Recovery and Recrystallization Kinetics in AA1050 and AA3003 Aluminium Alloys

Recovery and Recrystallization Kinetics in

AA1050 and AA3003 Aluminium Alloys

Recovery and Recrystallization Kinetics in

AA1050 and AA3003 Aluminium Alloys

PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus Prof. dr. ir. J.T. Fokkema,

voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 31 maart 2003 te 13.30 uur

door

Shangping CHEN

Master of Science

Xi’an Jiaotong University, China

geboren te Shaaxi province, China

Dit proefschrift is goedgekeurd door de promotor:

Prof. dr.ir. S. van der Zwaag

Samenstelling promotiecommissie:

Rector Magnificus, voorzitter

Prof. dr.ir. S. van der Zwaag, Technische Universiteit Delft, promotor

Prof. dr.ir. J.TH.M. de Hosson, University of Groningen

Prof. dr.ir. H.J. Huetink, University of Twente

Prof. dr.ir. M.J.L. van Tooren, Technische Universiteit Delft

Prof. dr.ir. A. Bakker, Technische Universiteit Delft

Prof. ir. L. Katgerman, Technische Universiteit Delft

Prof. dr. I.M. Richardson, Technische Universiteit Delft

This research was carried out in the framework of the Strategic Research Program of theNetherlands Institute for Metals Research in the Netherlands (www.nimr.nl).

Published and distributed by: DUP Science

DUP Science is an imprint of Delft University PressP.O.Box 982600 MG DelftThe NetherlandsTelephone: +31 15 2785678E-mail: [email protected]

ISBN 90-407-2382-6

Keywords: Aluminum alloys; Recrystallization kinetics; Modeling

Copyright © 2003 by Shangping Chen

All rights reserved. No part of the material protected by this copyright notice may be

reproduced or utilized in any form or by any means, electronic or mechanical, including

photocopying, recording or by any information storage and retrieval system, without

permission from the publisher: Delft University Press.

Printed in the Netherlands

v

Contents

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

1.1. Current status of recrystallization kinetics models ....................................... 1

1.2. Scope of dissertation......................................................................................... 3

References ................................................................................................................ 5

2. Quantification of the recrystallization behavior in Al-alloy AA1050 ................. 7

2.1. Introduction ...................................................................................................... 7

2.2. Experimental details......................................................................................... 8

2.2.1. Material preparation ................................................................................. 8

2.2.2. Microstructural characterization............................................................... 9

2.2.3. Composite image method ....................................................................... 10

2.3. Results.............................................................................................................. 11

2.3.1. Microstructural evolution ....................................................................... 11

2.3.2. Composite imaging analysis................................................................... 12

2.3.3. OIM observation..................................................................................... 14

2.3.4. Micro-hardness measurements ............................................................... 16

2.4. Discussion ........................................................................................................ 18

2.4.1. Composite image method ....................................................................... 18

2.4.2. Comparison of optical and OIM techniques........................................... 18

2.4.3. Effect of recovery ................................................................................... 18

2.5. Conclusions ..................................................................................................... 20

3. Modeling the kinetics of grain boundary nucleated recrystallization processes

after cold deformation using a single grain approach ....................................... 21

3.1. Introduction .................................................................................................... 21

3.2. The Kinetic Model .......................................................................................... 23

3.2.1. Grain Geometry ...................................................................................... 23

3.2.2. The deformed microstructure ................................................................. 25

vi

3.2.3. Nucleation............................................................................................... 27

3.2.4. Growth.................................................................................................... 28

3.3. Results and discussion.................................................................................... 31

3.3.1. The effect of grain geometry on the recrystallization kinetics ............... 31

3.3.2. Driving pressure for recrystallization..................................................... 35

3.3.3. Simulation of recrystallization kinetics .................................................. 38

3.4. Conclusions ..................................................................................................... 41

4. A refined single grain approach applied to the modeling of recrystallization

kinetics for cold rolled single-phase metals......................................................... 45

4.1. Introduction .................................................................................................... 45

4.2. Experiments .................................................................................................... 47

4.3. Modeling the recrystallization kinetics......................................................... 50

4.3.1. The orientation dependent microstructure in the deformed state ........... 50

4.3.2. Nucleation and Growth........................................................................... 51

4.3.3. Kinetics approach ................................................................................... 53

4.4. Results.............................................................................................................. 54

4.4.1. Effect of the grain geometry and the nucleation site density on the

recrystallization kinetics ......................................................................... 54

4.4.2. Effect of the strain on the recrystallization kinetics ............................... 57

4.4.3. Effect of the initial grain size prior to deformation on the

recrystallization kinetics ......................................................................... 58

4.4.4. Effect of the orientation on the recrystallization kinetics....................... 59

4.5. Application of the model to the experiment data ........................................ 60

4.6. Discussions....................................................................................................... 61

4.7. Conclusions ..................................................................................................... 64

5. A FE comparison study of hot rolling operation and PSC testing.................... 67

5.1. Introduction .................................................................................................... 67

5.2. Mathematical approach ................................................................................. 68

5.2.1. Constitutive model.................................................................................. 68

5.2.2. Hot rolling operation .............................................................................. 69

5.2.3. Plain strain compression......................................................................... 73

5.2.4. Friction condition ................................................................................... 73

5.3. Results.............................................................................................................. 73

5.3.1. Simulation of hot rolling process ........................................................... 73

vii

5.3.2. Simulation of PSC test............................................................................ 79

5.4. Discussions....................................................................................................... 80

5.4.1. Characteristics of hot rolling deformation.............................................. 80

5.4.2. Comparison of the hot rolling operation and PSC testing...................... 82

5.5. Conclusions ..................................................................................................... 83

6. Modeling recrystallization kinetics in AA1050 following simulated break

down rolling……………………………………………………………………….85

6.1. Introduction .................................................................................................... 85

6.2. The recrystallization kinetics model ............................................................. 87

6.2.1. The Single grain approach...................................................................... 87

6.2.2. Kinetics approach ................................................................................... 91

6.3. Experimental................................................................................................... 91

6.3.1. Material and experimental detail ............................................................ 91

6.3.2. Finite element analysis ........................................................................... 93

6.4. Results.............................................................................................................. 95

6.4.1. The deformation structure ...................................................................... 95

6.4.2. Softening behavior by micro-hardness indentation................................ 95

6.4.3. Recrystallization kinetics by optical microscopy ................................... 96

6.4.4. The relationship between static softening and recrystallization............. 98

6.4.5. The grain size of the fully recrystallized structure ................................. 99

6.4.6. JMAK and the S-F analysis of the recrystallization kinetics ............... 100

6.5. Application of the single grain model to predict the recrystallization

kinetics ........................................................................................................... 102

6.5.1. The subgrain size .................................................................................. 102

6.5.2. The calibration constant dC ................................................................. 102

6.5.3. Parameters for recovery........................................................................ 104

6.5.4. Activation energy for recrystallization................................................. 105

6.5.5. Textural components after hot deformation ......................................... 105

6.5.6. The critical size of a viable nucleus…………………………………...106

6.5.7. Prediction of recrystallization kinetics ................................................. 106

6.6. Conclusions ................................................................................................... 110

7. Effect of microsegregation and dislocations on the nucleation kinetics of

precipitation in aluminium alloy AA3003 ......................................................... 113

7.1. Introduction .................................................................................................. 113

viii

7.2. Mathematical description of the C-curve................................................... 114

7.2.1. Activation energy for nucleation .......................................................... 115

7.2.2. Nucleation site density ......................................................................... 117

7.2.3. The evolution of dislocation density .................................................... 118

7.2.4. Effect of micro-segregation of Mn ....................................................... 118

7.3. Experimental................................................................................................. 119

7.4. Results............................................................................................................ 120

7.4.1. The evolution of microstructure during homogenization..................... 120

7.4.2. The evolution of the conductivity and precipitation during isothermal

annealing............................................................................................... 120

7.4.3. The isothermal precipitation kinetics ................................................... 122

7.4.4. The softening kinetics during isothermal annealing............................. 125

7.5. Discussion ...................................................................................................... 126

7.5.1. Analysis of the experimental results..................................................... 126

7.5.2. Application the model to the experiment data...................................... 127

7.6. Conclusions ................................................................................................... 133

8. On the precipitation and recrystallization behavior in an AA3003 following

hot deformation ................................................................................................... 135

8.1. Introduction .................................................................................................. 136

8.2. Experimental................................................................................................. 136

8.3. Results............................................................................................................ 138

8.3.1. The evolution of microstructure during preheat treatment................... 138

8.3.2. High temperature mechanical behavior................................................ 139

8.3.3. The decomposition kinetics of the supersaturated matrix during

isothermal annealing............................................................................. 142

8.3.4. The softening kinetics........................................................................... 145

8.3.5. Nucleation mechanisms and recrystallized grain structure .................. 151

8.4. Discussion ...................................................................................................... 153

8.4.1. Effect of the deformation, recovery and recrystallization on

the precipitation kinetics....................................................................... 153

8.4.2. Effect of precipitation on the recrystallization kinetic ......................... 157

8.4.3. Interaction bewteen precipitation and recrystallization........................ 166

8.5. Conclusions ................................................................................................... 168

Appendix…………………………………………………………………….170

ix

Summary .................................................................................................................. 173

Samenvatting............................................................................................................ 177

Publications .............................................................................................................. 181

Acknowledgements .................................................................................................. 183

About the author...................................................................................................... 185

S.P. Chen

Chapter 1

Introduction

Recrystallization and related annealing phenomena which occur during the

thermomechanical processing of metals have long been recognized as being of great

importance in technological and scientific interest [1-3]. Recovery is the annealing

processes occurring in deformed metals without the migration of a high angle grain

boundary. It contributes to a balance of high strength and high toughness.

Recrystallization is the formation of a new grain structure in a deformed material owing

to the formation and movement of high angle grain boundaries driven by the stored

energy of deformation. It controls the grain structure of the final product. The

combination of recovery and recrystallization is largely responsible for the

microstructural evolution in metals during hot rolling and during annealing after cold

rolling. Metallurgical research in this field is mainly motivated by the requirements of

industry, and currently, a strong need exists for quantitative, physically-based models

which can be applied to metal-forming processes so as to control, improve and optimize

the microstructure and the texture of the finished products. Such models require a more

detailed understanding of both the deformation and annealing processes than we have at

present.

There are three major issues of significance with regard to recrystallization and related

phenomena, namely kinetics, microstructure and texture. Knowledge of the recovery

and recrystallization kinetics is essential in the aluminium industry for predicting rolling

loads and final rolled grain size. This dissertation aims to develop a physical model to

predict the kinetics of recovery and recrystallization in aluminium alloys, as a function

of deformation treatment, thermal history, alloy composition and precipitate content.

1.1. Current status of recrystallization kinetics models

The appropriate model would provide equations which describe the recrystallization

process in sufficient detail. The model should include the following parameters:

Chapter 12

Deformed microstructure: This determines the number, spatial distribution, and

viability of nucleation sites, and provides the driving force for growth of nuclei.

Microstructure: Grain boundaries and second phase particles will influence the

nucleation mechanisms and may also affect the growth.

Recovery: Recovery before and during recrystallization will affect the growth rate

and in certain cases the nucleation rate.

Up to now, this goal is far from being reached although models for recrystallization

have been developed over a very long time.

The basis for most analytical models to describe the kinetics of recrystallization is the

approach developed by Johnson, Mehl, Avrami and Kolomogorov (JMAK) [3]. In

JMAK theory it is assumed that the recrystallized nuclei form randomly in the pre-

existing microstructure and that the growth rate of these nuclei is constant and isotropic.

The ideal JMAK behavior is rarely exhibited by real materials. There have been several

attempts to improve the JMAK model, one of which is the Microstructural Path

Methodology (MPM) developed by Vandermeer and Rath [4]. MPM allows more

detailed information about nucleation and growth rates to be extracted from

experimental measurements than the original JMAK analysis permits. However, the

methodology is still based on the assumptions of a random nuclei distribution and

isotropic growth of the recrystallization boundaries. A theoretical method of treating

impingement caused by clustered nucleation at grain boundaries and grain edges was

presented a number of years ago by Cahn [5] and was further developed recently for

time dependent growth rates by Vandermeer and Masumura [6]. These types of

expressions have been shown to be successful in calculating the fraction of

recrystallization and the recrystallized grain size for specific circumstances. However,

one obvious problem concerning these models is the large number of empirical

parameters that have to be determined specifically for each material and pre-processing

history. Another disadvantage of these models is the lack of concern about the origin of

the recrystallized grains. Without the knowledge of the origin of the recrystallized

grains, the models remain at an empirical level and this limits the ability to understand

and control the process. Furthermore, none of the above analytical models describes the

effect of concurrent recovery on the recrystallization kinetics.

A recent development in modeling recrystallization is on the basis of an improved

understanding of the nucleation mechanism and how it is affected by microstructural

heterogeneities resulting from deformation. The topic has been reviewed by Nes and

Hutchinson [7] and Humphreys [8]. In a model developed by Nes and coworkers [9]

and a model by Sellars [10], the nucleation sites for recrystallized grains of different

Global introduction 3

crystallographic orientations, particle stimulated nucleation (PSN) and nucleation from

grain boundary regions have been incorporated. The microstructural parameters of the

deformed structure, subgrain size, sub-boundary misorientation, are used as transient

variables to correlate the recrystallization kinetics to the mechanical properties, such as

strain, stress, and strain rate. However, these models still apply the assumption that the

recrystallization kinetics follows the JMAK equation.

In the last decade, several microstructural models have been developed for simulating

the temporal evolution of the recrystallization microstructures, as well as for predicting

the recrystallization kinetics. These models can be grouped as cellular models [11-13],

computer Avrami models [14,15] and the models based on the Monte Carlo (MC) [16-

18], and Cellular Automaton (CA) [19] techniques. All these models involve inevitably

some assumptions about the nucleation of recrystallization. Although they may yield

realistic predictions of the recrystallization kinetics and the grain size distribution as a

function of various material and processing parameters, they provide little insight into

the mechanism of recrystallization and therefore lack predictive capability.

The points made here emphasize the need for a model that can predict the

recrystallization kinetics and has a firmer foundation in the physics of the nucleation

and growth of the recrystallization.

1.2. Scope of this dissertation

The overall goal of the project is to develop a physical model to predict the recovery

and recrystallization kinetics after cold and/or hot working of aluminium alloys when

the processing parameters such as the deformation strain, strain rate, deformation

temperature and the annealing temperature are known. The basic theory of this model is

described in Chapter 3. The model can be employed to describe the recrystallization

kinetics of polycrystalline metals, based on a single grain representation of the

deformed microstructure (characterized by a mean subgrain size and mean

misorientation of subgrain boundaries). A salient feature of the model is that the initial

grain size, the deformation geometry, the texture components and concurrent recovery

can be taken into account. The grain geometry is shown to have a large effect on the

recrystallization kinetics through a change in the impingement space. The

recrystallization kinetics in grains with different orientations are different because of the

inhomogenity of the deformed microstructure. The validation of the model is done on

the basis of dedicated experiments and using information from the literature. Chapter 4

is an application of the model to the recrystallization kinetics in commercial purity alloy

AA1050 after cold deformation. In Chapter 6 the model is adapted to treat the

recrystallization kinetics following hot deformation and relevant experiments to validate

the model are conducted on a commercial purity alloy AA1050.

Chapter 14

In aluminium alloys recovery and recrystallization processes are closely entangled,

since they not only often take place simultaneously, but also they have a distinct

influence on each other. As a consequence, the softening kinetics determined by means

of different methodologies may be different. In Chapter 2 several techniques are used to

quantitatively determine the softening and recrystallization kinetics. A mathematical

method is proposed to separate the effect of recovery and recrystallization.

In the literature, laboratory plane strain compression testing (PSC) is assumed to

adequately simulate hot rolling in terms of microstructure development. However, the

correctness of this assumption remains questionable. In Chapter 5 a comprehensive

study on the comparison of hot rolling and PSC by FEM calculation is reported. The

evolution of the key physical variables in both processes is found to be quite different. It

indicates that a further examination of the deformation history should be made when

applying the relationships established from PSC testing to real industrial hot rolling

operations.

Many aluminium alloys contain a mixed particle structure, which consists of a coarse

distribution of large inclusions resulting from the casting and a fine particle dispersion

due to subsequent hot rolling and annealing treatment. It has been well established that a

fine particle inhibits both the grain nucleation and growth rates while the coarse particle

(>1 µm) stimulates the nucleation rates by acting as nucleation sites. However, the

mutual interaction between the softening and decomposition processes is still poorly

understood. Quantitative description of these processes is far from being complete.

Chapter 7 and Chapter 8 deal with such phenomena taking place during

thermomechanical processes in a technological important alloy AA3003. The strain

induced precipitation kinetics are experimentally studied and modeled in Chapter 7. A

comprehensive experimental study on the softening kinetics following hot deformation

is described in Chapter 8, taking into account of the effect of the second phase particles

and the mutual interaction between the precipitation and softening. This thesis ends with

a summary in which the major results and conclusions are grouped together.

Global introduction 5

References

1. P. Cotterill and P.R. Mould, in 'Recrystallization and Grain Growth in Metals',

Surrey Univ. Press. London, 1976.

2. R.D. Doherty, D.A. Hughes, F.J. Humphreys, J.J. Jonas, D.J. Jensen, M.E. Kassner,

W.E. King, H.J. McQueen and A.D. Rollett, Mater. Sci. Eng. A, 238, 1997, 219-

274.

3. F.J. Humphreys and M. Haltherly, in 'Recrystallization and Related Annealing

Phenomena', London, Pergamon, 1996.

4. R.A. Vandermeer and B.B. Rath, Metall. Trans. A, 20A, 1989, 391-401.

5. J.W. Cahn, Acta Metall., 4, 1956, 449-459.

6. R.A. Vandermeer and R.A. Masumura, Acta Metall., 40, 1992, 877-886.

7. E. Nes and W.B. Hutchinson, in 'Proc. 10th Riso Int. Symp. on 'Materials

Architechure: the Scientific Basis for Engineering Materials'', J.Bilde-Sorensen

(Eds), Roskilde, Denmark, Riso National Labotatory, 1989, 233-245.

8. F.J. Humphreys, in 'Proc. Int. Conf. on 'Recrystallization' 90', T. Chandra (Eds),

Warrendale, PA, TMS, 1990, 113-122.

9. H.E. Vatne, T. Furu, R. Orsund and E. Nes, Acta Mater., 44, 1996, 4463-4473.

10. C.M. Sellars, in 'Thermomechanical Processing in Theory, Modeling & Practice

[TMP]2', B. Hutchinson et.al (Eds), Swedish Society for Metals Technology, 1996,

35-51.

11. H.V. Atkinson, Acta Metall., 36, 2001, 469-485.

12. V. Marx, F.R. Reher, and G. Gottstein, Acta Mater., 47, 1999, 1219-1230.

13. F.J. Humphreys, Scr. Met. Mat., 27, 1992, 1557-1562.

14. T. Furu, K. Marthinsen and E. Nes, Mater. Sci. Techn., 6, 1990, 1093-1102.

15. T. Saetre, O. Hunderi and E. Nes, Acta Metall., 34, 1986, 981-992.

16. A.D. Rollett, Prog. Mat. Sci., 42, 1997, 79-99.

17. A.D. Rollett, D.J. Srolovitz, M.P. Anderson and R.D. Doherty, Acta Metall., 40,

1992, 3475-3482.

18. D.J. Srolovitz, G.S. Grest and M.P. Anderson, Acta Metall., 34, 1986, 1833-1842.

19. H.W. Hesselbarth and I.R. Gobel, Acta Metall., 39, 1991, 2135-2152.

Chapter 16

S.P.Chen

Chapter 2

Quantification of the recrystallization

behavior in Al-alloy AA1050

A new methodology for the determination of the recrystallized volume fraction from

anodically etched aluminium alloys using optical microscopy is described. The method

involves the creation of a composite image from multiple micrographs taken at a series

of orientations. The results of quantitative analysis of images obtained by this new

method are compared with those obtained using the traditional single image optical

microscopy technique, orientation imaging microscopy (OIM) and microhardness

indentation. The multiple orientation image method is shown to consistently yield a

recrystallized volume fraction which is significantly higher than that determined from a

single image while multiple orientation imaging and OIM results are found to be in

good agreement. Furthermore it is shown that, after the subtraction of the effect of

concurrent recovery using the rule of mixtures, microhardness indentation can also be

used to determine the recrystallized volume fraction.

2.1. Introduction

In the study of the kinetics of recrystallization, it is important to determine the volume

fraction of recrystallized material as accurately as possible. Several techniques are

currently in use to quantify the recrystallization behavior of deformed metals. The most

widely used method is that of optical microscopy performed on a series of samples

recrystallized to different extents [1-4]. In addition to yielding quantitative information

regarding the extent of recrystallization, this technique provides some insight into other

microstructural features such as grain size as well as patterns of nucleation and growth.

However, when this method is applied to aluminium alloys such as AA1050, it turns out

Chapter 28

to be rather difficult to obtain precise data on the recrystallized fraction. Traditionally,

an anodic etching technique is used to reveal the grain structure and a number of

micrographs are taken of randomly selected areas. A point-counting technique is then

applied to obtain average values of the recrystallized fraction [1,3]. However, when

observed directly at a single orientation with respect to the polarisors the grain structure

of a partially recrystallized aluminium sample prepared in this manner is rarely clearly

and unambiguously visible in its entirety [5]. When the sample is rotated under

polarized light, the microstructure in a field of view seemingly undergoes a

metamorphosis in which apparently unrecrystallized regions begin to appear

recrystallized (as previously hidden boundaries become visible) and apparently

recrystallized regions begin to appear unrecrystallized (as internal substructural details

become visible). This situation leads to extra complications in the identification of

recrystallized grains with only a single micrograph. In order to eliminate such

uncertainties in the determination of the fraction recrystallized, in this paper, a new

method has been designed which is based on the construction of a composite image of a

single region produced from a set of single micrographs taken at a series of stage

rotations. This method enables the structure to be faithfully revealed and thus enables

the recrystallized fraction to be more accurately determined.

Recently, orientation imaging microscopy (OIM) has been employed to determine the

recrystallized fraction [6-9] in Al-base alloys. Although so far OIM has been

predominately applied to the determination of texture, grain boundary structure and

phase determination, an important potential application of OIM lies in the field of

recrystallization, in particular the determination of recrystallization kinetics and the

crystallographic relationships between recrystallized and unrecrystallized grains. By

OIM, it is possible to determine accurately whether an area is recrystallized. Other,

indirect methods (hardness indentations, x-ray diffraction, neutron diffraction and

electrical resistivity) have also been employed [1,10-12] to determine the

recrystallization fraction. These methods measure certain effects of microstructural

changes on the properties and provide only average values including both recovery and

recrystallization effects. Nevertheless, if the effects of recovery and recrystallization can

be separated such techniques may provide valuable additional information regarding

recrystallization behavior. In this study OIM and microhardness measurement have

been employed along with the optical microscopy techniques described above to

determine the recrystallization fraction in the commercially pure AA1050 alloy. The

results so obtained are compared and critically discussed.

