stabilized by the addition of elements, which promote the lowering of the beta-transus

temperature. Such elements are classified in two groups: the isomorphous and the eutectoid.

The former consists of those elements that are miscible in the phase: molybdenum, vanadium,

tantalum, and niobium. The second is formed by those whose form eutectoid systems with

titanium, having eutectoid temperatures as low as 550C, i.e., as much as 333C (600 F) below

the beta-transus for pure titanium. This group includes manganese, iron, chromium, cobalt,

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nickel, copper, and silicon. Besides the cited elements, Zr and Tin, due to their extensive solid

solubility, are also employed as strengthening agents and also to retard the rates of phase

transformation.

2.1.2 - Classification of Titanium Alloys

Titanium alloys are classified, basically, taking in account their chemical composition,

the weight % of the alloying elements, and its effect on the resultant microstructure at room

temperature. As an example, the reason why pure titanium is classified as -titanium is due to

the fact that at room temperature its microstructure is formed entirely by grains with hexagonal

close-packed (hcp) crystal structure. An example of dual phase alloy is Ti-6Al-4V, which

contains both alpha and beta stabilizers, and as a consequence alpha + beta alloys exhibit a

certain volumetric fraction of beta phase stabilized at room temperature. Table 2.2 presents a

number of commercially available alloys arranged accordingly to their classification in alpha,

near-alpha, alpha + beta and beta alloys [Reed-Hill, 1992].

2.1.2.1 - Alpha-Titanium Alloy

This group consists of both pure titanium (or unalloyed) and those alloys containing -

stabilizing elements such as Al, Ga and Sn, either singly or in combination. The commonly used

alloys are the several grades of commercially pure (CP) titanium, which are in effect Ti-O alloys,

and the ternary composition Ti-5AI-2.5Sn. As mentioned previously at ordinary temperatures

these are HCP materials [Collings, 1994]. As alpha alloys are single-phase materials, tensile

strengths are relatively low especially for low oxygen grades, although their high thermal

stability leads to reasonable creep strengths. These alloys are also characterized by good ductility

down to very low temperatures, reasonable strength, toughness and good weldability [Wood,

1972]. However, due to the fact that the alloys in this group are single phased with hexagonal

crystal structure they also exhibit a high rate of strain hardening, being the high content of

oxygen associated with its limited formability [Polmear, 1995].

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2.1.2.2 - Near-Alpha Titanium Alloys

Developed to meet demands for higher operating temperatures, this class of alloys,

possess higher room-temperature tensile strength than that exhibit by alpha alloys. They also

show the greatest creep resistance of all titanium alloys at temperatures above 400C. Usually,

near alpha-alloys are forged and heat treated in the alpha + beta field so that primary beta-

grains are always present in the microstructure.

Improved creep performance has been achieved in special compositions by carrying out

these operations at higher temperatures in the upper alpha + beta and beta fields resulting in a

change to a more elongated alpha microstructure. The two alloys, which currently show the

highest creep resistance with a maximum operating temperature about 600C (1112F) are the

Timetal 1100, and Timetal 834. Timetal 1100 is processed by forging just below the -transus

and the resultant microstructure exhibits a mixture of equiaxed and elongated alpha grains, whichprovides a balance of good creep and low cycle fatigue resistance [Polmear, 1995].

2.1.2.3 Alpha/Beta ( + ) Alloys

These alloys have both and phases in equilibrium at room temperature. They combine

the strength of alloys with the ductility of alloys, and their microstructure and properties canbe varied widely by appropriate heat-treatments and/or thermo-mechanical processing. The most

known and used ( + ) alloy is the Ti-6Al-4V or Ti-6-4. Other commercially available alloys in

this class are the Ti-6-6-2 and Ti-6-2-4-6 whose can exhibit, in certain cases, higher strengths

than the high temperature near-alpha alloys. Other characteristics of these alloys are the good

weldability, which is a function of-stabilizing contents, good combination of properties having

a wide processing window meaning less stringent processing requirements than those required

for other alloys types and their capability for applications up to 400C. They also can be

strengthened with a solution treatment to establish the hardenability followed by aging. The

amount of strengthening that can be achieved is a function of section thickness and chemical

composition of the alloy: as the -stabilizing content increases the hardenability increases.

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2.1.2.4 - Beta, Near-Beta and Metastable-Beta alloys

There is no clear-cut definition for beta titanium alloys. Conventional terminology

usually refers to near-beta alloys and metastable-beta alloys as classes of beta titanium alloys. A

near-beta alloy is generally one that has appreciably higher beta stabilizer content than a

conventional alpha-beta alloy such as Ti-6Al-4V, but is not quite sufficiently stabilized to readily

retain an all-beta structure with an air cool of thin sections. For such alloys, a water quench even

of thin sections is required. Due to the marginal stability of the beta phase in these alloys, they

are primarily solution treated below the -transus to produce primary alpha phase which in turn

results in an enriched, more stable beta phase. The Ti-10V-2Fe-3Al alloy is an example of a

near-beta alloy. On the other hand, the metastable-beta alloys are even more heavily alloyed with

beta stabilizers than near-beta alloys and, as such, readily retain an all-beta structure upon air-

cooling of thin sections. Due to the added stability of these alloys, it is not necessary to heat treatbelow the -transus to enrich the beta phase. Therefore, these alloys do not normally contain

primary alpha since they are usually solution treated above the -transus. These alloys are termed

metastable because the resultant beta phase is not truly stable, it can be aged to precipitate

alpha for strengthening purposes. Alloys such as Ti-15-3, B120VCA, Beta C, and Beta III are

considered metastable-beta alloys.

Unfortunately, the classification of an alloy as either near-beta or metastable beta is not

always obvious. In fact, the metastable terminology is not precise since a near-beta alloy is

also metastable, i.e., it also decomposes to alpha plus beta upon aging. There is one obvious

additional category of beta alloys: the stable beta alloys. These alloys are so heavily alloyed with

beta stabilizers that the beta phase will not decompose to alpha plus beta upon subsequent aging.

There are no such alloys currently being produced commercially. An example of such an alloy is

Ti-30Mo.

The interest in beta alloys stems from the fact that they contain a high volume fraction of

beta phase, which can be subsequently hardened by alpha precipitation. Thus, these alloys can

generate quite high strength levels (in excess of 200 ksi) with good ductility. Also, such alloys

are much more deep hardenable than alpha-beta alloys such as Ti-6Al-4V. Finally, many of the

more heavily alloyed beta alloys exhibit excellent cold formability and as such offer attractive

sheet metal forming characteristics.

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Table 2.2 Summary of commercial and semi-commercial grades and alloys of titanium [Reed-Hill, 1992].

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2.2 - Mechanical Behavior of Titanium and its Alloys

In hexagonal close-paked (hcp) metals the low number of easy slip systems, their

asymmetrical distribution, and the strict crystallographic orientation relationships for twinning

results in the formation of a strong deformation texture. The deformation mechanism together

with the texture is responsible for the strong anisotropy of the mechanical properties [Hosford

and Backofen, 1964]

In hcp alpha-titanium, slip occurs most commonly on the basal {0001}, prismatic {1010},

and pyramidal {1011} slip planes (figure 2.2). The actual dominant slip planes depend on the c/a

ratio, as well as alloy composition, temperature, grain size, and crystal orientation. In general,

slip will tend to occur on the plane having the largest inter-planar distance. For hexagonal

materials exhibiting c/a ratio less than 1.663 (considered the ideal ratio), the prismatic plane is on

average the most densely packed plane. For alpha-titanium (c/a = 1.587), the prismatic plane isthe most densely packed. As a consequence, the smallest resolved shear stress occurs at the

prismatic slip plane. This is the case of high purity alpha-titanium. If high interstitial levels of

oxygen and/or nitrogen are present, as it is the case in low purity alloys (i.e., CP titanium), all

three slip planes are activated, but the prismatic plane is still the one with lower resolved shear

stress required to initiate slip. Hexagonal materials, due to its 6-fold rotation symmetry do not

exhibit a complete set of slip systems. As a consequence of this limited number of slip system

capable of being activated, further deformation is accommodated either by pyramidal

glide or traction/compression twinning.

Twinning results when a portion of the crystal takes up an orientation that is related to the

orientation of the rest of the untwined lattice in a definite symmetrical way. The plane of

symmetry between the two portions is known as twinning plane. In titanium, the most common

twinning plane is (1012) and twinning direction is [1011] [Dieter, 1986].

2.2.1 Slip Modes in HCP Metals

The primary slip systems operative in HCP metals with c/a ratio less than the ideal

1.633are the prismatic {1010} planes in the basal directions. The other first order

possible slip systems are the basal (0001) and pyramidal {1011} planes with basal directions

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. These systems will provide combinations of 4 independent slip systems, since they all

occurs on the basal direction.

Figure 2.2 The hexagonal unit cell (a) and the first order slip and twinning planes for hcpmetals (b) [Dieter, 1986].

Differently from materials with cubic crystalline structure that posses 5 or more glide

systems, in hexagonal close packed metals the most common basal and prismatic glide modes

have only 2 or 3 independent glide systems respectively [Groves, 1963]. As a consequence, since

at least four or five independent slip systems are necessary to accommodate arbitrary plastic

strains, secondary systems like pyramidal glide with Burgers vector, or twinning systems

can be activated contributing to accommodate the imposed strain [Yoo, 1981 and Partridge,

1967]. Figure 2.3 shows the primary and secondary glide systems for Titanium.

Regarding hexagonal metals, the activation of slip and twinning systems is normally

affected by parameters like c/a ratio, interstitial constituent (i.e. Oxygen content principally inthe case of CP Ti), strain hardening, strain rate and temperature.

At room temperature, as a consequence of cold rolling, Ti deforms by prismatic glide

{1010}, pyramidal glide {1011} with Burgers vector and secondary

{1011} with Burgers vector, {1012} (and in some cases {1121} twinning

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in tension and {1122} twinning in compression [Rosi et al. 1956; Conrad, 1981; Chin,

1975].