Qualification of the recrystallization behavior 9

2.2. Experimental details

2.2.1. Material preparation

The chemical composition of the commercially pure aluminium alloy AA 1050 used is:

0.185 wt.% Fe, 0.109 wt.% Si and Al in balance. The material contains plate-shaped

intermetallic compounds FeAl3, which have an aspect ratio in the range of 1 to 6. The

size of FeAl3 particles ranges between 0.2 and 7 µm.

In order to produce material in a suitable form for further experimentation a cast ingot

of AA1050 was hot rolled in 19 passes, resulting in a reduction in thickness from 500

mm to 4 mm. The hot rolling processes started at 520°C and finished at 305°C. The hot-

rolled material was annealed at 600°C for 2 h and then quenched into water. The

material was again heated to 400°C and held at this temperature for 2 h to reduce the

content of iron in solid solution. The average grain size after this treatment was 90 µm.

The materials were finally cold rolled to a reduction in thickness of 50% (from 4 mm to

2 mm). The rolled sheet was cut into small samples of 20×15×2 mm. The samples were

annealed in a salt bath at 340°C for times ranging from 30 s to 3 h and then quenched

into water to obtain different extents of recrystallization. The temperature of the salt

bath was controlled to within an accuracy of 2°C. The time taken for a specimen to

reach the set temperature was approximately 5 seconds.

2.2.2. Microstructural characterization

The optical microscopy and OIM as well as micro-hardness examinations were carried

out on the section parallel to the RN plane (R rolling direction and N surface normal) to

encounter as many grain boundaries as possible. After standard sectioning and

polishing, specimens for optical metallography were etched anodically with Barker’s

reagent (1% HBF4 aqueous solution) [13] at 20V for approximately 120 s depending on

the annealing time. For OIM scanning, the specimens were etched with Keller’s reagent

for 30 seconds. In order to account for inhomogeneities in the microstructure in the

through thickness direction of the sheet, all the metallography and hardness

measurements were performed at locations along the center line of the sheet, in a band

covering approximately one third of the thickness of the sheet.

The optical microscopy examinations were performed using a NEOPHOT inverted

stage metallurgical microscope. Both a standard point-counting technique and a LEICA

QUANTIMET digital image analysis facility were used to evaluate the recrystallized

fraction. In all cases measurements were conducted on five separate areas. A Buehler

OMNIMET MHT automatic micro hardness tester was used for the microhardness

measurements. The micro hardness of each specimen was determined, using 50 g load

Chapter 210

and 15 s loading time. Hardness tests were made after polishing and 18 measurements

were taken on each specimen.

Orientation imaging microscopy (OIM), developed by TexSEM Laboratories Inc., was

integrated with a Philips XL-30s FEG scanning electron microscope (SEM) and

employed for the combined microstructural and crystallographic analysis. By a fast

procedure of capturing and processing electron back-scatter diffraction patterns

(EBSPs), the OIM system produces thousands of orientation measurements, linking

local lattice orientation with grain morphology. Each measurement is represented by a

pixel in the orientation micrographs, to which a color or gray scale value is assigned on

the basis of the local details of lattice orientation or the quality of the corresponding

EBSP image quality (IQ). Recrystallized grains are distinguished according to IQ and

orientation spread. The volume fraction of recrystallized material was determined from

the ratio of the area of the recrystallized grains to the area of the whole image. In this

work the area of a typical OIM scan was about 1800×600 µm², containing

approximately 130 original grains.

2.2.3. Composite image method

In order to construct composite images a total of four micrographs at 20° stage rotation

intervals were taken from each area. The rotation center of the stage should be aligned

perfectly with the objective lens so that no misalignment of the micrographs results. The

following step was to trace all visible recrystallized grains from each of the micrographs

onto an acetate sheet in order to construct a single representation containing all

recrystallized regions. To reveal the contribution to the total volume fraction made by

each rotation, a separate color was used to trace the visible recrystallized structure from

each individual micrograph. The border of each micrograph was also traced in order to

define a common overlaid area. Quantitative metallography (a standard point counting

technique) was then performed using the composite image. In this study the criteria used

to determine whether a grain is recrystallized were the following:

Size: Grains with an area considerably smaller (<10%) than the original deformed

average grains were taken to be potentially recrystallized grains (when the fraction

recrystallized is small).

Polygonality: Polygonal grains with sharp triple points and straight boundary edges

were counted as potentially recrystallized grains.

Equiaxiallity: Grains with low aspect ratios were taken to be potentially recrystallized

grains. In this study those grains having an arbitrarily chosen aspect ratio up to and


including 1.75 are considered to be recrystallized, the reminder being classed as

unrecrystallized.

Absence of a visible internal structure: Absence of deformation bands was taken as

an additional indication of recrystallized grain character.

A further development of the technique is to use color micrographs in an attempt toreduce the apparent metamorphosis. By adding a 1/4 filter, grains with different

orientation will exhibit different colors under polarized light. This facilitates a better

identification of the recrystallized grains. In this investigation, similar composite images

were composed from color digital micrographs according to the procedure described

above. The unrecrystallized material can be extracted from the digital image. The

fraction of the recrystallized area, the recrystallized grain sizes (defined as square rootof grain area divided by / 4 ) of individual grains and grain size distribution were then

calculated using automated areal analysis.

2.3. Results

2.3.1. Microstructural evolution

The deformed structures show that the grains in the central part of the rolled sheet

follow the same shape change as the whole specimen. The grains are elongated in the

rolling direction and exhibit a pancake shape with a thickness of 45 µm on average, i.e.

half of the initial grain size. The triple point angles among the grains are distorted and

not equal to 120°. The as-deformed grain structure showed non-uniform colors and

somewhat unclear grain boundaries. The non-uniform colors reflect the local variation

in the disturbance of the crystal created during deformation.

After annealing the deformed sheet for 5 minutes at 340°C, the triple point angles

tended to return towards 120°. A few nuclei were observed at the pre-existing grain

corners and along the grain boundaries. The newly formed recrystallized grains

exhibited a uniform color within the grains and sharp grain boundaries. There were

some indications of strain-induced migration of pre-existing grain boundaries (SIBM).

As the annealing time increased to 10 min, the pre-existing grain boundaries became

corrugated. More nuclei could be seen at the pre-existing grain corners or along the

grain boundaries. In some cases clusters of nuclei could be observed. Fig. 1 shows the

nuclei along the boundaries of the pre-existing deformed grains and the SIBM in

regions close to grain edges. After 20 minute annealing, evidence of strain-induced

migration of pre-existing grain boundaries is obvious. The number of nuclei is still

increasing and at the same time nuclei coalescence is observed. Some of the deformed

grains were found to have many more nuclei than others whilst some of them were still

Chapter 212

nuclei free. When annealed for 30 minutes or longer, the density of the nuclei remained

constant. However, the size of the recrystallized grains increased with annealing time.

No nuclei were found around FeAl3 particles.

Fig. 1. Optical micrographs showing the microstructure of AA 1050 recrystallized at

340°C for (a) 10 min and (b) 60 min. Arrows indicate the SIBM of region close to the

pre-existing grain boundaries.

2.3.2. Composite imaging analysis

Fig. 2 shows a series of color micrographs taken at 20° rotation intervals of a sample

after annealing 50 min at 340°C. As can be seen, the area labeled A in Fig. 2a seems to

be one grain but it is clear in Figs. 2b and 2c that it actually consists of two grains. One

may easily identify the grain boundary at position B on Figs. 2b and 2c, but it is hardly

visible in Fig. 2a. The difference among the individual micrographs is a clear

illustration that it is difficult to determine an accurate recrystallized fraction from a

single micrograph.

The areal fractions recrystallized obtained from single and composite images using the

point counting technique are displayed in Fig. 3. It can be seen that consideration of a

single image leads to a significant underestimate of the volume fraction of recrystallized

material. For all samples studied the composite images constructed from three

orientated micrographs yield a volume fraction about 50% higher than estimated from a

single micrograph. For the samples under investigation here, more than three

micrographs do not show a significant modification to the composite images and

consequently do not lead to a further increase of the measured fraction. The points in

Fig. 3 represent the mean values, and the error bars (shown only for the measurements

on the composite images composed from three micrographs) indicate the spread of the

individual measurements.


Fig. 2. Multiple micrographs showing the microstructural metamorphosis at place A

and B when rotating the sample annealed at 340°C for 50 min.

Chapter 214

Fig. 3. Recrystallized fraction determined from single and composite images using a

point counting technique, showing the effect of multiple orientated micrographs.

Fig. 4 shows a histogram of the recrystallized grain size at various times obtained by

composite images combined with the automated areal analysis technique. It seems that

the recrystallized grain sizes in the partly recrystallized material follow an

approximately log-normal distribution. The maximum equivalent diameter of the

recrystallized grain, dmax, and the mean grain size, dmean, are employed to describe the

grain size evolution, which is shown in Fig. 4d. Both curves demonstrate that the

growth rate of the recrystallized grains decreases with annealing time and the difference

between the two curves becomes larger. As can be seen in Fig. 1, the impingement

between the clustered grains may prevent them from growing further. This explains the

big difference between dmax and dmean at the later stages of recrystallization.

2.3.3. OIM observation

Fig. 5 shows an OIM image of the same sample as in Fig. 2. The left part in Fig. 5 is the

same region as in Fig. 2 in which there is a nearly perfect grain match with the

conventional microstructure (a mirror image). The black lines indicate high angle grain

boundaries with a misorientation larger than 15°, and the thin blue and gray lines in a

deformed grain represent sub-grain boundaries with a misorientation angle between 10-

15° and 4-9°, respectively. From the combined information of Inverse Pole Figure (IPF)

maps and IQ maps, recrystallized grains can easily be distinguished from non-

recrystallized grains. The recrystallized grains are uniform in color, which means that

the spread in orientation within the grains is quite small, and free from low angle sub-

grain boundaries. In contrast, the non-recrystallized grains contain deformation

substructures as revealed in slightly different colors and low angle sub-grain boundaries.

0

0.2

0.4

0.6

0.8

1

0 2000 4000 6000 8000Time (s)

Fra

ction r

ecry

sta

llized

Single micrograph

Two micrographs

Three micrographs


Correspondingly, the recrystallized grains are much brighter than the non-recrystallized

grains on the IQ map. This is indicative of a high degree of crystal perfection.

Fig. 4. Size evolution of recrystallized grains vs annealing time.

Fig. 5. OIM map (IPF) showing the microstructure and boundaries with different

misorientation levels in the same sample as in Fig. 2.

20 min

0

0.05

0.1

0.15

0.2

0.25

0.3

3 10 17 25 35 40 MoreGrain size (µm)

Fre

quency

a

90 min

0

0.05

0.1

0.15

0.2

0.25

0.3

8 26 48 70 92 MoreGrain size (µm)

Fre

qu

en

cy

c

50 min

0

0.05

0.1

0.15

0.2

0.25

0.3

4 16 30 44 58 MoreGrain size (µm)

Fre

qu

ency

b

0

20

40

60

80

100

120

0 1000 2000 3000 4000 5000 6000Annealing time (s)

dm

ax, d

me

an

(µ

m)

dmean

dmax

d

Chapter 216

Fig. 6 shows the recrystallized fraction determined from OIM for several annealing

times. The results obtained with the optical microscopy technique (the manual point

counting and automated area analysis) are also shown for comparison. Here the attachederror bars are only shown for the area analysis (error is within 5% ). The values from

the point counting technique are lower than those from the areal analysis and both

results are slightly lower than the data from OIM but still within the experimental

scatter of areal analysis data.

Fig. 6. Comparison of the fraction of recrystallized material determined by composite

image method and by OIM.

2.3.4. Micro-hardness measurements

The micro hardness (HV) measurements after isothermal annealing at 340°C for various

times are shown in Fig. 7a. From the error bars it can be seen that there is a large scatter

between the individual hardness measurements. However, a clear overall pattern of

annealing behavior can be observed. Similar to the optical microscopy determination,

the sizes of the error bars are larger in the intermediate time range, which demonstrates

the inhomogeneous character of the partially recrystallized material. It can be seen that

the hardness curve from the onset of annealing to the point that represents 2%

recrystallized material (according to optical microscopy examination) can be

represented by a straight line. The slope of the curve increases when the

recrystallization process starts. It indicates a rapid decrease of hardness with increasing

fraction of recrystallized material. The microhardness of a sample directly after coldwork (Hm) is 46.8 and that of the fully recrystallized material ( 0H ) is 24.5. The

0

0.2

0.4

0.6

0.8

1

0 2000 4000 6000 8000Time (s)

Fra

ctio

n r

ecry

sta

llise

d

By point counting

By area analysis

By EBSD


softening fraction, R , of an annealed sample could be calculated from these two values

as follows [1,15]

0

m

m

H HR

H H (1)

where H is the measured hardness. R calculated according to formula (1) is plotted in

Fig. 7b (diamond symbol). Obviously these data include the softening effects

contributed by both recovery and recrystallization, as will be discussed later.

Fig. 7. Microhardness measurement (a) and the derived fraction of recrystallized

material before and after subtraction of recovery effects (b).

0

0.2

0.4

0.6

0.8

1

10 100 1000 10000 100000Time (s)

Fra

ctio

n s

oft

en

ed

Including prior andconcurrent recovery

Excluding prior recovery

Excluding prior recoveryand concurrent recovery

b

102

103

105

104

101

20

25

30

35

40

45

50

10 100 1000 10000 100000Time (s)

Mic

ro H

V

lnrec mH H B t

a

104

101

103

105

102

Chapter 218

2.4. Discussion

2.4.1. Composite image method

The work was aimed at accurately quantifying the fraction of recrystallized material

using optical microscopy. As demonstrated in Fig. 2, taking only a single image can not

provide complete information of the partially recrystallized microstructure. If two

neighboring grains are oriented in such a way that certain lattice planes of both lie

roughly perpendicular to the direction of the polarized light, they will exhibit a similar

color. Therefore, some recrystallized grains may not be distinguished from the

deformed parent grain in a single micrograph. As a result, the measured fraction may be

underestimated. Here it is shown that this problem can be easily overcome by using

composite micrographs produced by repeatedly rotating the sample (by e.g. 20°) around

the optical axis. Those grain boundaries that are not shown in a single micrograph can

be revealed in others. By this new method the characterization of recrystallized grains is

more reliable and a more accurate recrystallization fraction is obtained.

2.4.2. Comparison of optical and OIM techniques

The recrystallization fractions determined by both optical microscopy and OIM show

the same course of transformation. Measurements by OIM yield slightly higher values

of the fraction, especially at shorter annealing times. This is due to the resolution

difference between the two measuring systems. The smallest recrystallized grain size

that can be clearly observed by the optical microscopy under polarized light is of the

order of 2-5 µm. However, the spatial resolution of EBSP is less than 1 µm. Recovery

will not affect the results obtained by OIM and optical microscopy, as (1) recovered

regions still contain fairly small subgrains; (2) no significant lattice reorientation occurs

(no new high angle grain boundaries are created); (3) only a misorientation angle larger

than 15° is classified as recrystallized. Recovered regions will therefore be registered as

deformed material.

2.4.3. Effect of recovery

The hardness measurements show a slightly different annealing behavior than derived

from the optical and OIM experiments. The hardness value is proportional to the flow

stress of the material, which is strongly dependent on the dislocation density. This

measurement should reflect both recovery and recrystallization effects. Determination

of recrystallization kinetics from hardness measurements requires separation of the

effects associated with concurrent recovery and recrystallization. Assuming that the

isothermal recovery kinetics follows a logarithmic decay relationship, the time

dependence of micro hardness is then given by the following expression [14,15]

lnrecH A B t (2)


Where Hrec is the hardness of an annealed sample if only recovery takes place, 1A H ,

the hardness at 1 s annealing, and B, the slope of the straight line, as seen in Fig. 7a.

If the measured hardness, H, of the partially recrystallized materials follows the rule of

mixtures it holds that

0(1 ) recH f H f H (3)

Where f is the volume fraction of recrystallized grains. Assuming that the kinetics of

concurrent recovery follow a straight line when recrystallization starts, by substituting

equation (2) into (3), the recrystallization fraction excluding prior and concurrent

recovery can be obtained as shown in triangle labels in Fig. 7b. If it is assumed that

there is no concurrent recovery taking place after recrystallization starts, by setting

Hrec=40.5 the hardness at the start point of recrystallization (doted line in Fig. 7a), one

may exclude the prior recovery effect and obtain the recrystallized fraction only, as

shown in Fig. 7b (square label). The recrystallized fraction with exclusion of only the

prior recovery effect is about 10% higher than the value with exclusion of both prior

and concurrent recovery effects. This also indicates that the concurrent recovery effects

are significant during the early stages of recrystallization. By comparing the three

curves in Fig. 7b, one can see that taking into account recovery is indeed necessary to

properly determine recrystallization fraction. After subtraction of the recovery effect the

resulting fraction recrystallized is in good agreement with the metallographic

measurements, as shown in Fig. 8. The difference is within 5%.

Fig. 8. Comparison of fraction of recrystallized material quantified by micro-hardness

tests and optical microscopy on the base of a composite image method.

0

0.2

0.4

0.6

0.8

1

10 100 1000 10000 100000Time (s)

Fra

ctio

n r

ecry

sta

llize

d

By Micro Hardnessexcluding prior andconcurrent recovery

By Optical Microscopy

102

105

104

103

101

Chapter 220

2.5. Conclusions

1. A new method for characterizing the microstructure in AA1050 by optical

microscopy is presented. Combining this method with an areal analysis technique yields

a more accurate measurement of the fraction of recrystallized material, which is in

excellent agreement with OIM measurements.

2. Microhardness measurements demonstrate another course of annealing behavior,

which includes both effects associated with concurrent recovery and recrystallization

processes. After subtraction of the effect of the concurrent recovery, the micro hardness

test can provide a reasonable quantification of the recrystallized fraction developed with

annealing time.

References

1. E.C.W. Perryman, Trans. AIME, J. Metals, Sept. 1955, 1053-1061.

2. P.L. Orsetti Rossi and C.M. Sellars, Acta Mater., 1997, 45, 137-148.

3. R.A. Vandermeer, Metal. Trans., April 1970, 1, 819-826.

4. P. Faivry, R.D. Doherty, J. of Mater. Sci. 1979, 14, 897-919.

5. C.N. Sparks, PhD thesis, University of Sheffield, 1993.

6. M.P. Black and R.L. Higginson, Scr. Mater. 41, (2), 1999, 125-129.

7. E. Woldt, D. Juul Jensen, Metall. Trans., 26A, July 1995, 1717-1724.

8. O. Engler, G.Gottstein, Steel Research, 63, 9, 1992, 413-418.

9. D.J. Dingley and K. Baba-Kishi, Scanning Electron Microsc., 1986, 27, 383-395.

10. T. Furu, R. Orsund and E. Nes, Acta Met. Mater., 43, 1995, 2209-2232.

11. D. Bowen, R.R. Eggleston and R.H. Kropschot, J. Applied Physics. 1952, 23 (6),

630-635.

12. K. Mukunthan and E.B. Hawbolt, Metall. Mat. Trans. A 27, Nov. 1996, 3410-3423.

13. F. Li and P.S. Bate, Acta Metall. Mater. 39, 11, 1991, 2639-2650.

14. J.G. Byrne: "Recovery, Recrystallization and Grain Growth", MacMillan Co., New

York, NY, 1965, 37-59.

15. F.J. Humphreys and M. Hatherly, “Recrystallisation and Related Annealing

Phenomena”, Pergamon, London, 1996.

S.P. Chen

Chapter 3

Modeling the kinetics of grain boundary

nucleated recrystallization processes after

cold deformation using a single grain

approach

A physical model to predict the recrystallization kinetics of single phase polycrystalline

metals, based on a single grain representation of deformed microstructure (characterized

by a mean subgrain size and mean misorientation of subgrain boundaries), is presented.

The model takes into account the grain geometry, the position and the density of the

nucleation sites. The selected geometry is a regular tetrakaidecahedron, combining

topological features of a random Voronoi distribution characteristic for polycrystalline

material with the advantages of a single-grain calculation. The model employs

empirically determined relationships from existing literature to describe the deformed

microstructure and in so doing, facilitates the prediction of the recrystallization behavior

when only the deformation strain and the recrystallization temperature are known. The

boundary mobility and the driving force as well as the nucleation density are related to

the true plastic strain of deformation through the microstructure. The model also

describes the effects of concurrent recovery on the overall recrystallization kinetics.

3.1. Introduction

Recrystallization has long been an important subject for metallurgical research [1]. The

microstructural evolution during recrystallization can be described phenomenologically

as a nucleation and a growth process. The basis for the theoretical treatment of the

recrystallization transformation kinetics is the JMAK equation which is based on the

Chapter 322

assumptions that the recrystallized nuclei form randomly in the pre-existing

microstructure and that the growth rate of these nuclei is constant and isotropic. The

modeling process is implemented by posing the nucleation and growth characteristics of

the product phase based on the relationship between the volume fraction transformed,X , and the extended volume fraction, exX :

(1 ) exdX X dX (1)

The ideal JMAK behavior is rarely exhibited by real materials [1-3].

It is widely recognized that the nucleation sites in recrystallization are non-randomly

distributed [4,5]. Nucleation tends to occur at preferred sites such as prior grain

boundaries, edges and corners or inhomogeneities as transition bands and shear bands.

Indeed, grain boundary nucleation has been frequently observed [1,6,7] and it plays a

dominant role in recrystallization kinetics when the initial grain size is small, or at lower

strains. Another important issue [8-10] that arises from the analysis of experimentally

determined recrystallization kinetics is that the average interface migration rate is often

found to decrease with time.

For cases where the critical assumption of randomness is violated, the JMAK equation

can not be employed directly and an alternative approach to the impingement problem is

necessary. A theoretical method of treating impingement caused by clustered nucleation

at grain boundaries and grain edges was presented a number of years ago by Cahn [11]

and was developed recently for time dependent growth rate by Vandermeer and

Masumura [12]. In his analysis, which was based on Eq. (1), Cahn considered two

separate types of impingement: first, impingement among nodules originating from the

same grain boundary (or edge), and second, the impingement between nodules

originating from different boundaries or edges. Eq. (1) is, however, only an

approximation which applies when nucleation occurs at the heterogeneities. A more

accurate description of the kinetics and the resulting microstructure in these cases can

be obtained by computer simulation [1,13].

The goal of the present program is to develop a physical model to predict the

recrystallization kinetics when only the deformation strain and annealing temperature

are known. Since deformation is a necessary precursor for both nucleation of

recrystallized grains and sustained growth during recrystallization, the distribution of

nucleation sites is strongly dependent upon both the deformed grain geometry and the

defect structure induced by deformation. As a first step, this chapter deals with

recrystallization kinetics of the grain boundary nucleation with a simplified model. The

influence of the grain geometry, nucleus site density and the relative positions of the

Modeling the recrystallization kinetics of the grain boundary nucleation 23

nuclei inside the original grain on the kinetics of the recrystallization will be considered

in detail. The boundary mobility and the driving force as well as the nucleation density

will be related to the true plastic strain of deformation through the microstructure. The

effects of concurrent recovery and the starting grain size on recrystallization kinetics

will be also explored.