Figure 2.3 Glide systems in alpha titanium.

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In the case of high purity titanium deformed in uniaxial compression at 20C, it was reported

(via EBSD analysis), the activation of three types of twins: {1122}, {1012} and

{1121}, in the proportions of 40%-30%-30% respectively [Salem 2002, Kalidindi et al.

2004].

Zaefferer investigated the relation between the formation of cold rolling textures and the

activated glide and twinning systems during deformation of polycrystalline Titanium [Zaefferer,

2003]. Samples of three different titanium alloys (Ti-6Al-4V and two commercially pure

Titanium grades designated in the work as T40 (1000ppm O) and T60 (2000ppm O)) were

deformed up to 5% by uniaxial or biaxial. Zaefferer observed a considerable activity of

and twinning in the case of the T40 alloy with a pronounced TD-type texture and for the T60

Alloy, the higher oxygen content completely suppressed twinning and strongly reduced

activity resulting in a less developed TD-type texture which was a result of a combination of and basal slip. The results reported by Zaefferer are summarized below in the tables 2.3

and 2.4 and the main textures observed are presented in the figure 2.4.

Table 2.3- Number of grains showing a specific glide system for different samplesSlip system TA3Z0 (%) TA3Z45 (%) TA3Z90 (%) TA1Z (%) TA1B (%) T401B (%) T601B (%)

-Basal 1 (6) 3 (28) 4 (16) 2 (5) 11 (35) 5 (14) 9 (37)-Prismatic 4 (27) -- 1 (4) 9 (23) 3 (10) 2 (6) 1 (4)

-Pyramidal 3 (20) 2 (18) 3 (12) 3 (7) 3 (10) 8 (21) 3 (13)

-Screw 3 (20) 4 (36) 5 (20) 26 (65) 9 (30) 6 (16) 4 (16)

-Pyramidal 4 (27) 2 (18) 12 (48) -- 5 (15) 13 (34) 5 (21)

Others T401B - -Prismatic 4 (9) and T601B- -Prismatic 2 (8)

Phillipe and Fundenberger [Phillipe et all, 1995, Fundenberger et all, 1997], working with

cp-Titanium grade 1(T35) and grade 2(T60) respectively, studied the activation of glide and twin

systems during cold rolling and observed the occurrence of {1010} prismatic slip and a

very low activity of Basal and pyramidal slip. They also observed activation of two twinning

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systems: {1012} tension twins and {1122} compression twins. In the case of second order

pyramidal slip it was observed a low activity of this type of gliding up to 50% of

deformation but from this point up to 80% reduction in thickness, twinning is suppressed and to

accommodate further deformation in direction, the pyramidal gliding was activated

instead of {1122} compression twinning.

Table 2.4 - The most important deformation systems in hcp metals and their influence on thetexture evolution [Zaefferer, 2003].Burgers vector or

shear direction

Glide or

shear plane*

Name Related cold-rolling

texture type

1/3 () {0001}

{1010}

{1011}

Basal glide

Prismatic glide

Pyramidal glide

r-type [Sakai and

fine,1974],c-type

[Conrad, 1981]

r-type [Philippe et al.

1988]

1/3 () {1011}

{1122}

Pyramidal (I) glide

Pyramidal (II) glide

t-type

t-type {1012} {1012} Twin Under tension c-type

(Ti); compression r-

type (Zn)

{1121} {1121} Twin Under tension c-type

{1122} {1122} Twin Under compression

t-type

* Glide plane in case of dislocations, shear plane in case of twinning.

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Figure 2.4 - Schematics of all investigations carried out and definition of sample short names.The starting texture of the different materials is given in the form of (0001) and { /1010/} X-raypole figures. Sample short names are composed as follows: (1) chemical composition; (2) sheetthickness in mm; (3) deformation mode; (4) angle between RD and tension direction (0, 45,90) or deformation degree (2%, 4%) [Zaefferer, 2003].

In titanium, Rosi et al. observed no twins of any type at 800C, while McHarque and

Hamond reported a small amount of {1122} and {1121} twinning at 815C. At room

temperature and below that titanium slips along the direction on the {1010}, (0001) and

{1011} planes. Changes in length along the c axis are not possible with slip alone,

requiring a slip direction lying out of the basal plane (0001). Such slip has been reported in

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commercially pure titanium as a result from the motion of the dislocations along the

. A length change along the c axis can also be accomplished by twinning. In titanium,

{1012}, {1121} and {1123} twins allow an extension along the c axis, while {1122}, {1124}

and {1010} twins allow a reduction in the c axis; whish generally becomes less important as the

deformation temperature increases. Paton and Backofen [Paton and Backofen, 1970]

investigating iodide titanium single crystals under compression at temperatures from 25C to

800C, have found that reduction of up to a few percent strain along the c axis is accommodated

almost entirely by {1122} twinning from 25C to 300C. According to their results, although

slip is not responsible for a significant amount of strain below 300C, it is important as a

means of accommodating the shear ahead of a propagating {1122} twin.