3.2. The Kinetic Model

3.2.1. Grain Geometry

The mathematical concept that models the recrystallized microstructure (equiaxed

grains) most ideally consists of a construction of randomly chosen Voronoi cells. The

construction of a Voronoi tessellation of space shows a strong resemblance to the

evolution of a real microstructure. It has been demonstrated that a reasonable

correspondence exists between the geometrical parameters for the average values for the

Voronoi tessellation and the tetrakaidecahedron single grain [14]. Therefore, the

microstructure of an well-annealed material can be described easily by a distribution of

tetrakaidecahedra of various sizes. A number of studies has shown that this single grain

representation with discrete nucleation sites offers a new approach to modeling the

phase transformations in steels and ferrous alloys and a model for treating growth of a

ferrite into austenite was developed [14-16]. It will now be briefly summarized and

adapted to recrystallization.

In this phase transformation model, the austenite grain is described as a regular

tetrakaidecahedra. The ferrite grains are assumed to nucleate from the austenite grain

boundary. After nucleation the growth of each nucleus is modeled as an expanding

sphere and is controlled by interface velocity, which depends on the driving force and

interface mobility. In the case of recrystallization, there is no phase transformation.

However, since the deformed metal stores some energy in the form of various types of

imperfections, the deformed state can be considered as a phase that has a higher energy

than the fully recrystallized phase. In this sense, the recrystallization can be treated as if

a new phase would form. For a phase transformation, the driving force is the difference

in the Gibbs energy between two phases, and for recrystallization, this comes from the

stored energy. On the other hand, the recrystallization process is recognized to be the

motion of a boundary. This is true not only for the growth stages but also for the

nucleation stage because the formation of a mobile recrystallized front, i.e. nucleation,

results from subgrain growth, subgrain coalescence or strain-induced grain boundary

migration. Therefore, that the concept of the transformation is controlled by boundary

velocity is also employed in the case of recrystallization. Furthermore, it is well known

that the new recrystallized grains are frequently nucleated from the original grain

boundaries, the area around the second phase particles and deformation inhomogeneities

Chapter 324

such as deformation bands [1]. For pure aluminium without second phase particles

under intermediate deformation strains the grain boundary nucleation should be the

dominant mechanism.

Based on the above analysis, the concept of the previous phase transformation model is,

in principle, also applicable to the recrystallization process. To investigate the kinetics

of recrystallization, we first study the behavior of a single deformed tetrakaidecahedron.

Homogenous deformation not only changes the shape of the tetrakaidecahedron but also

leads to an increase in the grain surface area per unit volume, while there is no change

in the number of grain corners per unit volume. The deformation of a

tetrakaidecahedron can be described by a 3 3 deformation matrix [17], which operates

on each vector in turn to generate a set of new vectors defining the new shape. Thus a

vector, u, becomes a new vector, v, as a consequence of a homogenous deformation S:

3

2

1

00

00

00

u

u

u

c

b

a

Suv (2)

where iu are the components of u; and a , b , and c are the principle distortions (ratios

of the final to initial lengths of the unit vectors along the principal axes ( , ,x y z )).

Therefore, aln , bln and cln are the true strains along the three principal axes of the

deformation and 1abc . Rolling involves plane strain deformation with 1b and

1ac .

The three-dimensional construction, as shown in Fig. 1, is used in the simulation. This

has a reference frame of rolling direction (RD), transverse direction (TD), normaldirection (ND) with RD and ND along x and z axes, respectively. The length of the

undeformed cube side, d , the distance between two opposite square faces of the

tetrakaidecahedron, is defined as the grain size prior to the deformation. In the

numerical calculations the cube is divided into volume elements that are mapped onto a

three dimensional matrix. The number of elements used to describe the cube is( NNN ,, ). After a deformation the number of the elements takes the integer of

( cNbNaN ,, ). The aspect ratio of grains in the direction RD/ND is ca / depending on

the strain of the rolling deformation. The recrystallized nuclei can be randomly

distributed in the deformed grain or be arranged on the grain boundary or corners. In the

model it is assumed that grains grow independently of one another. After eachnucleation event, the nucleus grows in a spherical manner with a growth velocity V

perpendicular to the interface, which is a constant in space but may vary with time. The

simulated geometric grain is permitted to grow beyond the faces of the

tetrakaidecahedron, but the volume of the new grain is computed only for the portion of


the grain contained within the tetrakaidecahedron. An approach, which is comparable

with the time cone method of Cahn [18], is used to calculate the volume fractiontransformed. We calculate the probability that the element ( , , )x y zX in the volume is

untransformed at time t . At each time step, the radius of each nuclei, ( , )R t , and the

distance between each element and the center of every nuclei, X' , are calculated andcompared. If the magnitude of the grain radius ( , )R t exceeds the distance between X'

and X :

22( , ) X X' 0R t (3)

the element X will have transformed before time, t . The number of untransformed sites

is then counted to calculate the transformed fraction and from that the transformation

rate.

Fig. 1. A deformed tetrakaidecahedron in a cuboid used as computer specimen.

3.2.2. The deformed microstructure

In a high stacking fault energy metal such as aluminium it is assumed that theapplication of a strain of ~ 0.2 results in a uniform equiaxed cellular structure [19,20].

These cells or subgrains are slightly misoriented with respect to each other. Forintermediate rolling strains ( 0.2 1.5 ), the substructure is inhomogeneous, with the

size and the shape of the cells/subgrains varying from region to region in the material.

x

z

y

RD

ND

TD

Chapter 326

At larger strains ( 1.5), the substructure becomes more complex but more spatially

homogeneous.

It has been demonstrated that mean subgrain size and misorientation depend on the true

deformation strain [19-21]. The variation in the average subgrain/cell size with strain

follows a relationship that is of a similar type for a range of materials, and independent

of the deformation mode. The relationship between the mean cell/subgrain size, , and

deformation strain in aluminium is given by [21]:

0.35 0.17 / (4)

where is the true strain, for rolling deformation ln a , and is the mean subgrain

diameter in m.

In the case of aluminium the average subgrain boundary misorientation, , is reported

to increase rapidly with strain, reaching about 2°-3° at a strain of about 1, after which it

remains constant up to rolling strain as high as 4 [20]. The following relationship

between mean subgrain diameter and misorientation has been established for pure

aluminium deformed by cold rolling [22]:

/ b C (5)

where b is the Burger’s vector and C is a constant between 60~80 rad.

For aluminium, the mean subgrain size and mean misorientation of the subgrain

structure as a function of the true strain are shown in Fig. 2. It can be seen that the

subgrain size decreases while the mean misorientation increases as strain increases.

Both saturate at higher strains.


Fig. 2. The variation of the mean subgrain size and misorientation with deformation

strain.

3.2.3. Nucleation

In this chapter, nuclei are assumed to be present at pre-existing high angle grain

boundaries after deformation and only the kinetics of grain boundary nucleated

recrystallization is considered. Other potential nucleation sites such as deformation

bands and shear bands increase in significance when the deformation true strain

becomes greater than 1.3 [18]. The present study on strictly grain boundary nucleated

recrystallization is limited, therefore, to the range of small to intermediate deformation

strains.

The probability of finding a critical-sized subgrain on the grain boundary depends on

the average subgrain size, , and the grain boundary area per unit volume, vS . The

nucleation site density vN in the grain boundary can be estimated using the expression

[23,24]:

2/V d vN C S (6)

where dC is a calibration constant which determines the potency of the grain boundary

as a nucleation site ( dC , in the present work, is taken as 42.5 10 ).

Deformation can increase the nucleation site density for recrystallization through anincrease in grain boundary area per unit volume, vS , of the deformed structure. The

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.5 1 1.5 2 2.5True strain

Mis

orie

nta

tio

n (

° )

0

0.5

1

1.5

2

2.5

Sub

gra

in s

ize (

µm

)

Chapter 328

value of vS is dependent on the rolling reduction strain and the initial grain size, and for

a tetrakaidecahedron, varies with rolling strain in a manner described by [17]:

2 2 2 213 1 2 / 3 2 / 2 2 /

2vS a a a a a a a

d (7)

where a e is the distortion along the rolling direction.

Fig. 3 shows the variation of the nucleation density with true strain for three differentinitial grain sizes. As the strain increases the nucleation site density vN increases

significantly as a result of both an increase in vS and a concurrent decrease in .

Fig. 3. The nucleation site density as a function of true strain.

3.2.4. Growth

The migration of low and high angle boundaries is the most important atomic-scale

mechanism that occurs during recovery and recrystallization of the deformed materials.The relation between the rate of migration of the interface, V , and the driving pressurefor boundaries with a specific energy and mobility, nM , moving into a uniformly

deformed matrix is given by the simple expression:

nV M P (8)

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

True strain

Nu

cle

atio

n s

ite

de

nsity (

a.u

)

d=50µm

100

150


where P is the driving pressure which is assumed to arise from the stored energy of the

dislocations in the subgrain boundaries and in the subgrain interior.

While a real deformed microstructure contains subgrains of a variety of sizes and

misorientations, in the present model, following the approach of Humphreys [25], we

simplify the analysis by considering the substructure to be adequately described using

two components. The major component is considered as an assembly of equiaxed

subgrains of mean equivalent radius, R , mean misorientation, , and with boundariesof mean energy and mobility, and M respectively. The minor component we

consider as “particular” subgrains (effectively sub-critically sized recrystallizationnuclei) which have a larger size ( nR ) and different boundary characteristics ( , ,n n nM )

to that of the “average” subgrain assembly.

The total energy of a particular subgrain as a function of radius R is given by:

34

3n n vE A R E (9)

where 2~ 4 nA R is surface area of the particular subgrain and vE is stored energy in the

average assembly estimated by [1]:

2/ / 2v iE R Gb (10)

The driving pressure for this particular subgrain to grow is given by the relative changeof the total energy when the dislocation density inside cell, i , can be ignored.

21 n

n n

dEP

A dR R R (11)

where is a geometrical constant and has a value of ~1.5 [1].

If the main mechanism of recovery in the subgrain assembly is subgrain coarsening, the

growth rate of a uniform subgrain assembly may be expressed in the form [1,26]:

dR M

dt R (12)

As a simple approach, the boundary energy is assumed to be dependent only on the

average misorientation angle and is given by [1]:

Chapter 330

1 lnm

m m

(13)

where m and m are the values of boundary energy and misorientation for high-angle

boundaries and which are commonly taken as 0.324 J/m² [1] and 15° respectively.

Subgrain boundary mobility is assumed to be related to and the limited data in the

literature shows that the M vs curve is sigmoidal [25,29,30]. In this analysis, the

following empirical form is taken:

( )

1n

m

B

nM M e (14)

where nM is the mobility of a high angle boundary at an annealing temperature given

by:

0 exp /n gM M Q R T (15)

where 0M =1.78 102 m4 /Js and Q =147 kJ /mol [27,28], gR , gas constant, 4n and

5B for aluminum [25]. It is suggested [25] that Eq. (14) should be applicable over the

temperature range 250 - 400 C. It should be noted that Eq. (14) is an empiricalrelationship and M is very sensitive to the values of B and n .

Up to now the model has dealt with the recrystallization kinetics of a single deformedgrain. As long as the physical parameters, dC , 0M , Q, B and n could be experimentally

determined, the recrystallization kinetics in an as-deformed grain with size, d , which is

given as a data array of the fraction recrystallized, f, vs time, t, could be predictednumerically provided the deformation strain, , and the annealing temperature, T , are

known.

Actually, the starting grain size distribution is also an important parameter for the

recrystallization kinetics. Experimental measurements [31,32] have shown the grain size

distribution in annealed materials to be approximately log-normal, where the maximum

grain size is typically 2.5~3 times mean size. To determine the overall recrystallization

kinetics in a log-normal polycrystalline aggregate, we treat it in the following way: thedistribution function of the starting grain size is divided into j equal size classes in the

interval max0 / 2.5 ~ 3d d (depending on the standard deviation). For each of the


size classes, the average grain size id and frequency i are calculated. A data array of

the fraction if vs time t for each of the j size classes is computed from the single grain

model. Each class is operating in a monolithic body without reference to other size

classes. The overall fraction recrystallized f is obtained by weighted summation over the

j classes through Eq. (16):

( )i i if f (16)

where i is the frequency of the initial grain with diameter of id , if is the fraction

recrystallized of a class of the grains with diameter of id at annealing time, t .

3.3. Results and discussion

3.3.1. The effect of grain geometry on the recrystallization kinetics

In real materials, the space into which the recrystallization nuclei can grow is effectively

defined by both initial grain size and the deformation geometry. Ideally, the starting

grain structure prior to deformation is isotropic and essentially uniformly sized. As

deformation ensues, the grain structure is constrained to conform to the deformation

geometry. In this section a number of simulations of the recrystallization kinetics will bepresented, in which all the nucleation takes place at time, 0t , and the interface

velocity is assumed to remain constant throughout the annealing cycle and to be

independent of the applied deformation strain. Thus, only the influence of the grain

geometry resulting from various degrees of deformation, the position of the nuclei and

the nucleation site density on the recrystallization kinetics will be considered anddiscussed. The results are presented as a function of the dimensionless reduced time *t ,

which is defined as:

* Vt t

d (17)

where d is the starting grain size. The reduced time parameter indicates that the actual

transformation time depends on the ratio dV / rather than solely on V .

Let us first consider an ideal case, assuming six nuclei either each located at one of the

corners of the six quadrilateral faces or at the center of the quadrilateral faces of a

tetrakaidecahedron, to see the effect of the grain geometry on the recrystallization

kinetics. The calculated kinetics curves, giving the recrystallized fraction as a function

of reduced time after various strains, are shown in Fig. 4. Curves in thin line represent

the cases where each of the six nuclei is at a corner of the quadrilateral faces while the

Chapter 332

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Reduced time t*

Fra

ction r

ecry

sta

llized

1 2

3 4

Fig.4. S.P.Chen

curves in thick line are the cases where each of six nuclei is at the center of a

quadrilateral face.

Fig. 4. Calculated recrystallization kinetics for six nuclei each at one of quadrilateral

faces after various rolling strains. 1. 0 , 2. 0.69 , 3. 1.1 , 4. 1.6 (for

details, see text).

It is clear from these simulations that the geometrical shape change caused by the

deformation exerts a large effect on the overall recrystallization kinetics with the rate

becoming slower as deformation strain increases. The reduced rate is a result of the

elongation of the tetrakaidecahedron in the rolling direction and the concomitant change

in the relative positions and distances between corners and faces. The distance between

two opposite quadrilateral faces of the tetrakaidecahedron along the rolling direction

increases whilst along the normal direction it decreases. Early impingement between

growing grains along the normal direction in the deformed structure, therefore, occurs

and this is intensified with increasing deformation strain.

The shape of the curves in Fig. 4 changes with increasing deformation strain. Anevolution of the shape of the curves from the typical sigmoidal form ( 0 ) to the

truncated sigmoidal shape (after a small deformation) and to a straight line with a kink

(at large strains) with increasing deformation is evident. In the case where there is no

deformation the nuclei are symmetrically distributed in space on the six quadrilateral

faces of the tetrakaidecahedron. It is this uniform distribution of nuclei which gives rise


to the sigmoidal shape of the kinetics curves. After deformation the distribution of

nuclei is no longer uniform in space because of relative change of the nucleation site

positions. Early impingement in the normal direction but late in the rolling direction,

therefore, is responsible for the transition from a three dimensional, to a two-

dimensional and finally to a one-dimensional growth, and gives rise to the observed

shape change of the recrystallization kinetics curve.

In Fig. 4 the recrystallization kinetics for random nucleation using the JMAK equationassuming an exponent 3k are also shown. For the cases of six nuclei, when each

nucleus is on one of the quadrilateral faces or at one corner of the quadrilateral faces thenumber of nuclei per tetrakaidecahedron, nN , is 3 and 1.5 respectively. The nucleus

density nn , is, therefore, given by 32 /n nn N d . The thick dotted line in Fig. 4

corresponds to 3nN while the thin dotted line applies to 1.5nN . Clearly,

impingement geometry resulting from deformation is quite complex and the JMAK

model is not capable of taking this into account.

Fig. 5. Calculated recrystallization kinetics for cases of 1 and/or 2 nuclei each at one of

quadrilateral faces in different directions after a rolling strain of 0.69 .

The dependence of the kinetic curves on the relative positions of the nuclei can be seen

in Fig. 5, which is calculated for a deformed grain geometry corresponding to a strain of

0.69. The number on each curve corresponds to the number of nuclei. The symbols R, N

and T refer to the nuclei located at the quadrilateral faces perpendicular, normal or

transverse to the rolling direction respectively. For example, when each of the two

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2Reduced time t*

Fra

ction r

ecry

sta

llized 2N 2T

1N

2R

1T 1R

Chapter 334

nuclei is at one of the two quadrilateral faces perpendicular to normal direction the rate

(curve marked 2N in Fig. 5) is steeper than those of the other two cases (2T and 2R).

The balance between early impingement and larger interface area of recrystallized /

unrecrystallized regions explains this relative order. When two nuclei start from two

opposing quadrilateral faces of the deformed tetrakaidecahedron in the direction of the

rolling deformation, the impingement occurs later but the interface area inside the

deformed grain during growth is comparatively small.

Fig. 6. Calculated recrystallization kinetics as a function of the number of nuclei after a

particular rolling strain. (a). 0 , (b). 0.69 .

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2Reduced time t*

Fra

ctio

n r

ecry

sta

llized

24 6 2N 1N

2R 1R

(b)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2Reduced time t*

Fra

ctio

n r

ecry

sta

llize

d %

24 12 6 2 1

(a)


It may also be inferred that nucleation density has a significant influence on the kinetics

of recrystallization. In Fig. 6 the calculated recrystallization curves for different numberof nuclei positioned within the deformed tetrakaidecahedron after strains of 0

(Fig. 6a) and 0.7 (Fig. 6b) are shown. The curves marked 24 represent the condition

that each of the 24 nuclei is positioned at a corner of the tetrakaidecahedron. Curves

marked 6 are the cases that each of the 6 nuclei is arranged at one of the quadrilateral

faces. For the case of 2 nuclei, each is located at one of the two parallel quadrilateral

faces. R, N, T have the same meaning as in Fig. 5. The greater the number of nuclei, the

faster the transformation rate becomes. Fig. 6b also suggests that, when the density of

nuclei are low, the relative positions of the nuclei in the deformed grain are more

significant for the overall recrystallization time than the number of nuclei present.

We must emphasize that the above simulations are based on the assumptions that the

growth rate is constant in time and equal for all conditions, and that the nuclei start from

the same equivalent position irrespective of the applied strain. These results, therefore,

just indicate that the grain geometry change resulting from deformation is of importance

in deciding the kinetics of recrystallization. In reality, larger deformation results in a

higher stored energy and a larger surface area per unit volume that make growth rate

and the nucleation site density increase with increasing deformation, hence the true

trend in the recrystallization kinetics is the opposite to that shown here.

In the following sections we will first establish the relationships between the driving

pressure and the deformation strain and then present the simulations of the

recrystallization kinetics in aluminium, in which the effect of plastic deformation is

taken into account properly. The positions of nuclei on the faces of the

tetrakaidecahedron are determined randomly assuming that there is an equal probability

that each of the 14 faces may act as a nucleation site. Each simulation is therefore

repeated a number of times and the average of the simulated kinetics taken as

representative.

3.3.2. Driving pressure for recrystallization

1. The influence of deformation on driving pressure and critical nucleus size

As demonstrated by Eq. (9), the driving pressure for recrystallized grain growth depends

on two terms: the boundary surface energy of the nuclei and the stored energy in theassembly. The surface energy term depends on the misorientation, n , (which is

assumed to be 10-15°) between the nucleus and its surroundings and radius of the

nucleus. The stored energy in the assembly is determined by the mean subgrain size and

misorientation in the assembly. When it is assumed that there is no concurrent recovery

Chapter 336

then the stored energy in the unrecrystallized material remains constant. The sum of the

two terms yields the driving pressure that increases nucleus size and approaches

asymptotically the value of the stored energy in the assembly as recrystallization

proceeds. When the driving pressure is positive, the "nucleus" may begin to grow. Thecritical nucleus size for recrystallization is dependent on ,R and n and is given by:

2 ( )

( )n

cR R (18)

Fig. 7 shows the variation of the driving pressure with the radius of the recrystallized

grain after various deformation strains. As the deformation increases, the subgrain size

decreases and misorientation increases, the driving pressure increases whilst the critical

nucleus size decreases.

On the basis of an earlier assumption, nM is a constant for high angle grain boundary

( =15°) at a given temperature [1]. The growth rate will, therefore, vary in the same

manner as the driving pressure as recrystallization proceeds. That is to say, it will

increase rapidly during the early stages of recrystallization and then, as the process

continues, approach asymptotically a value equal to the product of the stored energy in

the assembly with the mobility.

Fig. 7. The variation of driving pressure with recrystallized grain radius after various

deformation strains.

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10

Recrystallized grain radius (µm)

Drivin

g p

ressu

re (

MP

a)

=1.95

0.69

0.22

0.41

Rc=3.5µm


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4 5

Timex104(s)

Drivin

g p

ressure

(M

Pa)

No recovery

Recovery n=4

Recovery n=3.5

Fig.8, S.P.Chen

2. Effect of concurrent recovery on the driving pressure

The above results apply when there is no recovery in the assembly. When concurrent

recovery takes place the driving pressure may decrease as the recrystallizing grain

grows. The magnitude of this recovery effect on recrystallization will depend on the rate

of the recovery reaction as compared with migration rate of the interface. As proposed

previously (section 2.4) the dominant recovery mechanism is considered to be subgrain

growth, a situation expected during annealing of aluminium at higher temperature

[33,34]. If , and thus , are assumed to remain constant, which is reasonable if it is

assumed that no orientation gradient is present [1], the kinetics of subgrain growth will

be parabolic. As might be expected, the increase in deformation strain results in a higher and smaller R , and so, should increase the kinetics of subgrain growth. The effect of

concurrent recovery on the driving pressure for recrystallization after various degrees of

deformation is shown in Fig. 8. In contrast to the case of no recovery, where the driving

pressure continues to increase to a stable level, in the presence of concurrent recovery it

will first increase due to the decrease of the surface energy term as the recrystallized

grain grows and then decreases because of the decrease in the stored energy in the

assembly. The driving pressure is lower when concurrent recovery is active and this

effect increases with increasing strain.

Fig. 8. The effect of concurrent recovery on the driving pressure for recrystallization.

Chapter 338

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

Timex104 (s)

Fra

ctio

n r

ecry

sta

llize

d

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ev/E

0

No recovery

Recovery n=4

Recovery n=3.5

Fig.9, S.P.Chen

3.3.3. Simulation of recrystallization kinetics

1. The recrystallization kinetics by single grain approach

After considering the grain shape change, the microstructure (subgrain size,

misorientation) evolution and the nucleation site density change as a function of

deformation strain we are now in a position to simulate the kinetics of the

recrystallization process in aluminium by specifying various microstructures after

different degrees of rolling deformation and inserting appropriate values of driving

force and interface mobility from Eqs. (9) and (12) into Eq. (6). In this simulation we

consider a single grain, 100 µm in diameter. The number of nuclei after true strains of

0.22, 0.41 and 0.69 are 4, 10 and 20, respectively.

Fig. 9 shows the predicted recrystallization kinetics in aluminium after various strains

for the cases of no recovery and with recovery and also the variation in the storedenergy 0/ EEv ( vE and 0E being the stored energy in the assembly at time t and 0t

respectively) when concurrent recovery is active. The rate of recrystallization increases

with the amount of deformation. While the stored energy increases with deformation,the increase in nucleation site density resulting from vS as deformation increases could

account for a significant portion of the corresponding increase in recrystallization

kinetics. It can be noted that the effect of concurrent recovery on the recrystallization

kinetics increases as deformation strain increases. The magnitude of the effect of the

recovery on the recrystallization kinetics is strongly dependent on the value of theparameter n employed in Eq. (14).

Fig. 9. Simulation of recrystallization kinetics in aluminium after various degrees of

deformation by the single grain model.


0

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5Time x10

4(s)

Rre

x (

µm

)

No recovery

Recovery n=4

Recovery n=3.5

Fig.10, S.P.Chen

Fig. 10. Prediction of the change in the recrystallizing grain radius with time.

The predictions of the recrystallized grain radius as a function of annealing time are

illustrated in Fig. 10 for the case of no-recovery and the case of recovery. When the

condition of no recovery is assumed linear growth is predicted. This is a result of the

stored energy and hence driving pressure increasing throughout the nucleation stage and

then attaining a constant value during recrystallization. In the cases where recovery is

taken into account a departure from this linear growth occurs for all strains. As the

deformation strain increases the deviation from the linear growth condition becomes

more obvious with the extent of the deviation depending on the magnitude of the

concurrent recovery assumed.

2. The recrystallization kinetics when there is a grain size distribution

A more realistic simulation of recrystallization kinetics should consider the range of the

grain sizes before deformation. The initial grain size affects the recrystallization kineticsin three ways: by a change in the nucleation site density as a consequence of vS , which

decreases as grain size increases, by changing the relative distances among nuclei and

by a variation in stored energy resulting from variation in dislocation density after

deformation. The last factor is most complicated as it also depends on texture and

orientation with respect to deformation field and is not considered here.

Fig. 11 shows simulations of recrystallization kinetics for an assembly with a grain sizedistribution (of mean size d =100 µm and a standard deviation µ=1.5) in comparison

with the simulated kinetics from a single grain of mean size. In this case, the

Chapter 340

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5Timex10

4 (s)

Fra

ction r

ecry

talli

zed

Single grain

Multigrains µ=1.5

Fig.11, S.P.Chen

distribution function is divided into 12 equal size classes. The overall kinetics curves are

derived according to Eq. (16) by assuming that the subgrain size does not change with

grain size. The temperature in the simulation is kept at 340°C and it is assumed that

there is no concurrent recovery present. Fig. 12 shows the effect of the standard

deviation of the grain size distribution on the recrystallization kinetics after a strain of

0.69 for both the case of concurrent recovery and that of no recovery. As can be seen,

the rate of recrystallization of the assembly of grains (thick line) is slower than that of

the single grain of the mean size (thin line). The rate difference between the two cases

decreases with increasing deformation strain in the strain range studied. These can be

explained as follows. The recrystallization kinetics is accelerated by an increase in the

nucleation site density that is proportional to the grain boundary area per unit volume Sv.

Fine-grained materials recrystallize faster because of a larger Sv. Deformation also leads

to an increase in Sv. The increase in Sv by grain size will have a less effect as strain

increases. On the other hand, deformation makes the geometrical shape change of the

grain and leads to a change in the relative positions of the nuclei. As we have analyzed

in section 3.1, the relative positions of the nuclei have a stronger effect on the overall

recrystallization kinetics than the change in nucleus density with increasing strain.

Hence, recrystallization kinetics becomes less dependent on the original grain size asstrain increases. A notable and interesting fact is that the 0.68t of recrystallization for

both single grain and multigrain approaches are the same for all the grain size

distributions considered.

Fig. 11. Simulation of recrystallization kinetics in aluminum after various degrees of

deformation assuming a grain size distribution.


0

0.2

0.4

0.6

0.8

1

0 0.4 0.8 1.2 1.6

Timex104 (s)

Fra

ction r

ecry

sta

llized

Single grain

Multigrains µ=1.38

Multigrains µ=1.5

Recovery n=4

No recovery

Fig.12, S.P.ChenFig. 12. The effect of the width of the grain size distribution on the overall

recrystallization kinetics after a strain of 0.69.

3.4. Conclusions

A physical model to predict the grain boundary nucleated recrystallization kinetics of

single-phase metals based on the microstructure of the deformed state is proposed. The

model, notwithstanding its simplicity, provides a new insight into the kinetics of

recrystallization. The model includes the grain geometry of the deformed structure, the

nucleation density and the relative positions of the nuclei and these factors have been

shown to have a large effect on the overall recrystallization kinetics. By employing data

from the open literature, in conjunction with the deformed grain geometry, the

recrystallization kinetics could be simulated for a range of deformation and heat

treatment conditions. The influence of concurrent recovery on recrystallization kinetics

is shown to increase with increasing deformation strain. The potential of the model is to

be explored further for recrystallization kinetics following warm deformation.

Chapter 342

References

1. F. J. Humphreys and M. Hatherly, “Recrystallization and Related Annealing

Phenomena”, Pergamon, London, 1996.

2. D. Juul Jensen, N. Hansen and F. J. Humphreys, Acta Metall., 1985, 33, 2155-2255.

3. N. Hassen, T. Leffers and J. K. Kems, Acta Metall., 1981, 29, 1523-1622.

4. P. A. Beck and P. R. Sperry, J. Appl. Phys., 1950, 21, 150-157.

5. J. Hjelen, R. Orsund and E. Nes, Acta Metall., 1991, 39, 1377.

6. P.L. Orsettirossi and C.M. Sellars, Mater. Sci. and Techn.,1999, 15, 185-192.

7. P.L. Orsettirossi and C.M. Sellars, Mater. Sci. and Techn.,1999, 15, 193-201.

8. T. Furu and E. Nes, Materials Science Forum, 1993,113-115, 311-316.

9. R. A.Vandermeer and B. B. Rath, Metall. Trans. A, 1989, 20, 391-401.

10. R. A.Vandermeer and D. Juul Jensen, Acta Metall. 1994, 42, 2427-2436

11. J. W. Cahn, Acta Metall., 1956, 4, 449-459.

12. R. A. Vandermeer and R. A. Masumura, Acta Metall., 1992, 40, 877-886.

13. A. D. Rollett, Progress in Materials Science, 1997, 42, 79-99.

14. Y. Van Leeuwen, S. Vooijs, J. Sietsma and S. van der Zwaag, Metall. and Mat.

Trans. A, 1998, 29, 2925-2931.

15. S. I. Vooijs, Y. Van Leeuwen, J. Sietsma and S. van der Zwaag, Metall. and Mat.

Trans. A, 2000, 31, 379-385.

16. T.A. Kop, Y. Van Leeuwen, J. Sietsma and S. van der Zwaag, ISIJ International,

2000, 40, 713-718.

17. H. K. D. H. Bhadeshia: “Worked examples in the geometry of crystals”, 1987, The

Institute of Metals, London.

18. J. W. Cahn, Trans. Indian Inst.Metall.,1997, 50 (6), 573-580.

19. T. Furu, R. Orsund and E. Nes, Acta Metall. Mater. 1995, 43 (6), 2209-2232.

20. J. Gil Sevillano, P. van Houtte and E. A. D. Aernoudt, Progress in Materials

Science, 1980, 25, 69-412.

21. H. E. Vatne, T. Furu and E. Nes, Proceedings of the International Conference on

Recrystallization and Related Topics, Monterey, California, 21-24 Oct., 1996.

22. N. Hansen and D. A. Hughes, Phys. Stat. Solidii (b), 1995, 149, 155-172.


[TMP]2', B.Hutchinson et.al (Eds), Swedish Society for Metals Technology, 1996,

35-51.

24. H. E. Vante, T. Furu, R. Orsund and E. Nes, Acta Mater. 1996, 44, 4463-4473.

25. F. J. Humphreys, Acta Mater., 1997, 45, 4231-4240.

26. R. Orsund and E. Nes, Scripta Metall. 1989, 23, 1187-1192.

27. Y. Huang and F .J. Humphreys, Acta Mater., 1999, 47, 2259-2268.

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29. G. Gottstein and L. S. Shvindlerman, Scripta Metall., 1992, 27, 1515-1520.


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Chapter 344

S.P. Chen

Chapter 4

A refined single grain approach applied to

the modeling of recrystallization kinetics for

cold rolled single-phase metals

A comprehensive model for the recrystallization kinetics is proposed which incorporates

both microstructure and the textural components in the deformed state. The model is

based on the single grain approach proposed previously. The influence of the as-

deformed grain orientation, which affects the stored energy via subgrain size and sub-

boundary misorientation, is taken into account. The effects of the deformed grain

geometry, the nucleation site density and the initial grain size prior to deformation on

the recrystallization kinetics is assessed. The model is applied to the recrystallization

kinetics of cold-rolled AA1050 alloy.

4.1. Introduction

Recrystallization kinetics in simple metallic systems is frequently described by JMAK

type models [1]. The basis of the JMAK analysis, which yields the volume fraction

transformed, X, against time t, is:

1 exp( )nX kt (1)

The constant k involves the nucleation rate, or the nucleation site density, and the

growth rate. The exponent n relates to the time dependencies of nucleation rate, growth

and the dimensionality of the growth fronts. The crucial assumption made in deriving

Chapter 446

the various forms of JMAK is that the nucleation sites are randomly distributed in space

leading to n values to be 3 or 4. However, in almost all experimental studies of

recrystallization kinetics in aluminium alloys the exponent n is less than 2. Such a

deviation from ideal JMAK behavior is attributed to be the occurrence of concurrent

recovery [2], or a non-uniform distribution of stored energy [3,4].

The occurrence of the simultaneous recovery during recrystallization will reduce the

dislocation density and hence the stored energy. Consequently, the concurrent recovery

should retard recrystallization kinetics and cause a negative deviation from linear

JMAK behavior. However, both experiments [5,6] and theoretical predictions [3,7] have

shown that concurrent recovery can not fully explain the observed lower value for n.

Hence a more detailed analysis of the uniformity of stored energy in deformed metals

and its effect on the recrystallization behavior seems appropriate.

It is well known that the deformation in a polycrystal material is inhomogeneous [8,9].

The inhomogenity is of a dual nature. Firstly, the area along the original grain

boundaries undergoes a more severe plastic deformation than the average macroscopic

level. As a result, a deformation gradient will form in the grain; the original grain

boundaries are of highest dislocation density, and therefore, the main source of

recrystallization nuclei, especially for single-phase alloys. Like the effect of concurrent

recovery, this kind of the inhomogenity also leads to a time dependent growth rate of

recrystallizing grains and therefore reduces the slope of JMAK plot. However, it was

found that continuous reductions in growth rate by a factor of 10 were required to cause

the JMAK exponent to drop significantly below a value of 3. The variation in growth

rate by a factor of 2 in real experiments caused no detectable change in the simulated

exponent [10,11]. Furthermore, this explanation can not be applied to cases where the

nucleation is on a coarser scale than the distribution of the stored energy.

The second inhomogenity refers to the fact that the deformation microstructures varies

from grain to grain because of the initial crystallographic texture. As a result, the

nucleation sites for recrystallization in deformed grains with different orientations are

different [12,13]. Each as-deformed grain will recrystallize at a rate that depends on its

size as well as initial orientation with respect to the deformation field depending on the

accumulation of the stored energy.

Hence, the non-uniformity of the recrystallization behavior is most likely due to the

variation of the stored energy as a function of the grain orientation. Dillamore et al [14]

has pointed out that the variation of stored energy as a function of the texture is related

to the variation of the Taylor factor. The order of the stored energy of the deformationtextures in low carbon steel is [15], {110} 110 {111} {211} 011 {100} 011uvwE E E E . The

A single grain approach applied to the modeling of recrystallization kinetics 47

dislocation structures and density in aluminium alloys were observed to be dependent

on the crystallographic orientation too although there was a wide scatter in the

experimental data [16,17].

In addition, it has been realized that the recrystallization kinetics and the resulting grain

structures are determined by the ratio of the nucleation to growth rate, the density and

the spatial distribution of nucleation sites as well as the impingement space. All these

factors are strongly dependent on the microstructure, the substructures in individual

grains and the local orientations between grains, developed during plastic deformation.

Therefore, a more realistic recrystallization model should take into account all these

details of deformed microstructure.

The advanced techniques such as EBSD and hard x-rays from a synchrotron source

make it possible to examine the recrystallization behaviors at the level of an individual

grain [8,18,19]. However, theoretical models at this level are still lacking. In [7] we

have proposed a single grain approach to predict the recrystallization kinetics of a

polycrystal aggregate. The structure of the deformed metal is taken into account in the

model, which contains the information about the main features of rolled material, i. e.

on the degree of cold rolling, the grain size, and the grain shape. An as-deformed

tetrakaidecahedron is applied to describe the grain geometry. The grain shape change

with strain is introduced through a parameter of aspect ratio according to the

macroscopic change in the shape of the specimen (principle of the Taylor model).

Additionally, the subgrain size and the misorientation between subgrains belonging to

the same grain are related to the degree of deformation.

In the present Chapter, the physical model of the recrystallization kinetics, based on the

single grain approach, is further developed to incorporate both microstructural

inhomogeneities and the textural components in the as-deformed state. The influence of

the as-deformed grain orientation, which affects the stored energy via subgrain size and

sub-boundary misorientation, is taken into account. The effect of the initial grain size,

the grain geometry and the nucleation site density on the recrystallization kinetics is

assessed. The model is applied to the recrystallization kinetics in cold rolled AA1050.

4.2. Experiments

The AA1050 studied has a chemical composition of 0.185 wt.% Fe, 0.109 % Si, balance

Al. Cold-rolled plate with thickness of 5 mm was annealed at 600°C for 2 h and

quenched into water. The material was then subjected to a precipitation treatment at

400°C for 2 h to reduce and stabilize the content of iron in solid solution. The average

grain size after this treatment was 98±5 µm. Metallographic examination revealed

Chapter 448

coarse (2-4 µm) iron-rich constituent produced during casting and fine (0.1-0.3 µm)

precipitates resulting from annealing. Following a cold rolling treatment leading to areduction of 50% in thickness ( 0.69 ), the samples were annealed in a salt bath at

340°C for a range of times and then quenched into water to get different degrees of

recrystallization. All the metallographical and OIM observations were confined to the

center layer of the sample. The OIM facility performed a scan over an area of

1200 600 µm2, containing approximately 120 original grains. The degree of

recrystallization was determined quantitatively and accurately using the method

described in [20].

Five typical orientations of grains in the deformed state, namely, Cube {100}<001>,

Goss {110}<100>, Brass {110}<112>, S {124}<211> and Copper {112}<110> are

observed in OIM analysis. The volume fractions of the main texture components in as-

deformed state were calculated by allowing a 15° spread around the ideal orientations,

as listed in Table 1. The average fully recrystallized grain size after deformation to

strain of 0.69 and annealed at 340°C was 69±5 µm.

Figure 1 shows a partially recrystallized structure in a sample annealed at 340°C for 50

min, showing the characteristic of the inhomogeneous nucleation. Recrystallization was

found to occur faster in the grains with Copper and/or S orientations. Grains with a

Cube orientation are slowest in the recrystallization.

The overall recrystallization kinetics as measured by optical microscopy is shown in

Figure 2 by the square doted line. Also in this Figure we illustrate schematically the

approximate recrystallization kinetics examined in grains with different orientations by

OIM.

Table 1. The Talyor factor of the main texture components and related substructural

parameter values after cold rolling 50%

Texture

component

{hkl}<uvw> M value

(µm)

Frequency

%

RG {011}<110> 4.90 0.46 4.68 0

Copper {112}<111> 3.67 0.61 3.53 2

S {123}<364> 3.52 0.64 3.34 7

Brass {011}<211> 3.27 0.68 3.15 9

Goss {011}<100> 2.75 0.82 2.63 20

Cube {001}<100> 2.45 0.94 2.29 13

Random 3.06

average

0.73 2.92 49


Fig. 1. OIM of an AA1050 sample annealed at 340°C for 50 min, showing the

inhomogeneous nature of the recrystallization process.

Fig. 2. The overall softening kinetics measured by optical microscopy (square marks)

and approximate recrystallization kinetics of 3 texture components at 340°C after a

strain of 0.69.

0

0.2

0.4

0.6

0.8

1

0 2000 4000 6000 8000Time (s)

Fra

ction

Brass

Copper

S Cube

Random

Chapter 450

4.3. Modeling the recrystallization kinetics

4.3.1. The orientation dependent microstructure in the deformed state

The dislocation density within a given grain after a plastic deformation is primarily

related to the amount of crystallographic slip , which is defined as [21]:

M (2)

where M is the orientation factor of the as-deformed grain (Taylor factor), is the

macroscopic shape change (true strain) of the grain, which is assumed to be the same as

that of the specimen.

As an approximation, we assume that aluminium follows a linear strain hardening

behavior, i.e.:

0c (3)

where c is the resolved shear stress, 0 , the frictional part of c , , a constant.

The relationship between the internal stress and dislocation density can be written as:

'0c (4)

where ' is another constant. Combining equations (3) and (4) one obtains:

2 (5)

In most cases, and certainly for this aluminium grade, the dislocations are arranged in a

cellular substructure or into subgrain. If the substructure in a specific deformed graincan be described by a subgrain size and a sub-boundary misorientation then [1]:

(6)

The relationship between the deformation features ( , ) of an individual grain and the

average values ( , ) of the polycrystalline aggregate can be expressed as:

2

av

M

M (7)


where avM is the average value of the Taylor factor. In addition, an empirical

relationship between the average cell/subgrain size, , and deformation strain [22,23]

is employed in the present work:

0.35 0.17 / (8)

and

/ b C (9)

where is in m, b is the Burger’s vector and C is a constant between 60~80 rad.

Eq. (7) indicates that the stored energy is proportional to the square of the Taylor factor.

4.3.2. Nucleation and Growth

Based on the experimental data, nucleation is assumed to start from the original grain

boundaries separating grains of different orientations. In a deformed grain with a

specific orientation, the microstructure will have a variety of cell sizes and a distribution

of misorientations. We assume that the microstructure can be described using two

components [24]: an assembly of equiaxed subgrains and specific subgrains, effectively

sub-critically sized recrystallization nuclei. The former can be characterized by a meanequivalent radius, R , a mean misorientation, , and with boundaries of mean energyand mobility, and sbM , respectively. The latter have a larger size ( nR ) and different

boundary characteristics ( , ,n n nM ). In the present study we assume that the

misorientation angle between the particular subgrain and surroundings is 15°.

The effective nucleation site density for grain boundary nucleation is expressed by

[25,26]:

2/V d vN C S (10)

where dC is a calibration constant which determines the potential of the grain boundary

as a nucleation site. Sv is the grain boundary area per unit volume, which is related to theoriginal grain size d, rolling strain , and the grain shape tessellation in the space. For a

tetrakaidecahedron, Sv varies with rolling strain in a manner described by [7]:

aaaaaaad

Sv /22/23/2132

1 2222 (11)

Chapter 452

where ea is the distortion along the rolling direction.

The driving pressure acting on the recrystallized front is:

2 n

n

PR R

(12)

where is a geometrical constant and has a value of ~1.5 [1].

The growth rate of a recrystallized grain is expressed as:

2n nn n

n

dRM P M

dt R R (13)

where nM is the mobility of a high angle boundary at the annealing temperature and

which is given by:

0 exp /n gM M Q R T (14)

where 0M is a pre-exponential factor, Q, activation energy for boundary migration, gR ,

gas constant and T, the annealing temperature.

The boundary energy is assumed to depend on the average misorientation angle only

and is given by [1]:

1 lnm

m m

(15)

where m and m are the values of boundary energy and misorientation for high-angle

boundaries and which are commonly taken as 0.324 J/m² [1] and 15° respectively.

Combining Eqs (9), (13) and (15), the relationship between the growth rate of a

recrystallizing grain and the subgrain size of the assembly is approximately given by:

2 2

1n m n

m

dR bC M KV

dt (16)

with K being a temperature dependent constant.


4.3.3. Kinetics approach

The single grain model described in Chapter 3 and [7] is expanded along the line

described above and is used to calculate the recrystallization kinetics in individual

grains with a specific orientation. Nucleation event in an as-deformed grain is assumed

to be site-saturated. A growing recrystallized grain is modeled as an expanding semi-

sphere nucleated on grain boundaries. New grains grow independently of one another

until hard impingement occurs.

As the position of the nuclei has a significant effect on the recrystallization kinetics [7],

the position of nuclei on the faces of the tetrakaidecahedron are determined randomly

assuming that there is an equal probability that each coordinate on the 14 faces

(including grain edges and corners) may act as a nucleation site. Each simulation is

therefore repeated a number of times (40) in the model calculation and the average of

the simulated kinetics is taken as the representative, i.e. the time to obtain a given

fraction of recrystallization, f, is given by the average over the number of the

simulation, j.

, , /av i i jt t j (17)

where ,i jt is the time at a given fraction in the j th simulation for a grain with a specific

orientation labeled i.

Generally, a deformed polycrystal contains many grains of different orientations. Each

grain has a different set of surrounding grains. In this attempt at modeling

recrystallization kinetics in a polycrystal material we assume that there is no correlation

between the orientations of neighboring grains and all the grains of one crystallographic

orientation (within a certain small range) are represented by a single grain of the mean

size. (The simulation results in the next section will show that this assumption is

reasonable).

The overall kinetics of the polycrystalline aggregate can then be obtained by a weighted

summation of the kinetics on the main textural components in the as-deformed state.

The fractional recrystallization f of the assembly is given by:

( )i i if f (18)

where i is the fraction of a main textural component.

Chapter 454

4.4. Results

4.4.1. Effect of the grain geometry and the nucleation site density on

the recrystallization kinetics

Before we present the simulation results, two new concepts need to be introduced, i.e.,

the effective nucleation site density and the numerical nucleation site density. As long

as Cd can be experimentally determined the nucleation site density, Nv, can be calculated

using Eq. (10), which is called the effective nucleation site density. In reality, a

recrystallized grain at the original grain boundary may invade into two (or three)

adjacent grains, at which the growth of the recrystallizing grain in one side is faster than

in the other. However, in the single grain kinetics model the new grain is assumed to

start from the original grain boundaries and only the portion of the new grain inside the

objective grain is taken into account when fractional recrystallization is calculated. As a

consequence, only one half of a physical nucleus is considered. Therefore the numerical

nucleation site density, taken to be twice the effective nucleation site density, is

employed in calculating the nucleation sites in a single grain in the present model. In the

model the number of the nuclei in a single grain calculated might be a non-integer. For

such a case, the representative of the simulated recrystallization kinetics is treated as a

weighted summation on the simulation kinetics obtained from simulations for the two

nearest integer nuclei per grain values.