2.3 - Texture

The most commonly and important used materials for industrial practice, such as metals,

ceramics and some polymers are polycrystalline materials and their component units are referred

to as crystals or grains. Grain orientations in polycrystals are rarely random due to the processing

history that the polycrystalline materials are normally submitted to, such as solidification from

melting, hot rolling, cold rolling and annealing among other thermo-mechanical processes.Therefore, in most materials there is a pattern in the orientations, which are present and a

tendency for the occurrence of certain orientations. This tendency is known as preferred

orientation of crystals ortexture. The relevance of texture to materials lies in the fact that many

materials properties are texture-dependent. According to Bunge-1987, the influence of texture on

materials properties is, in many cases, 20-50% of the properties values. Some examples of

properties which depend on the average texture of a material are: Youngs Modulus, Poissons

ratio, strength, ductility, toughness, magnetic permeability, electrical conductivity and thermal

expansion (in non-cubic materials) [Randle and Engler, 2000].

Texture, in hexagonal materials, is represented by the Miller indices {hkil} where

{hkil} corresponds to the family of crystallographic planes parallel to the surface of the sample

and corresponds to the family of crystallographic directions parallel to the rolling

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direction (RD) of the sample. The resulting rolling textures, in the form of pole figures, as a

function of c/a ratio is shown in figure 2.5.

Figure 2.5 Sheet textures in hcp materials as a function of c/a ratios (schematically).

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The ideal cold rolling texture component is represented in figure 2.6 and other typical

textures in hexagonal materials are shown in figure 2.7.

Figure 2.6 Ideal cold rolling texture component for flat-cold rolled titanium: {2115} .

Texture can be determined by means of X-ray diffraction, neutron diffraction andelectron diffraction using Transmission Electron Microscope (TEM) or Scanning Electron

Microscope (SEM). X-ray diffraction is the most commonly applied technique but the neutron

and electron diffraction techniques are gaining interest because it permits one to correlate

microstructures, neighbor relations and texture [Kocks, 1998].

Among the ways to describe texture, pole figure (PF), inverse pole figure (IPF) and

orientation distribution function (ODF) are the most usual methods. Pole figure is a projection

[Cullity; 2001], more often represented as a stereographic projection, which shows the variation

of pole density with pole orientation for a selected set of crystal planes having the rolling

direction (RD), the transversal direction (TD) and the normal direction of the sample as reference

axis.

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Figure 2.7 Typical textures [Wang, 2003].

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Pole figures are measured using x-ray diffraction and in order to have a specific (hkil)

reflection, the following condition, known as Braggs law (equation (2.1)), must be satisfied.

n = 2 dhkl . sin (2.1)

During the pole figure measurement, to determine a pole density, the x-ray detector

remains stationary at the proper 2 angle, to receive the (hkil) reflections, while the specimen

rotates in two particular ways. These rotations permit a complete scanning of the specimens

surface and the positioning of the sample on the texture goniometer is shown in figure 2.8.

Figure 2.8 - Positioning and movement of the sample on the texture goniometer inside the X-raymachine (a). The relation between crystallite coordinates (Xc, Yc, Zc) and sample coordinates(Xs, Ys, Zs), (b), (c) and (d).

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The and angles, which are respectively the polar and the azimuthal angles, define the

movements of the sample during the pole figure measurement.

The inverse pole figure (IPF) is a pole density projection of the (hkil) planes referred to

the stereographic triangle. Inverse pole figure presents an advantage over the pole figure because

an IPF shows the density distribution of all planes within the stereographic triangle instead of

showing only the density of a specific crystallographic plane (see figure 2.9).

RD

(0002) (1010) (2110)

TD

a)

ND TD RD (1010)

(0002) (2110)

b)

Figure 2.9 As received material: a) Pole figures and b) Inverse pole figures.

The pole figure and the inverse pole figure are very helpful tools however principal

orientations of the texture cannot be precisely determined from them because they do not provide

information regarding the crystallographic directions in the plane of the sample. The figure 2.10

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exemplifies a situation where two different texture components, the cold rolling and the

recrystalization components, exhibit the same (0002) pole figure, which can be misleading if the

analysis is based only on basal pole figures.

Figure 2.10 Pole figure representation of the cold rolling and the recrystalization texturecomponents.

It has been well established that the orientation distribution in textured materials can be

qualitatively as well as quantitatively evaluated by the crystallite orientation distribution function

analysis (ODF) developed by Bunge and by Roe [Bunge, 1982; Roe, 1965]. The ODF describes

the frequency of occurrence of particular orientations in a three-dimensional orientation space.

This space is defined by three Euler angles (, , ) which are related to the macroscopic axis of

the sample, defined as rolling direction (RD) axis, transversal direction (TD) axis and normal

direction (ND) axis through a set of three consecutive rotations that must be given to each

crystallite in order to bring its crystallographic axes into coincidence with the specimen axes.