In the single grain kinetics model, the recrystallization fraction is proportional to3( / )N Vt d , in which N is the number of the nuclei in an as-deformed grain of diameter

d, t is real time, 2/V K is the growth rate. If the interface mobility is assumed to

remain constant throughout the recrystallization process and to be independent of the

applied deformation strain, and for a certain number of N, we can present the fraction of

the recrystallized material as a function of the dimensionless reduced time *t , which is

defined as:

*

2

Kt t

d (19)

The real time can be calculated provided the K is known at a given annealing

temperature.

Figure 3 shows the simulated recrystallization kinetics after a strain of 0.69 assuming

that six nuclei are located randomly on the surface of a tetrakaidecahedron. Each curve

represents one simulation. As can be seen, the simulated recrystallization kinetics

differs from each other because of the difference in the nucleus positions. The

difference in the kinetics reflects the characteristics of the nucleation nature. The six


nuclei can either be present as clusters or as being far apart. When they form as a

cluster, a slow recrystallization rate is obtained due to early impingement. Contrarily,

when nuclei form far apart from each other, a faster rate will result. In order to obtain

the representative of the recrystallization kinetics in a single deformed grain simulations

were run a number of times with the same number of nuclei but different nucleation

positions. 40 simulations were found to be an adequate number to yield a representative

with a stable kinetics curve. In Figure 3 this average behavior is indicated by a square

dotted curve. The representative curve shows a typical sigmoidal shape. The JMAK

equation is only applied to the representative of simulation kinetics to obtain least

square best fit of the JMAK exponent n, to determine the effects of underlying

assumption and parameter values on n.

Fig. 3. The determination of the representative of the simulated recrystallization

kinetics for six nuclei located randomly on the surfaces of a tetrakaidecahedron at a

strain of 0.69.

Figure 4 shows the simulated recrystallization kinetics for a tetrakaidecahedron of size100d µm at different strains, 0.41, 0.69, 1.10, 1.37, 1.61 and 2.3 with the number of

the nuclei per grain fixed at 6 and 1 µm . Because in this subset of the simulations

we assume that the subgrain size, , does not change with the strain, then the kinetics

after various degree of rolling strain will only reflect the effect of the grain geometry

resulting from deformation on the recrystallization kinetics. As deformation increases,

the aspect ratio of the grain increases, the resulting n decreases.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

t*=Kt/ ²d

Fra

ctio

n

n = 2.68

-5

-4

-3

-2

-1

0

1

2

-2.5 -2 -1.5 -1 -0.5 0Lnt*

Ln

(Ln

(1/(

1-x

)))

Chapter 456

Fig. 4. The simulated recrystallization kinetics for the case of six nuclei located

randomly on the surfaces of a tetrakaidecahedron at various strains. From leftmost to

rightmost, 0.2, 0.41, 0.69, 1.10, 1.37, 1.61, 2.30.

Fig. 5. Effect of the grain geometry resulting from deformation and the number of the

nuclei on the JMAK exponent.

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

0 0.5 1 1.5 2 2.5Strain

JM

AK

n

Real

N=2

N=6

N=12

N=24

n = 2.82

n = 1.78

-5

-4

-3

-2

-1

0

1

2

-3 -2 -1 0 1Ln t* (=Kt/ ²d)

Ln(l

n(1

/(1-x

)))


The preceding simulations are repeated to study the effect of the number of the nuclei in

an objective grain on the recrystallization kinetics. The resulting variations of n against

strain for various numbers of the nuclei are shown in Figure 5. It can be seen that the

JMAK exponent depends on the number of the nucleation sites. In a deformed

tetrakaidecahedron an increase in the number of the nuclei gives a rise in n. However, in

the case of the undeformed one, an opposite trend is found.

4.4.2. Effect of the strain on the recrystallization kinetics

In reality, as strain increases the sub-grain size decreases, Sv increases. Therefore, the

nucleation site density and the growth rate increases with strain. These factors favor the

recrystallization kinetics. Figure 6 shows the simulated recrystallization kinetics in a

tetrakaidecahedron with an initial size of 100 µm and average substructures after

various rolling strains. In these simulations, the subgrain size and the nucleation site

density at each strain are calculated according to Eq. (8) and Eq. (10), respectively. The

dC is taken as 41.5 10 (which is calculated from the current experiment and this value

will be used in the following simulations). The fractional recrystallization is plotted

against 21 100( / )Kt d with 1 being 1 µm. As can be seen, as the strain increases, the

kinetics curves shift to shorter times while the JMAK exponent n decreases (which is

shown in Figure 5 by thick line with legend ‘real’). The final variation of n with strain

reflects the combined effect of the grain geometry and the nucleation sites. On the one

hand, the increase in the number of the nuclei will increase n. On the other hand, the

grain geometry change resulting from increasing strain leads to a lower value of n.

Fig. 6. The simulated recrystallization kinetics in a single grain after various strains.

From leftmost to rightmost, =2.3, 1.61, 0.69, 0.41.

n= 2.76n = 2.65

n= 2.37

n= 2.22

-5

-4

-3

-2

-1

0

1

2

-6 -5 -4 -3 -2 -1 0ln t*(=Kt/ 1²d100)

Ln

(ln

(1/(

1-x

)))

Chapter 458

Fig. 7. The effect of the initial grain size on the recrystallization kinetics. (a), =0.2, (b),

=0.69, (c), =1.61, (d) =2.30.

4.4.3. Effect of the initial grain size prior to deformation on the

recrystallization kinetics

Apparently, the initial grain size affects the nucleation site density as it affects vS ,

which decreases as initial grain size increases. In addition, it can lead to a larger

variation in stored energy resulting from variation in dislocation density after

deformation. The second factor is most complicated as it also depends on texture and

orientation with respect to the deformation field and will be considered later. In order to

determine to which extent the initial grain size affects the recrystallization kineticsthrough vS , we assume that the subgrain size does not change with strain and fixes at

1µm, then the nucleation site density at a given strain is only a function of the initial

grain size. Figure 7 shows the simulated recrystallization kinetics, which is presentedagainst 2

1 100( / )Kt d , in the grains with 3 different initial diameters, i.e. 58 µm, 100 µm

and 200 µm for 4 strain levels. As can be seen, the effect of the initial grain size on the

recrystallization kinetics depends on the deformation strain applied. In an undeformed

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2t*=Kt/ 1²d100

Fra

ction

d=58µm

d=100

d=200

a

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2t*=Kt/ 1²d100

Fra

ction

d=58µm

d=100

d=200

b

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2t*=Kt/ 1²d100

Fra

ction

d=58µm

d=100

d=200

c

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3t*=Kt/ 1²d100

Fra

ctio

n

d=58µm

d=100

d=200

d


tetrakaidecahedron, smaller grain does recrystallize faster than coarser one. However,

this effect decreases as the deformation strain increases. For example, at larger strains

the recrystallization kinetics in a coarser grain is faster than that in the finer grains.These results indicate that the increase in vS resulting from deformation strain plays a

dominant role in the increase of the nucleation density.

4.4.4. Effect of the orientation on the recrystallization kinetics

Table 1 lists the Taylor factors of the main deformation textural components found in

metals with f.c.c. crystal structure [22,27]. The Taylor factor of the Goss component is

taken as an average of the values corresponding to the inclination of the shear band with

respect to the rolling direction being 30 deg. and 35 deg. respectively. The related

substructure parameters in grains with different orientations at a strain of 0.69 arecalculated using Eqs. (7)-(9) and listed in Table 1. We assume that dC has the same

value for all grain boundaries and equals 41.5 10 when calculating the nucleation site

density in grains with different textural components. Figure 8 shows the simulated

recrystallization kinetics in grains with different orientations but a fixed initial grain size

of 100 µm at a strain of 0.69. As can be seen, a grain with a larger Taylor factor

recrystallizes earlier and at a higher rate.

Fig. 8. The simulated recrystallization kinetics in grains with different orientations and

overall recrystallization kinetics in AA1050. Thin solid line: from leftmost to rightmost,

RG, Copper, S, Brass, Random, Goss, Cube. Square dotted line: the overall

recrystallization kinetics.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1t*=Kt/ 1²d100

Fra

ctio

n

Overall

Chapter 460

4.5. Application of the model to the experiment data

As indicated above, the overall recrystallization kinetics should include the effect of the

grain geometry, the nucleation site density, the starting grain size and the strain

variation from grain to grain as a consequence of the grain orientation. The grain size

distribution in annealed materials is approximately log-normal, where the maximum

grain size is typically 2.5~3 times mean size. To determine the overall recrystallization

kinetics of the grains with different sizes, we can apply the method specified in [7].

However, from the proceeding simulation results we have realized that the grain

orientation has a much stronger effect on the recrystallization kinetics than the initial

grain size. Hence, for the sake of simplicity, we assume in the overall recrystallization

kinetics that all the grains with a particular crystallographic orientation can be

represented by a single grain of mean size.

Now the single grain model can be applied to predict the recrystallization kinetics if the

input parameters, deformation strain, , the initial grain size, d, the calibration constant,

Cd, the mobility at a given annealing temperature, Mn, and the volume fraction of the

main texture components in the as-deformed state are determined.

In the current experiment on AA1050, the initial grain size prior to deformation as well

as the fully recrystallized grain size after deformation to strain of 0.69, annealed at

340°C are taken to be 100 µm and 69 µm, respectively. From these values the number

of the effective nuclei in a single grain of mean size is estimated to be 3, leading to a

calibration constant Cd is of 41.5 10 . The corresponding nucleation site density and

the subgrain size in a grain with a given orientation are determined according to the

section 4.4. Since Mn could not be determined from the present experiment we leave it

as a free parameter in the model. The simulated recrystallization kinetics in grains with

different orientations is shown in Figure 8.

The overall recrystallization kinetics can be obtained now using Eq. (18) with

considering the frequency of the texture components in the deformed state listed in

Table 1. The resulting curve is shown in Figure 8 by the square dotted line. The

Avrami-plots of the experimental data and the simulation is shown in Figure 9. The

JMAK exponents measured and simulated are 1.99 and 1.94, respectively. Obviously,

the present model could yield a much more practical prediction of the JMAK n, by

taking into account the deformation inhomogenity from grain to grain due to variation

of the Taylor factor.


Fig. 9. The comparison of the experiment with the prediction from the single grain

model by considering orientation dependent kinetics.

4.6. Discussions

The simulation results from the present model shows that the JMAK exponent of a

apparent recrystallization kinetics curve depends on the grain geometry, nucleation site

density, the initial grain size prior to deformation and the main textural components

after deformation. This result, which is one of the important features that the present

model reveals, implies that the recrystallization space (the space that the recrystallizing

grains grow into) is of importance in deciding the recrystallization kinetics. In real

materials, recrystallization space is effectively defined by both the initial grain size and

the deformation geometry. The change of the grain geometry due to deformation leads

to a change in the impingement space of the recrystallized grains. An increase in the

number of the nuclei will also cause a change in the impingement space.

Another salient feature that the present refined model shows is that the effect of the

texture components in the as deformed state on the recrystallization kinetics is

significant. This point will be elaborated further by simulations. Figure 10 shows the

simulated orientation dependent recrystallization kinetics at two strain levels, 0.69 and

1.61, respectively. We use the two starting structures with different fraction of texture

components as listed in Table 2. The overall kinetics curves of the two examples are

also indicated in Figure 10. In the same figure in the smaller box the Avrami plot for

these two examples are plotted. It can be seen that the shape and the slope of the JMAK

plot are related to the mixture of the texture components. When there are Goss and Cube

n = 1.94

n = 1.99

-4

-3

-2

-1

0

1

2

3

-5 -3 -1 1 3 5 7 9 11Ln t* for simulation, Ln t for experiment

Ln(ln

(1/(

1-x

)))

Simulation

Experiment

Chapter 462

texture components present in the as-deformed state, the JMAK plot shows a non-linear

behavior, i.e. two-stage n, which has been frequently reported in the literature.

Fig. 10. The simulated recrystallization kinetics in grains with different orientations and

overall recrystallization kinetics for the two examples. Thin solid line: from leftmost to

rightmost, RG, Copper, S, Brass, Random, Goss, Cube. (a), =0.69, (b), =1.61.

Table 2. The frequency of textural components in the as-deformed state for two

assumed structures

Texture

components

Copper S Brass Goss Cube Random

Example 1 16 20 23 0 0 41

Example 2 3 8 12 22 8 47

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1t*=Kt/ 1²d100

Fra

ctio

n

Example 1

Example 2

n = 1.88

n= 2.33

-4

-3

-2

-1

0

1

2

3

-3 -2 -1 0Ln t*

Ln(ln(1

/(1-x

)))

0

0.2

0.4

0.6

0.8

1

x

a

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35t*=Kt/ 1²d100

Fra

ctio

n

Example 1

Example 2

n = 2.10

n= 1.53

-4

-3

-2

-1

0

1

2

3

-5 -4 -3 -2 -1Ln t*

Ln(ln(1

/(1-x

)))

0

0.2

0.4

0.6

0.8

1

x

b


Many factors that control recrystallization are not specifically treated in JMAK model

but are casually lumped into the constants in Eq. (1). Two important factors that are

included in this category are the initial grain size prior to deformation and the amount of

deformation. These points can be clarified with the help of the present model.

Theoretical studies in the literature suggested that the finer grained material

recrystallizes after a shorter incubation time and within a shorter relative time period

than the coarser grained material although the experiments yielded controversial results

[28,29]. Generally, since recrystallization nucleates primarily along prior grain

boundaries, the rate of nucleation is directly proportional to Sv, which increases as the

grain size decreases. Therefore, after similar amounts of deformation and for similar

growth rates the rate of recrystallization should increase as the prior grain size

decreases. However, if it is assumed that the substructure in a fine grain and a coarse

one is equal (this is true for aluminum alloys at strain larger than 0.5 [30]) and the

fraction of the texture components is the same in both cases, present model shows that it

depends on the strain level whether small grains recrystallize faster or slower than large

grains. At lower strains, small grains recrystallize faster. However, at larger strains, the

recrystallization rate in coarser grain becomes higher. This is attributed to the change in

the impingement space due to the deformation geometry and the relatively large number

of nucleation sites in the coarser grains as strain increases. This prediction is in

excellent agreement with experimental study in Sellars’s group [31]. It should be

pointed out explicitly that the mechanism of intra-granular nucleation in larger strains

need not to be involved to explain this result, as in the current simulations nucleation

always occurs at the grain boundaries.

Fig. 11. The volume fractions of the texture components in the cold-rolled AA1050 as a

function of the degree of deformation. (a) fine initial grain size (50 µm), (b), coarse

initial grain size (350 µm) [32].

0

20

40

60

80

0 20 40 60 80 100Degree of cold rolling (%)

Textu

re c

om

po

ne

nts

(%

)

{110}<112>+{123}<634>+{112}<111>

{100}<001>

{110}<001>

a

0

20

40

60

80

0 20 40 60 80 100Degree of cold rolling (%)

Textu

re c

om

po

ne

nts

(%

) {110}<112>+{123}<634>+{112}<111>

{100}<001>

{110}<001>

b

Chapter 464

Finally, the initial grain size and the amount of strain are both known to have a large

effect on the texture components in the as deformed state. As can be seen from

Figure 11, which shows the textural component evolution with cold rolling strain in a

commercial aluminum alloy [32], the rolling texture components intensify while the

fraction of the soft components, Goss and Cube, decreases with increasing strain. The

variation of the textural concentration with strain in the fine grain material is different

from that in the coarse grain one. Therefore the initial grain size and the degree of the

deformation can affect the recrystallization kinetics through their effect on the texture

components. The model correctly predicts this changeovers in relative behavior.

4.7. Conclusions

The following conclusions are obtained from the refined recrystallization model which

incorporates both the deformed microstructure and the textural components.

(1). The kinetics model based on the single approach provides a new insight to gain a

better understanding of annealing phenomena in deformed polycrystals, particularly the

effect of non-random nucleation sites and the geometry by which recrystallized grains

fill space on recrystallization kinetics.

(2). The simulation shows that the JMAK exponent depends on the grain geometry,

nucleation site density, the initial grain size prior to deformation and the main textural

components after deformation. The effect of grain size on recrystallization kinetics

depends on the amount of the prior strain applied.

(3). After introducing the deformation inhomogenity from grain to grain due to

difference in the Taylor factor, the single grain approach could predict correctly the

JMAK exponent.


References



2. R.A. Vandermeer and P. Gordon, in 'Recovery and Recrystallization of Metals',

New York, Interscience, 1962.



5. E.C.W. Perryman, Trans. AIME, J. Metals, 203, 1955, 1053-1061.

6. R.A. Vandermeer and B.B. Rath, in 'Simulation and theory of evolving

microstructures', M.P. Anderson and A.D. Rollett (Eds), TMS, 1990, 119-126.

7. S.P. Chen, I. Todd and S. van der Zwaag, Metall. Mater. Trans. A, 33A, 2002, 529-

537.

8. L. Delannay, O.V. Mishin, D.J. Jensen and P. van Houtte, Acta Mater., 49, 2001,

2441-2451.

9. V. Randle, H. Hansen and D.J. Jensen, Phil. Mag. A, 73, 1996, 265-282.


11. A.D. Rollett, D.J. Srolovitz, R.D. Doherty and M.P. Anderson, Acta Metall., 37,

1989, 627-639.

12. E. Nes and J.K. Solberg, Mater. Sci. Techn., 2, 1986, 19-23.


14. I.L. Dillamore and H. Katoh, Met. Sci., 8, 1974, 73-83.

15. W.B. Hutchinson, Met. Sci., 8, 1974, 185-196.

16. D.P. Field and H. Weiland, Mater. Sci. For., 157-162, 1994, 1181-1188.

17. X.Y. Wen and W.B. Lee, in 'Sheet Metal Forming Technology', M.Y. Demeri (Eds),

The minerals, Metals & Materials Society, 1999, 233-243.

18. E.M. Lauridsen, D.J. Jensen, H.F. Poulsen and U. Lienert, Scr. Mat., 43, 2000, 561-

566.

19. R.D. Doherty, D.A. Hughes, F.J. Humphreys, J.J. Jonas, D.J. Jensen, M.E. Kassner,

W.E. King, H.J. McQueen and A.D. Rollett, Mater. Sci. Eng. A, 238, 1997, 219-

274.

20. S.P. Chen, D.N. Hanlon, S. van der Zwaag, Y.T. Pei and J.Th.M. de Hosson, J. Mat.

Sci., 37, 2002, 989-995.

21. J. Gil Sevillano, P. van Houtte and E. Aernoudt, Scr. Met., 10, 1976, 775-778.

22. J. Gil Sevillano, P. van Houtte and E.A.D. Aernoudt, Pro. Mat. Sci., 25, 1980, 69-

412.

23. N. Hansen and D.A. Hughes, Phys. Sta. Sol. (b), 149, 1995, 155-172.

24. F.J. Humphreys, Acta Mater., 45, 1997, 4231-4240.

Chapter 466



35-51.


27. H.J. Bunge, in 'Texture Analysis in Materials Science', London, Butterworths, 1982.

28. J.C. Blade and P.L. Morris, in 'Proc. 4th Int. Conf. on Textures', Cambridge, 1975,

171-178.

29. L. Ryde, W.B. Hutchinson and S. Jonsson, in 'Recrystallisation '90', T. Chandra

(Eds), Warrendale, PA, TMS, 1990, 313-318.

30. P. Cotterill and P.R. Mould, in 'Recrystallization and Grain Growth in Metals',

Surrey Univ. Press. London, 1976.

31. P.L. Orsetti Rossi and C.M. Sellars, in 'Aluminium Alloys, Their Physical and

Mechanical Properties, Proc.of 6th Int. Conf. on Aluminium Alloys, ICAA-6', T.

Sato, S. Kumai, T. Kobayashi, and Y. Murakami (Eds), The Japan Institute of Light

Metals, 2,1998, 1227-1232.

32. N. Hansen and D.J. Jensen, Metall. Trans. A, 17, 1986, 253-259.

S.P. Chen

Chapter 5

A FE comparison study of hot rolling

operation and PSC testing

A comprehensive study on the comparison of hot rolling and Plane Strain Compression

testing (PSC) is reported in the present chapter. The evolution of the key physical

variables during hot rolling and PSC testing are quantified and compared, using Finite

Element Method (FEM) calculation. Calculated results show that the process variables,

i.e. local stress, strain, strain rate and temperature, vary from entry to exit and

throughout the thickness of work piece during hot rolling, while the evolution patterns

of these parameters during PSC testing are quite uniform under constant nominal strain

rate. Therefore, for a given amount of nominal deformation (reduction), the loading

history, the actual distributions of these process variables in time, is quite different. It is

indicated that a further examination of the deformation history should be made when

applying the relationships established from PSC testing to real industrial hot rolling

operation.

5.1. Introduction

Current studies in the literature assumed that PSC adequately simulates hot rolling in

terms of microstructure development and hence has the potential to allow laboratory

scale experiments to be performed rather than carrying out expensive, full scale plant

trials [1,2]. The principal reason is that both deformation processes approach plain strain

compressive deformation. However, application of the relationship of the basic physical

metallurgy of hot working obtained from PSC testing to industrial rolling operation

Chapter 568

requires accurate knowledge of the time variation of the temperature and the strain rate.

Although many investigators have developed mathematical models to predict the

temperature distribution and the strain distribution of the slab during hot rolling [2-4],

very few publications paid attention to the deformation history and the variation of the

strain rate pattern during hot rolling, which have been recognized as two important

parameters for hot deformation. In practical hot rolling conditions, where the nominal

strain rate ranges from 0.1 to 100 /s, the yield stress characteristics of the material are

strain rate rather than strain dependent. In addition, significant differences in

microstructure occur depending on the deformation history, which are reflected in the

subsequent recrystallization kinetics and the final grain size [5].

The variation in local strain and strain path in samples deformed in PSC testing has

been studied in detail using analytical slip-line-field theory [6], grid analysis [7], FEM

[8-10] with the aim of determining the optimum sample geometry and friction

conditions for as uniform deformation conditions in as large a part of the sample as

possible. Although the shape similarity is not maintained in PSC testing as it is in

rolling it was found that an acceptable uniform deformation pattern was obtained forcases in which the tool width to specimen thickness ( /w h ) ratio is larger than 1.5.

In this chapter, the deformation behavior and heat transfer phenomena in AA1050

alloys during hot rolling and in PSC testing at elevated temperature were studied using

FEM simulation to obtain the simultaneous variations of the field quantities. The

detailed strain, stress, strain rate and the temperature history of elements throughout the

rolling strip and inside the specimen in the PSC testing are quantified and analyzed.

Such a study provides a better understanding of the hot rolling and the PSC test, and

therefore, gives guidance to procedures to determine appropriate values for the

experimental parameters in PSC test to simulate hot rolling operation more accurately.

5.2. Mathematical approach

5.2.1. Constitutive model

A simple AA1050 commercial aluminium alloy is used as the model material in this

study. The physical properties of this alloy as a function of temperature are listed in

Table 1. The constitutive equation for the material used in the calculation was derived

from hot, plain strain compression tests on a Gleeble 3500, with the ratio of the tool

width to sample thickness being 2. The shape of stress strain curve at elevated

temperatures is well described by a generalized Voce type of equation [11]:

0( ) exp( / )ss ss ssC (1)

A FE comparison study of hot rolling and PSC testing 69

with 0 1 1lnA Z B , 2 2lnss A Z B , 13 3sinh ( )mC A Z B , exp( / )def g defZ Q R T .

where is nominal strain, is nominal strain rate; 0 , ss and are yield stress, the

steady-state stress and the flow stress, respectively; Z is the Zener-Hollomonparameter, 156 kJ/moldefQ , the apparent activation energy of the deformation process

during plastic flow, gR , the universal gas constant, T , the absolute temperature; iA , iB ,

m are fitting parameters whose values are given in Table 2.

Table 1. Physical properties of AA1050 used to run FEM simulation in Marc

Temperature

(°C)

Coefficient of

thermal

expansion

(x10-5) K-1

Specific heat

(N/mm2 K-1)

Thermal

conductivity

(W/m K-1)

E

(GPa)

20 2.20 900 230 71.8

100 2.39 938 232 68.8

200 2.43 984 234 65.1

300 2.53 1030 236 61.3

400 2.65 1076 237 57.6

500 2.82 1116 238 53.8

Table 2. Fitting parameters for the constitutive model used

1A 2A 3A 1B 2B 3B m

3.4 4.92 111.6 1.47x109 5.58x108 1.91x107 0.28

5.2.2. Hot rolling operation

1. Analytical analysis

The nominal strain during rolling deformation, when assumed to be homogeneously

distributed through the thickness of the strip, is given by:

02ln

3 f

h

h (2)

Chapter 570

while the instantaneous strain rate is defined as:

2 1

3

dh

h dt (3)

Using the geometric relation as shown in Fig. 1, Eq. (3) can be reformulated into:

2 2 sin

2 (1 cos )3R

f

V

h R (4)

Eq. (4) indicates that the instantaneous strain rate depends on both the roll gap geometry

and the roll speed. The strain rate reaches a maximum value at the entry and decreases

to zero at the exit. The corresponding average value of the nominal strain rate is given

by integrating Eq. (4):

0( )R

f

V

R h h (5)

where R is the radius of the roll, 0h and fh are the entry and exit thickness of the work-

strip, respectively, h is current height and t is time, RV R with the rotation rate

of the roll, and is defined in Fig. 1.

Fig. 1. Roll gap geometry and sampling plane evolution during hot rolling.


2. Finite element analysis

As the rolling operation is performed at elevated temperature, a heat-transfer analysis is

included and coupled with the deformation analysis. The temperature profile in the slab

during hot rolling was calculated under the following assumptions. 1). Since the length

of the piece is much greater than either its width or thickness, modeling of this

particular process can be reduced to a 2D approach. 2). Oxide formation does not

significantly affect the heat transfer between the strip and the work roll. 3). The work

rolls are assumed not to change shape or size during the deformation process, and

remain perfectly spherical in cross section.

The mathematical model for a nonlinear thermomechanical process is given by the

following coupled equation [12]:

( , )K Tu u f (6)

p

T TU C k q

x y y (7)

1r rr pr r

T TC rk

t r r r (8)

Eqs. (7) and (8) are, respectively, the governing equation for heat flow in the slab and inthe work roll. K is the material stiffness matrix, u is the nodal displacement vector,T

is the nodal temperature, f is the prescribed force vector, k , pC , are the thermal

conductivity, specific heat and the density of the slab respectively. rk , prC , r are the

thermal conductivity, specific heat and the density of the roll respectively. U is the stripentry velocity in mm per second ( x Ut ). q is a heat-generation term representing the

heat released due to plastic work and is given by:

q (9)

where is the equivalent flow stress, is the equivalent strain rate, and is the

efficiency of conversion of strain energy to heat which is taken to be 0.95, as has been

determined empirically for aluminium alloys [1].

The finite element formulation of the mathematical model given by Eqs. (6), (7) and (8)

leads to a set of non-linear equations which in MARC [13] are solved through the full

Newton-Raphson method. The flow formulation method and an Arbitrary Eulerian

Langragian Technique are used. Continuum mechanics is employed as the deformation

behavior is on a macro scale.

Chapter 572

Due to symmetrical geometry, computations are carried out only in top half after

assuming equal conditions on top and bottom. Initial conditions are listed in Table 3.

Furthermore, the following boundary conditions are imposed:

(1). Symmetrical cooling occurs at the centerline of the strip, the temperature change inthe work roll is confined to a thin layer ( =25 mm):

0, 0 0T

t y ky

(10)

0, 25 0r

Tt r R k

r (11)

(2). heat transfer between the strip and the work roll

0, ( ) / 2, ( )con strip roll

Tt y Y t k h T T

y (12)

(3). heat transfer to the environment at surface boundaries exposed to air

0, ( ) / 2, ( )env strip room

Tt y Y t k h T T

y (13)

where ( )Y t is the strip thickness at each zone.

(4). the velocity at the center of the strip in the y direction is zero due to the geometric

symmetry.0, 0, 0t y V (14)

Table 3. Variables used for simulation

Work roll Initial temperature °C 50

Strip initial temperature °C 390

Environment sink temperature °C 20

Work roll thermal conductivity (W/m K) 36

Work roll heat capacity (N/mm2K) 3.77Film coefficient to environment (strip and roll) envh 0.01

Contact heat transfer coefficient (roll) kW/m2K conh 20

radius of the rolls (mm) 100

Shear friction for rolling 0.6


5.2.3. Plain strain compression

Plain strain compression is simulated under an imposed constant nominal strain rate

which is controlled by the velocity of the upper tool. The displacement boundary

conditions can be summarized in terms of the applied strain in the y direction:

0 0xU at x (symmetry)

0 (1 exp( ))yU h t at y h (15)

0 0yU at y

where xU and yU are, respectively, the displacement along x and y direction. For

geometric symmetry reasons, simulation is only done in one half of the specimen to

save computational time.

5.2.4. Friction condition

Friction conditions at the roll-workpiece interface greatly influence the metal flow and

the formation of surface. In the present study, the method used to model the interface

friction is the interface shear factor approach, defined by [14]:

0

2[ arctan ]s

s s

uf µk

u (16)

where sf is the frictional stress, sk is the shear strength of the work strip, su is the

relative velocity between roll and work slab, 0u is a constant, usually takes a value of

(0.01 0.1) su . The interface shear factor, µ , represents the ratio of the interface shear

stress of the object being deformed to the actual material shear strength ( /i sµ k ),

where i is the interface shear stress.

5.3. Results

5.3.1. Simulation of hot rolling process

In the hot rolling deformation study it is assumed that the radius of the rolls is 100 mm.

Simulations are performed under 4 different conditions, i.e., two different reductions,50% ( 0 60h , 30fh ) and 37.5% ( 0 48h , 30fh ) with nominal strain rates of

2.5 /s and 0.25 /s. The interface friction coefficient and the relative velocity are selected

by sensitivity analysis. It is found that an increase in friction coefficient from 0.4 to 1.0

does not significantly affect the pattern of the equivalent strain and equivalent strain ratedistribution during hot rolling. Variation of the parameter 0u in Eq. (16) in the range of

Chapter 574

(0.01 0.1) su has little effect on the strain distribution and the roll force. In the

following hot rolling simulations, µ=0.6 and 0 0.1 su u are applied.

1. The deformation history during hot rolling

We first consider the case of 50% nominal reduction ( 0 60h , 30fh ) rolled at a

nominal strain rate of 2.5 /s. Let us follow a perpendicular plane of nodes throughthickness (C-S) pass through the roll gap (see Fig. 1). When it enters the roll gap, theplane bends gradually. The temperature profile, equivalent strain, equivalent stress, andequivalent strain rate along this row at 6 increments are shown in Fig. 2a-d.

Fig. 2. The equivalent strain, strain rate, stress and temperature profiles of the

sampling plane from surface to center at various positions for the case of a nominal

reduction of 50% and a nominal strain rate of 2.5 /s.

310

320

330

340

350

360

370

380

390

400

410

0 0.2 0.4 0.6 0.8 1Relative distance from surface

Te

mp

era

ture

°C

S1-C1

S2-C2

S3-C3

S4-C4

S5-C5

S6-C6

0

0.2

0.4

0.6

0.8

1

1.2


Eq

uiv

ale

nt str

ain

S1-C1 S2-C2S3-C3 S4-C4S5-C5 S6-C6

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5


Equ

ivale

nt str

ain

ra

te (

1/s

)

S1-C1 S2-C2

S3-C3 S4-C4

S5-C5

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

Relative distance from surface

Eq

uiv

ale

nt

str

ess (

MP

a)

S1-C1 S2-C2S3-C3 S4-C4S5-C5 S6-C6


At position C1-S1, about 5 mm ahead of the entry plane, the surface node S1 is not yet

in contact with the roll. Due to surrounding material being sucked into the roll gap, the

equivalent strain, stress, and strain rate begin to increase while the temperature shows a

little parabolic fall towards the surface due to air cooling of the stock on leaving the

reheating furnace.

At C2-S2, 5 mm behind the entry plane, the test plane is in the roll gap. The temperature

begins to drop quickly near the surface because of the chilling effect of the cold roll.

The equivalent strain rate, equivalent strain and equivalent stress now increase

significantly throughout the thickness.

At C3-S3, 15 mm from the entry, the test plane is at about one third of the contact arc

length, the temperature near the surface continues to drop but at center the temperature

increases due to heat generation from deformation. This temperature rise is obliterated

nearer the surface by the chilling effect of the rolls. The strain rate at the surface

decreases but reaches a maximum level at the center. Correspondingly, the stress

reaches its highest value in the center.

At C4-S4, 30 mm from the entry, the test plane is at about two third of the contact arc

length, the temperature begin to increases both at the surface and in the center. The

strain rate drops to a lower level, which is rather uniformly from surface to center. The

stress also decreases.

At C5-S5, 49 mm from entry, the test plane comes to the exit of the roll gap, the strain

rate drops to zero. The stress continues to decrease. The temperature near the surface

keeps increasing but decreases in the center.

At longer times (C6-S6, 16 mm away from the exit plane) the strain distribution does

not change. The stress drops down to a lower level, which may be considered as

residual stress. The temperature rebounds rapidly near surface but drops monotonically

in the center due to high thermal conductivity of aluminum alloy. The temperature

difference between surface and center of the slab becomes smaller.

2. The effect of the nominal strain rate and nominal reduction

In order to demonstrate the deformation history more clearly, the traces of two sampling

points C (at center of the strip) and S (on the surface) are followed. Fig. 3 and Fig. 4

show the evolution of equivalent strain, strain rate and temperature and resulting

equivalent stress during rolling operation under the 4 deformation conditions considered

(see section 3.1).

Chapter 576

The equivalent strain increases as sampling points pass through the roll gap (Fig. 3a and

Fig. 4a). The strain at point S is larger than that in point C, which is approximately same

value as the prediction by Eq. (2). However, the equivalent strain is also affected by the

nominal strain rate (see Fig. 3a). The same level of strain at point C occurs earlier in

case of a higher nominal strain rate. This may be a general characteristic of the rolling

operation.

Fig. 3. The effect of the nominal strain rate on the deformation history of the sampling

points C and S with a nominal reduction of 50%, (a). equivalent strain, (b). equivalent

strain rate, (c). temperature, (d). equivalent stress.

0

10

20

30

40

50

60

70

-20 0 20 40 60 80 100Distance from entrance (mm)

Eq

uiv

ale

nt str

ess (

MP

a)

Point C,2.5/s

Point S,2.5/s

Point C,0.25/s

Point S,0.25/s d

0

0.2

0.4

0.6

0.8

1

1.2

-20 0 20 40 60 80 100

Distance from entrance (mm)

Equ

ivale

nt str

ain

Point C,2.5/s

Point S, 2.5/s

Point C,0.25/s

Point S,0.25/s

eq.(2)

a

230

270

310

350

390

430


Tem

pera

ture

°C

Point C,2.5/s Point S,2.5/s

Point C,0.25/s Point S,0.25/sc

0

1

2

3

4

5

6

7

8


Equ

ivale

nt

str

ain

rate

(1

/s)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Eq

uiv

ale

nt str

ain

ra

te (

1/s

)

eq.(4)

Point S,2.5/s

Point C,2.5/s

Point S,0.25/s

Point C,0.25/s

b


Fig. 4. The effect of the nominal reduction on the deformation history of the sampling

points C and S at a nominal strain rate of 2.5/s, (a).equivalent strain, (b). equivalent

strain rate, (c). temperature (d). equivalent stress.

FEM simulation shows that the strain rate evolution is much more complicated (Fig. 3b

and Fig. 4b). A very high strain rate (~3 times of the nominal strain rate) exists just

beneath the surface at the entry into the roll gap. However at the center position the

strain rate increases first and then decreases as the strip moves forwards from the entry

to exit. The peak value is slightly larger than the nominal strain rate. The peak position

appears at one third to the two third of the contact arc length and is affected by the

nominal strain rate and the nominal reduction. A lower nominal strain rate and a larger

nominal reduction shift the peak towards the exit. At the exit of the roll gap the strain

rate decreases to zero. There is a large difference between the average strain rate

derived from the analytical Eq. (4) and the values from the FEM simulation over much

of the contact distance while the shape of the two curves is more or less comparable.

0

0.2

0.4

0.6

0.8

1

1.2


Eq

uiv

ale

nt str

ain

eq.(2),37%

Point C,37%

Point S,37%

eq.(2),50%

Point C,50%

Point S,50%

a

0

1

2

3

4

5

6

7

8


Eq

uiv

ale

nt str

ain

ra

te (

1/s

)

eq.(4),37%

Point C,37%

Point S,37%

eq.(4),50%

Point C,50%

Point S,50%

b

290

310

330

350

370

390

410


Te

mp

era

ture

°C

Point C,37%

Point S,37%

Point C,50%

Point S,50%

c

0

10

20

30

40

50

60

70


Eq

uiv

ale

nt str

ess (

MP

a)

Point C,50%

Point S,50%

Point C,37%

Point S,37%

d

Chapter 578

Temperature profiles demonstrate the strong cooling effect of the rolls, the rotation rate

of the roll (nominal strain rate) and the nominal reduction. The surface temperature of

the strip experiences large variations as it passes through the roll gap. The larger

reduction, the greater is the drop in slab surface temperature. The temperature in the

center point C is more sensitive to the nominal strain rate as can be seen by comparing

the Fig. 3c and Fig. 4c. At a higher strain rate the process in the point C becomes almost

adiabatic and the heat generated raises the temperature whereas at low strain rate the

temperature in point C will decreases since there is enough time for the internal heat

generated by inelastic deformation to be dissipated towards surroundings.

Fig. 5. The effect of the friction coefficient on the deformation history of the sampling

points C and S with a nominal reduction of 50%, (a). equivalent strain, (b). equivalent

strain rate, (c). temperature, (d). equivalent stress.

0

0.2

0.4

0.6

0.8

1

1.2

1.4


Eq

uiv

ale

nt

str

ain

Point S,µ=0.8

Point C,µ=0.8

Point C,µ=0.5

Point S, µ=0.5

eq.(2)

a

290

310

330

350

370

390

410


Tem

pera

ture

°C

Point C,µ=0.8

Point S,µ=0.8

Point C,µ=0.5

Point S,µ=0.5

c

0

10

20

30

40

50

60

70


Equ

ivale

nt str

ess (

MP

a)

Point C,µ=0.8

Point S,µ=0.8

Point C,µ=0.5

Point S,µ=0.5

d

0

1

2

3

4

5

6

7

8


Equiv

ale

nt str

ain

ra

te (

1/s

)

Point S,µ=0.8

Point C,µ=0.8

Point S,µ=0.5

Point C,µ=0.5

eq.(4)

b


As a combined effect of the strain rate, strain, and the temperature, the equivalent

stresses at each point also presents a peak evolution during the rolling process. The

position of the maximum values for the Point C corresponds to that of the equivalent

strain rate, and is affected by nominal strain rate and nominal reduction. At the entry of

the roll gap, although the strain rate is very high the equivalent stress at the surface is

low, because the strain is small and the steady state has not been reached yet.

3. Effect of the friction between the roll and strip

In hot rolling it is usually assumed that the friction is sufficiently high to give 'sticking'

friction over the whole arc of contact. As shown in Fig. 5, an increase in the friction

coefficient leads to a rise of the strain level near surface but has less effect on the strain

in the center. The strain gradient throughout the thickness increases significantly.

However, the friction has little effect on the strain rate history and no effect on the roll

strip temperature in the center of the strip.

5.3.2. Simulation of PSC test

For plain strain compression, the initial temperature is set at 390°C for both sample and

tools. We first ignore the heat generation due to the deformation which is the actual

experimental condition on Gleeble, where the temperature is controlled by a sensitive

temperature controlling unit involving a direct current heater and a water cooling

circuit. Because the heat loss by radiation and convection from the specimens to the

atmosphere is negligible, an isothermal deformation can be assumed. Then a

comparison was run for specimens with heat generation from plastic deformation. Weapply this simulation for the case / 2w h , where the geometrical effect of the tool to

specimen dimensions can be avoided. In an ideal frictionless compression test the

distribution of the local strain, strain rate and stress in all parts of the specimen is

considered to be uniform. The applied strain rate is constant as the nominal reduction

increases. However, the presence of friction will decrease the strain, strain rate at the

specimen surface while increasing these values in the center. Fig. 6 shows the

distribution of the equivalent strain rate along the y-y symmetrical plane for the cases of

a nominal strain rate of 2.5 /s with an interface fraction coefficient µ=0.2. As the

friction coefficient increases, the equivalent strain and strain rate gradient between the

surface and the center increases. Obviously, in PSC testing and hot rolling friction has a

different effect on the evolution of the process parameters.

When heat generation by deformation is considered the temperature will increase during

deformation. The temperature gradually rises as the test proceeds and reaches its highest

value at the end of the test, i.e. when the maximum deformation is attained. The

Chapter 580

temperature increase depends to a large extent on the nominal strain rate applied. For

example, at a nominal strain rate of 0.25 /s, the temperature increases only 1.7°C after

deforming to a nominal reduction of 65%, while for a higher strain rate of 2.5 /s, the

temperature increase is 10°C. Fig. 7 shows the effect of the nominal strain rate on the

temperature in the specimen center and its effect on the equivalent stress. The

temperature distribution is more or less uniform in the deformation area. This

temperature increase slightly lowers the equivalent stress.

5.4. Discussions

5.4.1. Characteristics of hot rolling deformation

The above results show the equivalent strain experienced by the stock during the rolling

pass is completely non-uniform. It increases with time to a maximum at the exit of roll

gap and varies through thickness. The strain level at the slab center corresponds closely

to the values of nominal true strain for each reduction and it is affected by the nominal

strain rate applied. The excess strain near the surface represents the redundant shear

brought about by constraining the metal to flow through the roll gap.

The variation of the local equivalent strain rate is more complicated and varies from

entry to exit and from point to point through the thickness. The values in both the

surface and center of the sheet are significantly different from the analytic prediction

given by Eq. (4). The strain rate pattern in the center of the rolling strip experiences a

slow increase at first and then decreases gradually to zero.

The importance of a knowledge of the accurate temperature of the slab within the

deformation zone of the rolling operation has been recognized [15,16]. Most of the

literature documented only pays attention to the temperature at the exit of the roll gap.

Of course this temperature is important because the following softening procedure is

strongly dependent on it. However, use of this exit temperature in the constitutive

equations for whole deformation process may not be so reasonable, especially for lower

nominal strain rates. In the condition of a higher nominal strain rate the temperature at

which the peak stress is reached is approximately the exit temperature. However, for

lower nominal strain rates, the temperature at which the equivalent stress reach its

maximum differs considerably from the temperature at the exit.

As can be seen, a severe deformation (lower temperature, larger equivalent strain and

higher initial strain rate) occurs near the surface of the rolling strip. So the

characterization of the deformation and the following softening behavior should be

applied to the center area.


Fig. 6. The distributions of the equivalent strain, strain rate and stress along the

symmetrical line in the tool direction at several nominal reductions for the case of a

nominal strain rate 2.5 /s, no heat generation, µ=0.2.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 2 4 6 8 10Distance from upper tool (mm)

Eq

uiv

ale

nt str

ain

0.71

0.63

0.5

0.39

0.3

0.16

a

30

35

40

45

50

55

0 2 4 6 8 10Distance from upper tool (mm)

Eq

uiv

ale

nt

str

ess (

MP

a)

0.71 0.63

0.5 0.39

0.3 0.16

c

0

0.5

1

1.5

2

2.5

3

3.5

4

0 2 4 6 8 10

Distance from upper tool (mm)

Eq

uiv

ale

nt

str

ain

ra

te

0.71 0.63 0.5

0.39 0.3 0.16

b

Chapter 582

Fig. 7. The effect of the nominal strain rate on the temperature and the equivalent stress

(µ=0.2).

It should be noted that a considerable strain, strain rate and stress are present before the

material enters into the roll gap. The equivalent stress reaches a maximum stress at

about one third to two third of the contact arc length from the entry. The fact that the

equivalent stress reaches a maximum value early in the roll gap could have important

practical implications. This will influence the following processes such as recovery and

recrystallization, as they both depend on the stored energy from the prior deformation

and the recrystallization temperature. This result is in good agreement with experiments

by Sheppard et. al [17]. They observed an imperfect cellular structure with ragged walls

and considerable dislocation activity on the spot which is still outside the quasi-static

deformation zone. At the onset of observable macroscopic plastic deformation the

electron micrograph indicated a structure consisting of well developed subgrain walls

but containing a multiplicity of microcells within each subgrain. Within one third of the

contact arc length from entry, the subgrain cell decreases. Beyond this point the

subgrain size is stabilized and no further change in size could be observed.

5.4.2. Comparison of the hot rolling operation and PSC testing

It can be concluded from the present simulation that only the history of equivalent strain

in the center of the rolling slab is comparable with that in the PSC testing, especially for

lower strain rates. However, as a key parameter for hot deformation, the evolution of

equivalent strain rate during deformation in both cases is quite different. In the PSC

testing a constant equivalent strain rate during deformation is obtained under a constant

30

35

40

45

50

0 0.2 0.4 0.6 0.8Nominal reduction

Equ

iva

len

t str

ess (

MP

a)

385

390

395

400

405

410

Te

mp

era

ture

°C

0.25/s 1/s 2.5/s


nominal strain rate experiment. In real industrial rolling operation the equivalent strain

rate in the center of the slab presents a peak variation.

It is assumed that the experimental PSC testing is carried out in an isothermal way. In

practice this is true at low strain rates since the internal heat generated by friction and

inelastic deformation is dissipated towards the surroundings, and therefore the specimen

temperature is not raised. However, at a high strain rate the process becomes almost

adiabatic and the heat generated raises the specimen temperature. For hot rolling

operation under higher strain rates the temperature in the sampling point in the center of

the strip will increase, but under lower strain rate the temperature drops significantly.