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Figure 2.11 shows the rotations where represents a rotation around the ND axis, represents a

rotation around the TD axis and represents a second rotation around the ND axis.

Figure 2.11 - Three consecutives Euler rotations defining an orientation.

ODF is a three dimensional description of texture but direct measurement of ODF is not

possible since conventional texture goniometry is only capable of determining the distribution of

crystal poles of diffracting planes normal, i.e., pole figures. Mathematical models have beendeveloped which allow the ODF to be calculated from the numerical data obtained from several

pole figures. Therefore, in order to compute the orientation distribution function for a

polycrystalline sample, pole figures measurements are required. The number of pole figures

needed for ODF calculation depends upon the crystal symmetry of the sample that is being

measured. For HCP materials, as it is the case of titanium, a minimum of five pole figures are

needed. The most widely adopted methods for calculating ODFs are those proposed

independently by Roe (1965) and by Bunge (1982), who used generalized spherical harmonic

functions to represent the crystallite distributions. The three Euler angles employed by Bunge to

describe the crystal rotations are 1, and 2, whereas the set of angles proposed by Roe are

referred to as , and respectively. The relationships between the Bunge and the Roe angles

are the following:

1 = /2 - ; = ; 2 = /2 -

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According to Roe, 1965, an ODF may be expressed as a series of generalized spherical

harmonics in the form of equation (2.2):

l l (, , ) = Wlm Zlmn (cos ). exp (-im). exp(in) (2.2)

l=0 m=-1 n=-1

Where Wlmn are the series coefficients and Zlmn (cos) is a generalization of the associated

Legendre functions, the so-called augmented Jacobian polynomials.

For hexagonal/orthotropic crystal/specimen symmetry, a three-dimensional orientation

volume may be defined by using three orthogonal axes for , and with each of the Euler

angles ranging from 0 to 90. The value of the orientation density at each point in this volume is

simply the intensity of that orientation in multiples of random units. Regions of higher and lower

orientation density are separated by three-dimensional contour surfaces and it is usual to take a

series of parallel sections through this space for ready visualization of the data contained in the

three-dimensional plot. In the case of hcp materials, due to their crystal symmetry, the

fundamental space can be reduced to the space spanned by the Euler angles (from 0 to 90),

(from 0 to 90) and (from 0 to 60) with sections every 5 or 10 degrees. Davies [Davies et al.,

1971] published a set of charts for hexagonal materials designed to aid on the task of indexingthe texture components of rolled materials with hexagonal symmetry. In this development

Davies has used a definition of Euler angles by Roe and has taken crystal directions

parallel to ND and parallel to RD (see figure 2.12).

Figure 2.12 Relationship between sample and crystal axis directions.

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The charts published by Davies are shown in the figure 2.13 and in figure 2.14 an

example of the advantage of using the ODF in texture analysis.

Figure 2.13 Constant sections through the Eulerian space: a)0, b)20, c)30, d)40 and e)60

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Figure 2.14 Location of the cold rolling and recrystalization components on the constant phi

sections of the Euler space using Roes definition [Roe, 1965].

2.3.1- Cold Rolling Texture

Hexagonal materials, such as, titanium and zirconium, have a limited number of slip

systems and generally develop a strong texture after cold rolling. Knight, 1978; investigated the

texture evolution of commercially pure titanium sheets after cold rolling at 21.4% and 89.4% of

reduction and observed that the most intense texture component, for both degrees of reductionwas the (2115) [0110]. Guillaume et al., 1981, when working with cold rolled titanium sheets,

found the same result. The (2115) [0110] orientation is 35 around the (0002) pole in the

transversal plane, which involves a rotation of the (0002) pole around the rolling direction, in the

plane defined by the transversal and the normal directions. Philippe et al. [Philippe, 1984], have

also found the same texture components after cold rolling of titanium and zirconium alloys.

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Inagaki [Inagaki, 1991] working with hot rolled and annealed pure titanium presenting a very

strong texture, found that after cold rolling reductions below 30% the textures were weakened by

twinning and slip rotations. At cold rolling reductions between 30 and 50% twinning occurred

less frequently and at rolling reductions above 50%, crystal rotation about //RD axis is

induced by slip deformation. Orientations located near the {0001} were rotated toward

the {2115} orientation, becoming stable at this orientation at rolling reductions above

80%. Inagaki also found that the [0001]//ND fiber texture increased remarkably at rolling

reduction between 30 and 50% and that it decreased rapidly at rolling reductions above 50%. The

[0110]//RD fiber, on the other hand, developed at rolling reductions above 50%.

2.3.2- Hot and Warm Rolling Texture

In the past, hot rolling textures in titanium have been studied by only few investigators

and warm rolling textures in titanium have called even less attention from the researchers.

Inagaki [Inagaki, 1990] investigated the effect of hot rolling temperature (750, 800, 850, 900 and

950C) on the development of hot rolling textures on commercially pure titanium plates.

According to Inagaki, the textures observed in the specimens hot rolled at temperatures below

800C are essentially the same as the cold rolling texture and their main orientation is {2115}

. Hot rolling at temperatures between 800 and 850C enhances the development of the

{2110} and {2118} main orientations, which seem to be formed by the

recrystallization that occurs during and after hot rolling. Hot rolling at temperatures above 880C

results in the formation of a strong transformation texture where the {2110} texture

component, derived from the BCC phase rolling texture, is the main orientation.

2.4 X-ray Peak Profile Analysis

In order to improve and to control the mechanical proprieties of any material it is

important to understand and to explain how variables such as dislocation density, dislocation

type and slip system activation affect the formation and evolution of certain microstructures

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during the deformation process. The study and determination of the dislocations slip systems

type is usually carried out with conventional techniques such as TEM. However, when the

material is highly deformed and the dislocation density reaches values as high as 1010/cm2, TEM

analysis is rather difficult. Also, throughout the sample preparation process required for TEM

experiments the original microstructure may change. Other alternatives on investigating the

microstructure are X-ray and neutron diffraction techniques. In recent decades, new applications

for the X-ray diffraction method (traditionally used for phase identification, quantitative analysis

and the determination of structure imperfections), have extended its usage to new areas, such as

the determination of crystal structures and the extraction of microstructural properties of

materials. Recent works have shown that X-ray diffraction peak profile analysis (XDPPA) is a

powerful alternative to transmission electron microscopy for describing the microstructure of

crystalline materials and providing information about the dislocation densities and dislocationtype extracted from the X-ray pattern [Ungr, 1999; Ribrik, 2001; Dragomir, 2002; Scardi,

2002; Glavicic, 2004; Scardi 2004; Ungr, 2004; Dragomir, 2005a and 2005b]. Besides that,

since the parameters provided by the two different methods are never identical, XDPPA is also

complementary to TEM enabling a more detailed understanding of microstructures.

X-ray diffraction peaks broaden when the crystal lattice becomes imperfect. The

microstructure means the extent and the quality of lattice imperfectness. According to the theory

of kinematical scattering, X-ray diffraction peaks broaden either due to crystallites smallness

(1 ), lattice defects are present in large enough abundance ( in terms of dislocations this means

a dislocation density larger than about 5x1012m-2), stress gradients and/or chemical

heterogeneities.

Peak broadening is caused by crystallite smallness, lattice defects, stress gradients and/or

chemical heterogeneities. As a consequence of these deviations from perfect crystalline lattice

the shape of the X-ray diffraction lines no longer consists of narrow, symmetrical, delta-function

like peaks, such as the diffraction lines given by an ideal powder diffraction pattern. The

aberrations from the ideal powder pattern can be conceived as: (i) peak shift, (ii) peak

broadening, (iii) peak asymmetries, (iv) anisotropic peak broadening and (v) peak shape. The

main correlation between these peak aberrations and the different elements of microstructure are

summarized in table 2.5.

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Table 2.5 The most typical correlations between diffraction peak aberrations and the differentelements of microstructure (Ungr, 2004).Sources of strain Peak aberrations

shift broadening asymmetry Anisotropic

broadening

shape

Dislocations

Stacking faults

Twinning

Microstresses

Long-range internal stresses

Grain boundaries

Sub-boundaries

Internal stresses

Coherency strains

Chemical heterogeneities

Point defects

Precipitates and inclusions

Crystallite smallness

The effect of these defects can be divided into two main types of broadening: size- and

strain broadening. The first depends on the size of coherent domains and may include effects of

stacking and twin faults and sub-grain structures (small-angle grain boundaries) whereas the

latter is caused by different lattice imperfection, especially dislocations. The two different effects

interplay with each other and very often are not easy to separate. Krivoglaz [Krivoglaz, 1969]

has shown that strain broadening can be described, in general, in terms of broadening caused by

dislocations. In the case of single crystals or coarse-grained polycrystalline materials, strain

broadening caused by dislocations can be well described by a special logarithmic series

expansion of the Fourier coefficients [Krivoglaz, 1969; Wilkens, 1970; Groma et al., 1988,

Ungr et al., 1989]. When grain size plays a role, the two effects (i.e. size and strain broadening)

overlap. In such cases the grain size or the properties of the dislocation structure can only be

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determined by the correct separation of the two effects. Two classical procedures are employed

in order to separate the strain and domain-size components of the broadening: Williamson-Hall

method and Warren-Averbach method. The first procedure [Williamson and Hall, 1953] is based

on the full width at half maximum (FWHM) and the integral breadths while the second is based

on the Fourier coefficients of the profiles [Warren and Averbach, 1950; Warren, 1959]. The

particle-size and dislocation microstrains are convoluted but can be separated, because the

particle-size broadening is independent of the order of the diffraction line, whereas the strain

broadening is not. In the Warren-Averbach method, the diffraction line profile is transformed

into its Fourier components and processed in order to separate the two broadening effects (after

correction for instrumental broadening). Evaluations carried out with both methods provide

apparent size parameters of crystallites or coherently diffracting domains and values of the mean

square strain but grain shape anisotropy and also strain anisotropy can turn difficult andcomplicate the evaluation process [Lour et al., 1983; Caglioti et al., 1958]. In practical terms,

strain anisotropy means that neither the full width of half maximum (FWHM) in the Williamson-