The friction in both cases also presents a different effect. In the hot rolling condition,

the increasing friction coefficient between the roll and the strip has little effect on the

distribution and the evolution of the deformation parameters in the center point.

However, increasing the friction coefficient results in an increase in the equivalent

strain, strain rate in the center sampling point in the case of PSC testing.

5.5. Conclusions

The evolution of the equivalent strain rate and temperature during hot rolling operation

and in constant nominal strain rate PSC testing is quite different. The highest strain rate

distribution is concentrated just beneath the surface of the slab around the entry point.

The strain rate in the center of the rolling slab experiences a peak variation, i.e. first

increases and then decreases as the slab passes the roll gap.

The equivalent stress, which is a combined effect of the strain rate, strain and the

temperature, reaches a maximum values at about one third to two third of the arc contact

length from the entry, the position of the peak corresponding to that of the equivalent

strain rate.

It is suggested that in order to simulate the rolling operation more accurately in a PSC

experiment the strain rate should be designed to impose a peak variation instead of a

constant value. The peak value is the equivalent value of the nominal strain rate. In PSC

testing the friction between the sample and the tools should be as small as possible. The

comparison should be made in a confined area which is the center of the strip or the

PSC specimen. For lower strain rates, the temperature drop in the hot rolling operation

must be taken into account.

Chapter 584

References

1. G.E. Dieter, in 'Mechanical Metallurgy', New York, McGraw-Hill, 1986.

2. M.A. Wells, D.J. Lloyd, I.V. Samarasekera, J.K. Brimacombe and E.B. Hawbolt,

Metall. Mater. Trans. B, 29, 1998, 709-719.

3. D.Q. Jin, V.H. Hernandez-Avila, I.V. Samarasekera and J.K. Brimacombe, in

'Modeling of Metal Rolling Progresses', J.H. Beynon, P. Ingham, H. Teichert, and

K. Waterson (Eds), The Institute of Materials, 1996, 36-58.

4. R. Colas, Model. Simul. Mater. Sci. Eng., 3, 1995, 437-453.

5. T. Furu, H.R. Shercliff, C.M. Sellars and M.F. Ashby, Mater. Sci. For., ,

1996, 453-458.

6. R. Hill, in 'The Mathematical Theory of Plasticity', Clarendon Press, Oxford, 1950.

7. J.H. Beynon and C.M. Sellars, J. Test. and Eval., 13, 1985, 28-38.

8. R.J. Hand, S.R. Foster and C.M. Sellars, Mater. Sci. Techn., , 2000, 442-450.

9. C.H. Lee and S. Kobayashi, Int. J. Mech. Sci., 12, 1970, 349-370.

10. S.I. Oh, S.L. Semiatin and J.J. Jonas, Metall. Trans. A, 23A, 1992, 963-975.

11. C.M. Sellars, F.J. Humphreys, E. Nes and D.J. Jensen, in 'Numerical Predictions of

Deformation Processes and the Behaviour of Real Materials', S.I. Andersen, J.B.

Bilde-Soresen, T. Lorentzen, O.B. Pedersen, and N.J. Sorensen (Eds), Riso National

Laboratory Roskilde, Denmark, 1994, 109-134.

12. O.C. Zienkiewicz, in 'The Finite Element Method', McGraw-Hill, London, 1977.

13. Marc, in 'Users' manual, Marc Analysis Research Corporation', Palo Alto, CA,

1996.

14. S. Kobayashi, in 'Numerical Analysis of Forming Processes', J.J. Pitman, O.C.

Zienkiewicz, R.D. Wood, and J.M. Alexander (Eds), Wiley, 1984, 45-71.

15. C.M. Sellars, Mater. Sci. Techn., 1, 1985, 325-332.

16. C. Devadas, I.V. Samarasekera and E.B. Hawbolt, Metall. Trans. A, , 1991, 307-

319.

17. T. Sheppard, M.A. Zaidi, P.A. Hollinshead and N. Raghunathan, in 'Microstructural

Control in Alminium Alloys: Deformation, Recovery and Recrystallization', E.

Henry Chia and H.J. McQueen (Eds), The Metallugical Society, Inc., 1985, 19-43.

S.P. Chen

Chapter 6

Modeling recrystallization kinetics in

AA1050 following simulated break down

rolling

A physical model based on the single grain approach is refined and developed to predict

the recrystallization kinetics of a single-phase metal following hot deformation. The

model involves the original and deformed grain geometry, the characteristics of the

dislocation networks formed and the average mobility of the moving interfaces. The

model accounts properly the effect of the concurrent recovery and textural components

in the deformed microstructure on the recrystallization kinetics. Experimental work is

conducted on the recovery and recrystallization kinetics in AA1050 following plastic

deformation at elevated temperatures as encountered during break down rolling to

extract the physical input parameters and to validate the model. The Plane strain

compression (PSC) test is used to simulate the hot rolling deformation. The predictions

are in good agreement with the experimental results.

6.1. Introduction

Hot working is a process of changing the shape and the microstructure of metals and

alloys at elevated temperatures. The important microstructural changes occurring during

hot working are recovery and recrystallization, which are the restoration processes that

result from the release of the internal stored energy during deformation [1-3]. Modeling

of thermomechanical processing is well established as a valuable procedure for

optimizing processing conditions and is currently the subject of major research efforts in

the aluminum industry.

Chapter 686

Several attempts have been proposed to model the recrystallization kinetics physically

[3,4]. These models consider the nucleation positions and growth behavior of the

recrystallized grains by linking them to the deformation microstructure through internal

state variables. The results have shown that such models describe the experimental

results for specific alloys well, though only for uniform processing conditions.

However, these models could not take into account of the effect of concurrent recovery

on the recrystallization kinetics properly. The contribution of the recovery on the

softening kinetics could be separated using different techniques [5] and in some cases

this effect is quite large and could not be ignored [6]. Moreover, recent experimental

investigations have revealed that the recrystallization is inhomogeneous from grain to

grain [7,8]. Such a general characteristic is believed to be caused by an inhomogeneous

distribution of stored energy. The deformed grains with different orientations have

different microstructures and stored energy. In this case not only will the growth rate

decrease during recrystallization, but growth of the grains will be restricted by

impingement with their neighbors within the original grains [1]. It appears that a more

sophisticated and fundamental description of the recrystallization should start from

considering the microstructure evolution in a single grain. An approach based on such a

principle has been proposed by the present authors to predict the recrystallization

kinetics after cold deformation in Chapters 3 and 4 [9,10]. In the single grain model, the

structure of the deformed metal is taken into account, which contains the information

about the main features of rolled material, i. e. on the degree of cold rolling, the grain

size, and the grain shape. An as-deformed tetrakaidecahedron is applied to describe the

grain geometry. The subgrain size and the misorientation between subgrains are related

to the degree of deformation. Additionally, the effect of recovery on the recrystallization

kinetics can be nicely predicted. The model has been further developed in Chapter 4 to

incorporate both microstructural inhomogeneities and the textural components in the as-

deformed state [10].

In this Chapter the model based on the single grain approach is adapted and elaborated

to predict the recrystallization kinetics following hot deformation. As a physical model,

all the input parameters are related to the microstructure and can be determined from the

experiment only leaving the pre-exponent factor of the mobility as an adjustable

parameter. Experimental study is carried out on the recovery and recrystallization

kinetics in AA1050 following simulating hot deformation in PSC to extract the input

parameters and to validate the model. The actual strain, strain rate and temperature

distribution in the PSC specimens were analyzed using FEM simulation. The model

predictions show a good agreement with the experimental results in a wide range of

annealing conditions.

Modeling recrystallization kinetics following hot deformation 87

6.2. The recrystallization kinetics model

6.2.1. The Single grain approach

A physical model to predict the recrystallization kinetics of single-phase metals, based

on deformed microstructure, after cold deformation, has been developed in Chapters 3

and 4 [9,10]. The model can be used to predict the recrystallization kinetics when only

the deformation strain and the annealing temperature are known. It takes into account

the grain geometry, the position and the density of the nucleation sites as well as stored

energy evolution. The effect of the concurrent recovery and the microstructural

inhomogeneous and textural component in the deformed state can also be incorporated.

With some adaptation we will apply it to modeling recrystallization kinetics after hot

deformation.

In the single grain approach, it is assumed that the microstructure of an well-annealed

material can be described by a distribution of regular tetrakaidecahedra of various sizes.

After rolling deformation, the grains elongate along the rolling direction. If it is further

assumed that each grain under the same strain as the macroscopic one, then a deformed

tetrakaidecahedron with a size distribution will represent the main features of rolling

deformation, degree of rolling, the grain size and grain shape.

For aluminum alloys the microstructure after hot deformation can be adequately

described by subgrain size and the average misorientation between the subgrains. In a

deformed grain with a specific orientation, the microstructure will have a variety of cell

sizes and a distribution of misorientations. We assume that the microstructure can be

described using two components [11]: an assembly of equiaxed subgrains and specific

subgrains, effectively sub-critically sized recrystallization nuclei. The former can becharacterized by a mean equivalent diameter, , a mean misorientation, , and withboundaries of mean energy and mobility, and sbM , respectively. The latter have a

larger size ( n ) and different boundary characteristics ( , ,n n nM ). In the present study

we assume that the misorientation angle between the particular subgrain and

surroundings is 15°.

The nucleation site density in a single-phase material is proposed to be given by [3]:

1 2 3 42 30 0 0

V Vv V

L S PN c P c c c (1)

where VP is the number of grain corners, where four grains meet per unit volume, VL is

the line length per unit volume where three grains meet, and vS is the grain boundary

Chapter 688

area per unit volume between two grains, P depending on finding a mobile boundary

inside an original grain, a deformation band boundary of 10 , 0 is the averagesubgrain size in the as-deformed state.

1c to

4c are constants. From the present

experimental observation and literature studies it can be concluded that nucleation at

grain boundary is the dominant term for a large regime of deformation.

Assuming grain boundary nucleation being dominant mechanism, the nucleation site

density, Nv, in a grain with a specific orientation, can be expressed as:

20 0/v d vN C S N (2)

where dC is a calibration constant,

0N , the nucleation site density contributed from

other mechanisms, 0 , the average subgrain size in the as-deformed state in a deformed

grain. The value of vS is dependent on the rolling strain and the initial grain size, and

for a tetrakaidecahedron, varies with rolling strain in a manner described by [12]:

2 2 2 213 1 2 / 3 2 / 2 2 /

2vS a a a a a a a

d (3)

where a e is the distortion along the rolling direction, d is the grain size of the

starting material.

The growth velocity of a recrystallizing grain front is described by:

nn

dRV M P

dt (4)

where V is the migration rate, nM is mobility, P is the overall driving pressure.

The mobility of a high angle boundary at an annealing temperature given by:

0 exp( )n rex gM M Q R T (5)

where rexQ is the activation energy for grain boundary motion, 0M is the pre-exponent

factor, gR is gas constant, T is the annealing temperature.

The overall driving pressure for a viable nucleus to grow is given by the sum of the

driving pressure from stored energy and the retarding pressure from boundary

curvature:


4 n

n

P (6)

Where is a geometrical constant and has a value of ~3 [1], is the subgrain

boundary energy, which is assumed to depend on as follows:

(1 ln )m

m m

(7)

where m is a value of misorientation at which the boundary approaches high-angle

character (taken to be 15° [1]) and m is high angle boundary energy per unit surface

(taken to be 0.324 J/m2 for Al [1]).

A deformed polycrystal contains many grains of different orientations. In this model,

based on the experimental fact, we assume such a deformation pattern that

microstructure may change from grain to grain and it can be related to the Taylor factorof the individual grain. The relationship between the deformation features ( , ) of an

individual grain and the average values ( , ) of the polycrystalline aggregate can be

related to Taylor factor as [10]:

2

av

M

M (8)

where M and avM are the Taylor factors of a grain with a specific orientation and the

average value of all the orientations, respectively.

We need to know the evolution of microstructural parameters: and with the flow

stress. Generally is considered to be a constant value of about 4° [2] when the steady

state has been reached and this value is assumed not to change during recovery.

A relationship exists between equivalent flow stress and , i for a variety of

deformation conditions and during recovery process, i.e. [13,14]:

1 2 /i av i avM Gb M Gb (9)

where i is the initial flow stress, G is the shear modulus, avM is the Taylor factor of

the polycrystal, b is the value of Burgers vector, and 1 and 2 are constants. During

steady state deformation the principle of similitude applies [2,3]:

Chapter 690

/i C (10)

where C is a constant of typical value of the order 5 [2]. Now the average subgrain

size in the steady state is predicted to be a function of flow stress:

0 3 ( )av s iM Gb (11)

where 3 1 2C , with 3 being of the order 3 [2], s is the flow stress in the

steady state. The subscript in 0 denotes the subgrain size in the as deformed state.

The subgrain size will increase during recovery stage. A widely used equation to

describe the recovery in Al alloys is expressed as [13]:

( )1 ln(1 )gi

rec

s i

R Tt tR

A (12)

with relaxation time parameter

0 exp rec

g

Q

R T (13)

where recR is the fraction residual strain hardening at time t , ( )t , the instantaneous

flow stress during recovery, recQ , an activation energy for recovery. A and 0 are

constants both dependent on the composition and deformation microstructure of the

material.

Combining Eqs. (11) and (12), the subgrain size evolution in a deformed grain with a

average Taylor during recovery is now given by:

0 / recR (14)

By analogy to Eq. (14), the subgrain size evolution in a specific deformed grain during

recovery can be written as:

0 / recR (15)


6.2.2. Kinetics approach

The single grain model described in Chapters 3 and 4 is elaborated along the line

described above and is used to calculate the recrystallization kinetics in individual

grains with a specific orientation. Nucleation event in an as-deformed grain is assumed

to be site-saturated. A growing recrystallized grain is modeled as an expanding semi-

sphere nucleated on grain boundaries. New grains grow independently of one another

until hard impingement occurs.

The position of nuclei on the faces of the tetrakaidecahedron are determined randomly

assuming that there is an equal probability that each coordinate on the 14 faces

(including grain edges and corners) may act as a nucleation site. Each simulation is

therefore repeated a number of times (40) and the average of the simulated kinetics is

taken as the representative, i.e. the time to obtain a given fraction of recrystallization, f,

is given by the average over the number of the simulation, j.

, , /av i i jt t j (16)

where ,i jt is the time at a given fraction in the j th simulation for a grain with a specific

orientation labeled i.

In the attempt at modeling recrystallization kinetics in a polycrystal material we assume

that there is no correlation between the orientations of neighboring grains and all the

grains of one crystallographic orientation (within a certain small range) are represented

by a single grain of the mean size.

The overall kinetics of the polycrystalline aggregate can be obtained by weighted

summation of the kinetics on the main textural components in the as-deformed state.

The fractional recrystallization f of the assembly is given by:

( )i i if f (17)

where i is the fraction of a main textural component.

6.3. Experimental

6.3.1. Material and experimental detail

The material used in the study is a commercial purity aluminium alloy AA 1050. The

as-received material is a sheet of transfer gauge of 25 mm in thickness that had been DC

Chapter 692

casted, homogenized, break-down-rolled and annealed. It has a fully recrystallized

microstructure with a grain size 104.4 µm in the center of the sheet and 95.2 µm in the

surface layer. Rectangular PSC specimens, the dimensions 10 20 15 mm, were

machined from the middle thickness of the sheet with the deformation in PSC being

parallel to the rolling direction. The deformation is applied on the sample thickness (of

10 mm) direction and the tool width is 5 mm. The PSC tests were carried out on a

Gleeble 3500. All tests were conducted in an atmosphere of vacuum. A water based

graphite lubricant was used to reduce the friction between the specimen and the tools.

A series of tests were conducted at the following combinations of temperature and strain

rate: 340°C and 0.25 /s; 390°C and 2.5 /s and 450°C and 26.2 /s, all to a strain of 0.62.

These process parameters, leading to an equal Zener-Hollomon parameter, were

calculated using an activation energy for deformation 156 kJ/mol. After deformation to

the defined strain the samples were unloaded and held at the annealing temperature,

which is the same as the deformation temperature, for various time intervals, followed

by jet water quenching.

Taking into account the deformation inhomogeneities in PSC samples, the

microstructural and micro-hardness analyses were carefully confined in three prescribed

areas. Sampling points are shown in Fig. 1. Their co-ordinates are, A (0, h/4, 0), B

(1.25, 0, 0), C (0, 0, 0) and D (0, h/8, 0), with h being the instantaneous thickness of thespecimen, for current deformation strain 5.8h . The actual distribution of the strain,

strain rate pattern is estimated from FEM simulation.

Fig. 1. Schematic illustration of the deformed specimen geometry and the sampling

points.


A composite image method [5] was applied to determine the volume fraction of the

recrystallized material. A Buehler OMNIMET MHT automatic micro hardness tester

was used for the micro-hardness measurements, using 100 g load and 15 s loading time.

Hardness tests were made after polishing and 18 measurements were taken for each

experimental point.

6.3.2. Finite element analysis

To decide the actual local strain stress distribution, the FEM program MARC [15], was

utilized. The deformation patterns (strain, stress and strain rate) are strongly dependenton two factors: the ratio ( /w h ) of the tool width to the sample thickness and the

fraction coefficient between specimen and the tool surface. The extent of the non-uniformity increases as the /w h decreases and/or the friction factor increases.

However, when the ( /w h ) is less than 1, which is the present case, the simulation

results do not show very much difference in the distribution of the strain, strain rate and

stress for various assumed frictional conditions at the sample-tool interface. Therefore

the deformation pattern is considered to be only dependent on the sample geometry.

Fig. 2 shows the simulated local equivalent strain, strain rate and stress distribution

along the vertical symmetrical line (yy in Fig. 1) for the specimens deformed to a

nominal reduction of 42%. As can be seen, the deformation patterns are identical for the

case of the deformation at 390°C, nominal strain rate 2.5 /s and the case of 340°C,

nominal strain rate 0.25 /s.

The local equivalent strain in the center of the specimen is about 1.18, which is almost

two times as that of the nominal strain (0.62). The strain rate gradient is even larger, for

example, for the case of 390°C×2.5 /s, equivalent strain rate in the center is 7.8 /s,

which is 3 times more than the nominal strain rate.

Chapter 694

Fig. 2. The distributions of a). the equivalent strain, b). strain rate, and c). stress along

the sampling line yy (Fig. 1) at a nominal strain of 0.62.

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6Distance from upper tool (mm)

Eq

uiv

ale

nt str

ain

rate

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Eq

uiv

ale

nt str

ain

rate

390°C,Nominalstrain rate 2.5/s


0

0.2

0.4

0.6

0.8

1

1.2

1.4


Equ

ivale

nt

str

ain



30

35

40

45

50

55

60


Equ

ivale

nt

str

ess (

MP

a)




6.4. Results

6.4.1. The deformation structure

The microstructure in the deformed region of a PSC sample is not homogenous. A

single X’s shaped deformation zone along the diagonals between the tool edges can be

observed. The grains become elongated along the rolling direction of the sample.

Obviously, the grains in the center of the specimen experience a larger deformation. An

area in contact with each tool over a certain length near the center undergoes a much

lower strain. In the deformed grains one or two sets of deformation bands are observed

to form within 5° of an angle of 35° to the ‘rolling plane’. Fewer deformation bands

were observed in a distance away from the center (for example, point A in Fig. 1).

Fig. 3 shows the true stress versus the true strain plots for specimens tested at the 3

conditions to a strain of 0.62, with the data corrected using a friction coefficient µ=0.08.

At a strain of 0.4 the stress has reached the steady state level of about 44 MPa. The

difference between the steady state stresses of 3 sets of testing conditions with

combined temperatures and strain rates is within 3 MPa. The overlap in the strain stress

curves indicates that the condition of equal Z value has indeed been obtained.

Fig. 3. The strain stress curves of the PSC tested at three different conditions.

6.4.2. Softening behavior by micro-hardness indentation

The softening kinetics as measured by micro-hardness testing is presented in Fig. 4. We

confined our HV measurement to an area not more than 100 µm of the point C and point

A. Clearly, the softening kinetics exhibits two stages: the first stage corresponding to

recovery, followed by an accelerated softening regime associated with recrystallization.

0

10

20

30

40

50

0 0.2 0.4 0.6 0.8Strain

Str

ess (

MP

a)

340°Cx0.03/s

390°Cx2.5/s

450°Cx26.2/s

Chapter 696

In addition, the softening kinetics due to recovery is faster in the point C where the

deformation is largest.

The percentage of softening is defined by:

def trec

def rex

H Hf

H H (18)

where defH is the hardness of the material in as-hot-deformed state, tH is the

instantaneous hardness after annealing time t , and rexH is the hardness of completely

recrystallized material. As the microhardness value is linearly proportional to stressvalue, 1rec recf R . The fraction softened in point C and A as determined by Micro-

hardness testing is shown in Fig. 4 by dotted lines.

6.4.3. Recrystallization kinetics by optical microscopy

The recrystallization kinetics in point C, annealed at various temperatures, as monitored

by the metallographic measurements, is shown in Fig. 5, which is similarly measured in

the central region within a size of 500 500 µm. Fig. 6 shows partially recrystallized

structures obtained after an anneal for 6000 s at 340°C (Fig. 6a) and after an anneal for

100 s at 390°C (Fig. 6b). As can be seen, recrystallization was initiated primarily at

prior grain boundaries; recrystallization is not homogenous, in some deformed grains

recrystallized grains appear rather large, but in some other deformed grains there are

still no nuclei to be found.

Fig. 4. Microhardness measurements of the specimens annealed at different

temperatures for various times. Solid symbols refer to position C; open symbols refer to

position A; dotted lines is the best fit of Eq. (12).

23

25

27

29

31

33

35

1.E-01 1.E+01 1.E+03 1.E+05Time (s)

Mic

roH

V

HV in point C

HV in point A

Recovery line

450°C 390°C 340°C

10-1 10

510

310

1


Fig. 5. The recrystallization kinetics measured by quantitative optical microscopy and

the softening kinetics by Microhardness test in point C. Solid symbols indicate the

optical microscopy data; open symbols indicate microhardness data.

(a) (b)

Fig. 6. Optical microstructures in the center of PSC specimens (a) deformed at 340°C,

0.25 / s to a strain of 0.62, annealing at 340°C for 6000 s, (b) deformed at 390°C,

2.5 / s to a strain of 0.62, annealing at 390°C for 100 s, showing grain boundary

nucleation and the deformation bands in the deformed grains.

0

0.2

0.4

0.6

0.8

1

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05Time (s)

Fra

ctio

n r

ecry

sta

llize

d(s

oft

en

ed

)

BY HV

BY OPM

450°C 390°C 340°C

10-1

10410

310

210

110

010

5

Chapter 698

6.4.4. The relationship between static softening and recrystallization

Fig. 7 shows a plot of the fractional recrystallization determined by optical microscopy

against the softening fraction measured in point C. As can be seen, the fractional

recrystallization was non-linearly related to the softening (after subtraction of the

recovery the softening follow a straight line as the open marks shown in the Fig. 7).