Hall plot [Williamson and Hall, 1953] nor the Fourier coefficients in the Warren-Averbach

analysis [Warren and Averbach, 1952; Warren, 1959] are smooth functions of the diffraction

vector g. Ungr proposed that a way to interpret strain anisotropy is to assume that dislocations

are one of the major sources for lattice distortions [Ungr and Borbly, 1996]. Two different

approaches can well account for the phenomenon, especially in the case of random

polycrystalline or powder specimen. One is a phenomenological approach assuming that the

random displacements of atoms are weighted by the anisotropic elastic constants of the crystal

[Stephens, 1999] and the FWHM is scaled by the fourth order invariants of the hklindices, given

for different crystal classes e.g. by Nye, (1957) or Popa, (1998). The other approach operates

with the anisotropic diffraction contrasts of dislocations [Stokes and Wilson, 1944; Ungr and

Borbly, 1996]. In the case of randomly oriented polycrystalline or powder specimen the

dislocation model has been shown to be formally equivalent to the phenomenological approach

[Ungr and Tichy, 1999] and the model is able to provide quantitative results, which have

physical relevance to the microstructure of the crystal [Cheary et al., 2000]. An advantage of this

model is that it also works in the case of a heavily deformed polycrystalline material or a single

crystal [Mohamed et al., 1997; Cheary et al., 2000; Borbly et al., 2000], situations in which a

strong preferred orientation is present.

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In polycrystalline material populated with dislocations the anisotropic line broadening

can be taken into account by using that the dislocation model of the mean square strain, ,

(where L is the Fourier length and g is the distortion tensor component in the direction of the

diffraction vector, g) [Wilkens, 1970a and 1970b]. In this model the dislocations are assumed to

have a restrictedly random distribution within a region defined by Re as the effective outer cut-

off radius [Wilkens, 1970a]. Here the anisotropic effect can be summarized in the average

contrast factors, C, which depends on the relative orientations of the line and Burgers vectors of

dislocations and the diffraction vector [Ungr and Borbly, 1996; Wilkens, 1970b; Klimanek and

Kuzel, 1988; Kuzel and Klimanek, 1988 and 1989; Ungr and Tichy, 1999; Dragomir and

Ungr, 2002]. The contrast factor of dislocations is a measure of the visibility of dislocations

in the X-ray diffraction experiments. Since, the contrast effect is mainly a characteristic of

dislocations, the theoretical values of the contrast factors and those obtained from the profile

evaluation enable the determination of the active dislocation slip system(s) in the studied sample

[Klimanek and Kuzel, 1988; Kuzel and Klimanek, 1988 and 1989; Ungr and Tichy, 1999;

Dragomir and Ungr, 2002].

Because of the complexity of the mechanical properties of hexagonal crystals [Chung &

Buessem, 1968; Gubicza et al., 2000; Solas et al., 2001; Tom et al., 2001] for a better

understanding of the bulk dislocation structure and the Burgers vector populations it is desirable

to complement TEM studies by X-ray diffraction profile analysis. When comparing thehexagonal crystal to the cubic crystal it becomes evident the higher level of complexity involved

when dealing with the hexagonal systems. Instead of three elastic constants hexagonal crystals

exhibit six elastic constants and two lattice constants (c and a) while cubic systems have only one

(a). Moreover instead of one, hexagonal crystal present two different types of anisotropy: shear

and compression [Chung and Buessem, 1968]; and finally, while in cubic systems there is one

major slip system, in hexagonal there are three different major slip systems related to the three

glide planes: basal, prismatic and pyramidal. If it is taken in account the different glide directions

and dislocation character (i. e., edge and screw) it is possible to group the slip systems into

eleven sub-slip-systems as shown in Table 2.6 [Yadav and Ramesh, 1977; Jones and Hutchinson,

1981; Honeycombe, 1984; Castelnau et al., 2001]. Dragomir and Ungr (2002) have recently

published a modified methodology to obtain the contrast factors for both cubic and hexagonal

crystals. They concluded that, at the actual state of the art regarding line broadening analysis, it

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is impracticable to compile the dislocation contrast factors for hexagonal systems in a similar

manner as it was done for cubic crystals and instead, they proposed to compile the average

contrast factors of the sub-slip-systems. The average contrast factor of a specific sub-slip-system

in a hexagonal crystal can be given by three parameters versus the fourth-order invariant of the

hkil Miller indices [Ungr and Tichy, 1999]: Chk.0, q1 and q2 and once these parameters are

determined all average contrast factor corresponding to the sub-slip-system in question can be

obtained [Dragomir and Ungr, 2002].

Table 2.6 - The most common slip systems in hexagonal crystals: (a) Edge dislocations and (b)Screw dislocations.(a) Edge dislocations:

Major slipsystems

Slip-systems Burgers vector Slip plane Burgers vectortypes

Basal BE >< 0112 }0001{ a

PrE >< 1102 }0101{ a

PrE2 >< 0001 }0101{ cPrismaticPrE3 >< 1132 }0101{ c + a

PyE >< 0121 }1110{ aPy2E >< 1132 }2112{ c + a

PyE3 >< 1132 }1211{ c + aPyramidal

PyE4 >< 1132 }1110{ c + a

(b) Screw dislocations:

Slip-systems Burgers vector Burgers vector types

S1 >< 0112 a

S2 >< 1132 c + a

S3 >< 0001 c

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2.4.1 - X-ray Peak Profile Analysis from MWP and Methodology for Determining the

Burgers Vector Populations

It is well known that the Fourier coefficients of the physical profiles can be written as a

multiplication of the Fourier coefficients corresponding to the size and distortion effect [Ungr

and Tichy, 1999]:

AL = ALS AL

D = ALS exp [- 22L2g2 ] (2.3)

where S and D indicate size and distortion, g is the absolute value of the diffraction vector,

is the mean square strain and L is the Fourier variable. As shown in [Wilkens, 1970a and

1970b; Krivoglaz, 1996] in a dislocated crystal the mean square strain can be written in terms of

dislocation density and the strain anisotropy, which can be taken in account by introducing the

dislocation contrast factors, ( 2Cb /4) f(), where and b are the density and the

modulus of the Burgers vectors of dislocations and 2Cb is the average contrast factor of the

dislocations present in the sample multiplied by the square of the dislocations burgers vector.

f()-function is the Wilkenss function, where =L/Re, Re is the effective outer cut-off radius of

dislocations, L is the Fourier length defined as n/2(sin2-sin1) with n being an integer starting

from zero, the x-ray wave length and (2-1) the angular range of the measured profile

[Wilkens, 1970a].

Contrast effect of dislocations depends not only on the material, but also on the relative

orientation of the diffraction vector, g, line vector, l, and Burgers vector, b [Klimanek and Kuzel,

1988; Kuzel and Klimanek, 1988 and 1989; Dragomir and Ungr, 2002]. Due to this in the case

of hexagonal crystals the three major slip systems (basal, prismatic and pyramidal) have to be

divided in 11 sub-slip systems by taking into account the different slip system types and the

dislocation character (i.e. edge or screw). These eleven sub-slip-systems are illustrated in figure2.16 and listed in table 2.6. It has been shown in earlier studies that in the case of hexagonal

crystals for a given sub-slip system the average contrast factor of dislocation can be written as

[Dragomir and Ungr, 2002]:

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Chk.l = Chk.0 [1 + q1x + q2 x2 ] (2.4)

where x = (2/3)(l/ga)2, q1 and q2 are parameters which depend on the elastic properties of the

material, Chk.0 is the average contrast factor corresponding to the hk.0 type reflections, a is thelattice constant in the basal plane, g is the diffraction vector and l is the last index of the hk.l

reflection for which the Chk.l is evaluated. The equation (2.4) is valid only when it can be

assumed that within a sub-slip system the dislocation can slip with the same probability in all

directions permitted by the hexagonal crystal symmetry. This equation also means that the

average contrast factors corresponding to a specific sub-slip-system and material constants

(elastic constants C11/C12, C13/C12, C33/C12, C44/C12 and the lattice constant c/a) have to follow a

parabola as a function of x having the parameters Chk.0

, q1

and q2

as parameterization

parameters. In the case of Titanium, the parabolas corresponding to the eleven sub-systems

described in table 2.6 and shown in figure 2.16 are presented in the figure 2.15.

Figure 2.15 The parabolas describing the average contrast factors for the eleven slip systems,in the case of Titanium, as a function of x = (2/3)(l/ga)2 [Dragomir et al., 2002].

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BASAL

{0001}

PRISMATIC

{0110} {0110} {1100}

PYRAMIDAL

{1011} {1011} {1121} {2112}

Figure 2.16 - Slip systems in hexagonal crystal systems [Honeycombe, 1984; Klimanek andKuzel,1988].

As it has been shown by Dragomir and Ungr, in the case of hexagonal crystal the measured

average)(

2

m

Cb characteristic to the examined sample can be written as follows [Dragomir and

Ungr, 2002]:

)(2

m

Cb ==

N

i

i

i

i bCf1

2)( (2.5)

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where N is the number of the different activated sub-slip systems,)(i

C is the theoretical value of

the average contrast factor corresponding to the ith sub-slip system and fi are the fractions of the

particular sub-slip systems by which they contribute to the broadening of a specific reflection.

On the left hand side of the equation (2.5) the m superscript refers to the measurable strainanisotropy parameter, 2Cb . For the hexagonal crystal structure, equation (2.5) can be written for

the three fundamental Burgers vectors types defined in the hexagonal systems: b1=1/3,

b2=, and b3=1/3:

)(

.2

m

lhkCb =2

1b >+

2005_Efffect of Thermo Mechanical Treatment on Texture Evaluation of Polycrystalline

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