This phenomenon was previously reported by Sellars group [16]. A large amount of

softening was involved even before the recrystallization start, which is identified by

optical microscopy, and this nonlinearity becomes insignificant as the annealing

temperature increases. Because both of the recovery and the recrystallization processes

are thermally activated, these processes are strongly temperature-dependent. Since, in

general, recovery proceed with lower activation energies than those which characterize

the recrystallization process, low temperatures will favor recovery over

recrystallization. At higher temperatures, thermal fluctuations assist both the initiation

and the progress of recrystallization. The earlier start of the recrystallization process

will reduce the relative softening due to recovery. Therefore, the extent of recovery is

more significant at lower isothermal annealing temperatures.

Fig. 7. The relationship between recrystallization fraction (measured by OPM) and

static softening (by HV). Filled marks: the static softening from HV before subtraction

of recovery. Open marks: after subtraction of recovery.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1Fraction recrystallized

Fra

ctio

n s

oft

en

ed

340°C

390°C

450°C


6.4.5. The grain size of the fully recrystallized structure

In a space-filled assembly of fully-recrystallized grains, the relationship between thenumber of grains per unit volume, vN , and the average grain size, d , is given by [17]:

1/31/ vd N (19)

where is a constant having a value of unity for the case of site saturation [17]. Fig. 8

shows the grain size distribution near the different locations in a fully recrystallized

specimen annealed at 390°C. it shows the expected log-normal behavior. As the

distance away from the center point C increases the range of the grain size becomes

larger as a consequence of the deformation inhomogenity. Table 1 lists the quantitativemeasurement of mean grain size (which are given by both arithmetic mean ( d ) and

square root mean ( sd ) and calculated vN (from d ) in the fully recrystallized samples

after annealing at different temperatures. The data in the table indicate that the grain

size, hence the nucleation site density, is a strong function of the location (strain and

stress vary with location) in the sample but is not much affected by the annealing

temperature. There is surprisingly little difference in arithmetic mean grain size after

annealing at different temperatures while the square root mean grain size increases asthe annealing temperature decreases. We will use the arithmetic mean d to characterize

the grain size, and consequently, assume that the final grain size does not depend on the

annealing temperature in the range studied.

Fig. 8. The grain size distribution in the different sampling points in a fully

recrystallized specimen annealed at 390°C.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

10 30 60 90 120 150 180 210 240 More

Equivalent diameter (µm)

Fre

quen

cy

Point C,d=75.4µm

Point D,d=85.3µm

Point A,d=101µm

Chapter 6100

Table 1. The mean grain size and the number of the grains per unit volume in the

sampling points

Annealing

temperature

Point C Point D Point A

d (µm) 76.8 86.5 103.6

sd (µm) 96.5 107.5 129.4340°C

vN 2.2 10-6 1.5 10-6 9.0 10-7

d (µm) 75.4 84.3 101.5

sd (µm) 89.4 97.8 121.4390°C

vN 2.3 10-6 1.7 10-6 9.7 10-7

d (µm) 74 81.5 102.2

sd (µm) 83.6 93.7 113.5450°C

vN (µm-3) 2.4 10-6 1.8 10-6 9.4 10-7

Average ( , )VN (µm-3) 2.3 10-6 1.7 10-6 9.3 10-7

6.4.6. JMAK and the S-F analysis of the recrystallization kinetics

The JMAK equation and the Speicl-Fisher (SF) equation are used to characterize the

recrystallization kinetics. JMAK equation is [1,18]:

1 exp( )nX bt (20)

and SF equation [1,19]:

1mX

ktX

(21)

where X is the volume fraction recrystallized in time, t, and n, b m, k are constants.

The JMAK description of isothermal kinetics used a least-squares analysis to obtain abest-fit line for ln ln(1/(1 ) lnX vs t . The SF description of same kinetics used for

ln( /(1 ) lnX X vs t . Both equations describe the experimental data reasonably well in

the range of 10 to 90% of recrystallization with 2R values in the range of 0.93 to 0.98.

The resulting kinetic parameters are listed in Table 2 and Table 3.


Table 2. The fitting parameters of JMAK and SF analysis in Point C

Temp °C n lnb m lnk

340 2.138 -18.949 3.145 -27.090

390 2.222 -12.390 3.339 -7.343

450 2.313 -5.239 3.130 -6.146

Table 3. The fitting parameters of JMAK and SF analysis in Point A

Temp °C n lnb m lnk

340 1.983 -18.872 3.030 -28.017

390 2.124 -12.645 2.747 -15.749

450 2.297 -6.232 2.888 -7.2569

As can be seen, the time exponents, n and m, remain relatively constant over the test

temperatures, while the parameters lnb and lnk, being reflective of the temperature

dependence of the nucleation and the growth rates, were strongly dependent on

temperature. In the present work the kinetics data are analyzed in terms of the averagevalues of n and m (listed in Table 4), the corresponding lnb and lnk values being

recalculated at each test temperature to characterize the isothermal recrystallization

kinetics. The kinetic curves resulting from this analysis, together with the experimental

measurements are shown in Fig. 9.

Fig. 9. The JMAK and S-F analysis of the experimental data.

0

0.2

0.4

0.6

0.8

1

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05Time (s)

Fra

ctio

n r

ecry

sta

llize

d

Point C

Point A

JMAK fit

S-F fit

450°C 390°C 340°C

10-1

102

100

101

104

103

105

Chapter 6102

Table 4. The average values of n and m and the appropriate lnb and lnk recalculated

n lnb m lnk

Point C 2.22 -59345/T+77.276 3.20 -85782/T+112.5

Point A 2.13 -57871/T+74.326 2.89 -78408/T+101.4

6.5. Application of the single grain model to predict the

recrystallization kinetics

We now turn to predict the recrystallization kinetics after hot deformation using the

single grain approach. To use the model we firstly specify the determinations of the

physical parameters from the experimental results.

6.5.1. The subgrain size

From Eq. (11) the average subgrain size could be decided if we know the value of theinitial flow stress i and the steady state stress s at each sampling point. In this study

the 16i MPa determined from strain stress curve is used and taking 3 2.6 in

Eq. (11) the resulting average subgrain sizes in two sampling points are listed in

Table 5, which are in a very good agreement with the literature data [20].

The subgrain size in a grain with a specific orientation is obtained from Eq. (8) with an

assumption that the subboundary misorientations are equal in the grains with different

orientations and take to be 4°.

Table 5. The local equivalent strain, stress, subgrain size and vS in the sampling points

Strain s (MPa) 0 (µm) vS lnZ

Point C 1.16 54.4 1.56 0.062 30.34

Point A 0.44 46.2 1.99 0.037 28.96

6.5.2. The calibration constant dC

From Eqs. (3) and (11), vS and 0 depend on, respectively, the equivalent strain and

the equivalent stress at the sampling points and both are a function of the positions in

the sample. As shown in Fig. 10, along the y-y sampling line, the calculated 20vS

increases linearly with local strain and varies with local stress in a manner of power


function. The values of ( , )vN measured at 3 sampling points against strain and stress

present an identical trend, which conforms that the 20vS vs ( , )vN presents a linear

relationship, as shown in Fig. 11. The slope and the intercept of the plot ( , )vN

against 20vS yield the calibration constant 5

9 10dC = × and 9

0 7.85 10N /µm3,

respectively, which indicates that the nucleation sites from other mechanisms such as

grain corners and deformation bands are neglectable compared with that of the grain

boundary nucleation.

Fig. 10. The variation of 20vS against a). strain and b) stress.

0

2

4

6

8

10

12

14

35 40 45 50 55Stress (MPa)

Sv/

0²x

10

3

0

2

4

6

8

10

12

14

Nv(

,)x

10

7

Sv/ 0²

Nv( , )

0

2

4

6

8

10

12

14

0 0.2 0.4 0.6 0.8 1 1.2 1.4Strain

Nv(

,)x

10

7

0

2

4

6

8

10

12

14

Sv/

0²x

10

3

Nv( , )

Sv/ ²

Chapter 6104

The nucleation site density can now be calculated from Eq. (2). The effective number of

the nuclei in a deformed single grain can then be decided. Since the nuclei are assumed

to initiate from the original grain boundaries only half of the new grain grows inwards

the objective grain. So we need to modify the number of the nucleation site calculated

by a factor of 2, hereafter, this is referred as the numerical number of the nucleation

sites and used in simulation.

Fig. 11. A linear relationship between ( , )vN and 20vS .

6.5.3. Parameters for recovery

By applying Eqs. (12) and (18) to the present experimental data, the best fit solutions

are shown in Fig. 4, and the fitting parameters are listed in Table 6. The reaction is

associated with an activation energy of 192 kJ/mol, which is a typical value of

activation energy for Fe diffusion in aluminium [21]. It suggests that the very smallamount of iron in this alloy control the recovery process. 0 is temperature dependent

and it is given by:

0 1 2bT b (22)

Table 6. Fitting parameters for recovery

A (kJ/mol) b1 b2 Qrec( kJ/mol)

Point C 95.8 3.6374x10-18 2.1304x10-15 192

Point A 95.8 2.4033x10-17 1.4144x10-14 192

0

2

4

6

8

10

12

0.002 0.004 0.006 0.008 0.01 0.012 0.014

Sv/ 0²

Nv(

,)x

10

-7 (µ

m-3

)


6.5.4. Activation energy for recrystallization

An apparent activation energy can be found by assuming that recrystallization process

follows an Arrhenius relationship with respect to the temperature, i.e., that

exp rexR

Qt

RT (23)

where rexQ is the activation energy related to the recrystallization processes.

The time Rt , required for 25%, 50%, 68% and 75% recrystallization was calculated for

each annealing temperature using the JMAK analysis; the RQ value was estimated from

linear plots of ln(1/ )Rt vs 11/ ( )T K , resulting in an apparent activation energy of

230 kJ/mol in the center point C and 225 kJ/mol in the point A. These values are in

good agreement with that of 234 kJ/mol reported by Perryman's study [22] and that of

230 kJ/mol by Sellars at al [23].

6.5.5. Textural components after hot deformation

A wide matrix of experimental plane strain compression data on the microstructure and

texture in hot worked AA1050 has been generated in the literature. The main texture

components found in AA1050 after hot deformation are: cube {100}<001>, Goss

{110}<001>, brass {110}<112>, copper (C) {112}<111> and S {123}<634>. The

concentrations of the textures are a function of the strain but not a function of Zener-

Holloman parameter. The amount of Cube component after hot working decrease as the

strain increases. The overall amount of rolling texture components increases, while the

ratio of the four rolling textural components Goss, brass, copper and S in AA1050 was

found not to alter in a range of hot working conditions. In this study we take the data

from Vernon-Pary et al [24], as listed in Table 7, as inputs for the modeling and

assuming that the grain with an orientation which has the larger Taylor factor

recrystallizes faster. The Taylor factors of the different texture components are

determined from H.J. Bunge [25].

Table 7. The input for the volume concentrations of texture components in two

sampling points

Sampling points Volume fraction (%)

Cube Goss Copper brass S Random

Point A 5.2 9 8.1 11.6 20.4 45.7

Point C 3 8.9 8.9 12.8 24.3 42.1

Chapter 6106

6.5.6. The critical size of a viable nucleus

Another input parameter is the critical size of a viable nucleus. According to

Thompson [26], the condition for instability, leading to the growth of a potential

nucleus is:

0nn

dR dRR R

dt dt (24)

where R and Rn are the average subgrain radius in the assembly and radius of the

potential nuclei, respectively.

Using Eqs. (4), (6), (12) and (15) and taking 2R , Eq. (24) is solved for the

condition that the left-hand side equals zero and the critical diameter of the potential

nuclei is given as:

2( ) 2criticl

n n n n

d dM M

dt dt (25)

where

0

2( )g

rex

R Td

dt A t R (26)

In the present modeling, it is assumed that the nucleation is site saturated and the

boundary between the potential nuclei and surroundings is high angle boundary, the sizeof the critical nuclei can be calculated by using Eq. (25) taking 0t .

6.5.7. Prediction of recrystallization kinetics

Until now we have determined all the input parameters except the pre-exponent factor

of the mobility, which is used as a tuning parameter. This constant is determined to be60.68 10 m4/Js by adjusting it to produce coincidence between the prediction and the

experimental points for the case of point C annealing at 390°C.

The predicted recrystallization kinetics in point C and point A for the case of 390°C

annealing by single grain model is shown in Fig. 12a and Fig. 12b. The recrystallization

kinetics in a grain with different orientations is taken as average of 40 simulations with

different nucleation site positions. The overall recrystallization kinetics is the weighted

summation of the kinetics from various textural components.

The overall recrystallization fraction against time in logarithmic scale depicted in the

small box of these figure shows a two-stage JMAK n behavior. The apparent derivation


occurs as the recrystallization increases up to 90%. The derivation is more obviously

seen in recrystallization rate in point A (Fig. 12b) where is subjected to a smaller

deformation. This is the contribution of the Cube texture component, which has been

predicted to have a sluggish recrystallization kinetics.

(a)

(b)

Fig. 12. The predicted recrystallization kinetics in point C (a) and in point A (b) at

annealing temperature 390°C, thin curves: from leftmost to rightmost, the

recrystallization kinetics in grains with Copper, S, B, Random, Goss and Cube. Square

pointed curve: the overall recrystallization kinetics.

0

0.2

0.4

0.6

0.8

1

0 400 800 1200 1600

Annealing time (s)

Fra

ction

-4

-3

-2

-1

0

1

2

3.5 4.5 5.5 6.5

ln time (s)

lnln

(1/(

1-x

))

0

0.2

0.4

0.6

0.8

1

0 400 800 1200 1600 2000 2400

Annealing time (s)

Fra

ction

-5

-4

-3

-2

-1

0

1

2

4 5 6 7 8

ln time (s)

lnln

(1/(

1-x

))

Chapter 6108

Fig. 13 shows the effect of the concurrent recovery on the recrystallization kinetics. The

effect of concurrent recovery on the recrystallization kinetics decreases as annealing

temperature increases and is slightly affected by the degree of the deformation.

(a)

(b)

Fig. 13. The predicted recrystallization kinetics in point C (a) and in point A (b) at 3

different annealing temperatures, cross-dotted curves: no concurrent recovery, circle

solid curves: with concurrent recovery.

0

0.2

0.4

0.6

0.8

1

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05Annealing time (s)

Fra

ction

450°C 390°C 340°C

100

105

104

103

102

101

10-1

0

0.2

0.4

0.6

0.8

1

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05Annealing time (s)

Fra

ction

450°C 390°C 340°C

10-1

104

103

102

101

100

105


Fig. 14 is a comparison of the prediction by the single grain model with the

experimental data at three annealing temperatures, together with JMAK fit. It can be

seen that the single grain model yields a good prediction of the experimental data in the

whole range of the recrystallization.

Fig. 14. The comparison of the experiment with the predicted recrystallization kinetics

at 3 different annealing temperatures, solid points: experiment in point C, empty points:

experiment in Point A. thin solid curve: JMAK fit, Point dotted curves: prediction by

single grain model.

6.6. Conclusions

1. The static softening kinetics of a commercial aluminum alloy AA1050 following hot

deformation includes both recovery and recrystallization. The rate of the static

restoration increases with increasing local equivalent strain and strain rate, i.e. with

increasing local equivalent stress.

2. The recovery kinetics could be modeled by the logarithmic decay relationship. The

contribution of the recovery to the softening kinetics decreases as the annealing

temperature increases.

0

0.2

0.4

0.6

0.8

1

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05Time (s)

Fra

ctio

n r

ecry

sta

llize

d

450°C 390°C 340°C

10-1

105

104

103

102

10110

0

Chapter 6110

3. The nucleation site density was found to be a strong function of the local strain and

stress but weakly affected by the annealing temperature.

4. A relatively novel approach of the single grain model was applied to model the

kinetic aspects of recrystallization. Following this method, the recrystallization kinetics

was related to the evolution of microstructure and the grain geometry. The effect of the

concurrent recovery and the textural components on the recrystallization can be

included.


References



2. E. Nes, H.E. Vatne, O. Daaland, T. Furu, R. Orsund and K. Marthinsen, in 'The 4th

International Conference on Aluminium Alloys', 2000, 18-49.



35-51.


5. S.P. Chen, D.N. Hanlon, S. van der Zwaag, Y.T. Pei and J.Th.M. de Hosson, J. Mat.

Sci., 37, 2002, 989-995.


7. E.M. Lauridsen, D.J. Jensen, H.F. Poulsen and U. Lienert, Scr. Mat., 43, 2000, 561-

566.

8. L. Delannay, O.V. Mishin, D.J. Jensen and P. van Houtte, Acta Mater., 49, 2001,

2441-2451.

9. S.P. Chen, I. Todd and S. van der Zwaag, Metall. Mater. Trans. A, 33A, 2002, 529-

537.

10. S.P. Chen and S. van der Zwaag, Metall. Mater. Trans. A, submitted, 2002.

11. F.J. Humphreys, Acta Mater., 45, 1997, 4231-4240.

12. H.K.D.H. Bhadeshia, In 'Worked examples in the geometry of crystals', London,

The institute of metal, 1987.

13. E. Nes, Acta Metall. Mater., 43, 1995, 2189-2207.

14. C.M. Sellars, Iron. and Steelm., 22, 1995, 459-464.

15. Marc, in 'Users' manual, Marc Analysis Research Corporation', Palo Alto, CA,

1996.

16. D.R. Barraclough and C.M. Sellars, Met. Sci., 13, 1979, 257-268.

17. E.E. Underwood, in 'Quantitative Microscopy', McGraw-Hill Book Co.,N.Y., 1968.

18. W.A. Johnson and R.F. Mehl, Trans. Am. Inst. Min (Metall. ) Eng., 135, 1939, 416-

443.

19. G.R. Speich and R.M. Fisher, in 'Recrystallization, Grain Growth and Textures', H.

Margolin (Eds), ASM, Cleveland, OH, 1966, 563-598.

20. E. Nes and T. Furu, Scr. Met. Mat., 33, 1995, 87-92.

21. G.M. Hood, Phil. Mag. A, 21, 1969, 305-324.


23. C.M. Sellars, A.M. Irisarri and E.S. Puchi, in 'Microstructural Control in Alminium

Alloys: Deformation, Recovery and Recrystallization', E. Henry Chia and H.J.

McQueen (Eds), The Metallurgical Society, Inc., 1985, 179-196.

Chapter 6112

24. K.D. Vernon-Parry, T. Furu, D.J. Jensen and F.J. Humphreys, Mater. Sci. Techn.,

12, 1996, 889-896.

25. H.J. Bunge, in 'Texture Analysis in Materials Science', London, Butterworths, 1982.

26. C.V. Thompson, H.J. Frost and F. Spaepen, Acta Metall., 35, 1987, 887-890.

S.P. Chen

Chapter 7

Effect of microsegregation and dislocations

on the nucleation kinetics of precipitation in

aluminium alloy AA3003

Nucleation kinetics of the precipitation of MnAl6 and MnAl12 in the aluminum alloy

AA3003 alloy has been investigated experimentally and theoretically. The results show

that cold rolling enhances the rate of precipitation and this effect increases as the

annealing temperature decreases. Micro-segregation of the Mn solute atom at the

dislocation network during cold deformation is found to have a significant effect on the

nucleation kinetics of the precipitation of MnAl6 and MnAl12 in AA3003 in addition to

the effect of dislocations, which increase the nucleation site density and reduce the

nucleation barrier. A model to predict the start times of the precipitation during

isothermal holding is constructed by considering the effects of dislocations and recovery

as well as micro-segregation of Mn on the nucleation kinetics of precipitates. The

predictions of the model are in good agreement with the experiment data.

7.1. Introduction

Generally, AA3003 alloys contain a mixed particle structure consisting of a coarse

distribution of large intermetallic inclusions due to the casting and a fine dispersion of

small particles precipitated during subsequent hot rolling and annealing treatments.

Unlike in the precipitation-hardening alloys such as Al-Cu, where homogenous

precipitation can take place, the precipitation in the dispersoid-containing non-heat-

Chapter 7114

treatable Al-Mn alloys only occurs heterogeneously. Therefore, the precipitation

processes will be influenced by defects related to deformation and softening processes.

While the decomposition of the supersaturated solid solution of the undeformed binary

alloy Al-Mn [1,2] and AA3003 [3,4] has been studied extensively, relatively few reports

document the decomposition kinetics from the deformed state. Limited data show that

the precipitation rate in deformed material is appreciably higher and the MnAl6 phase is

formed at lower temperature [5]. However, since recovery, recrystallization and

precipitation overlap and interact strongly, the general qualification of the effect of the

deformation and softening processes on the precipitation kinetics, especially on the

initial stage of the process, is complicated.

The effect of deformation on precipitation kinetics has often been attributed to the

increase of dislocation density, which increases both the nucleation site density and

diffusivities of precipitation-forming elements in the material [6,7]. However, as will be

seen in the current experiment, the precipitation process can still be promoted even

when the dislocations formed during deformation have largely been eliminated before

the precipitation starts. Consequently, the effect of deformation on the precipitation can

not be explained from dislocation density consideration only.

Many experimental observations have suggested that deformation can cause the

redistribution of the precipitate-forming elements between the dislocation cell walls and

the cell interiors [8-10]. Therefore, segregation of the solute atoms to the potential

nucleation sites may contribute to precipitation kinetics and provide an explanation for

the observed acceleration of the precipitation in a recovered structure.

In this chapter the early stages of the precipitation kinetics in an AA3003 alloy has been

investigated. The decomposition kinetics is monitored by conductivity measurements

and microstructural analysis. An Avrami type equation is applied to analyze the

precipitation kinetics. Thereafter, the precipitation-time-temperature relationship (C-

curve) is established. The contributions from the dislocations, recovery and micro-

segregation of solute to the nucleation kinetics of the precipitation are assessed by using

a model on the basis of classical nucleation theory. The model predictions and the

experimental observations are compared and a good agreement is obtained.

7.2. Mathematical description of the C-curve

In AA3003, precipitation only occurs heterogeneously by nucleation at high energy

defect sites such as grain boundaries, inclusions and dislocations. Thus, the precipitation

sequence is strongly influenced by deformation and recovery. In this paper, classical

nucleation theory is developed to take these effects into account. Assuming that the

Effect of microsegregation and dislocations on the nucleation kinetics 115

grain boundary and dislocation nucleations are the main mechanisms, the total steady

state nucleation rate can be written as:

I I Igb dis (1)

where Igb and Idis are grain boundary nucleation rate and dislocation nucleation rate

respectively. In the spirit of classical nucleation theory the steady state nucleation rate

per unit volume I can be expressed as [11]:

2 *( / ) exp( / )I N C a D G RTv

0 eff (2)

where G is the energy barrier for nucleation (J/mol). Nv, the number of available sitesfor the nucleation. a, the lattice parameter of the matrix phase. 0C , the solute

concentration, T, the temperature, R, the gas constant, and Deff, the diffusion coefficientof the rate controlling element, eff 0 dexp( / )D D Q RT , with D0, the pre-exponential

factor, Qd, the activation energy for diffusion.

During the early stages of a precipitation reaction, the reaction rate is controlled by the

nucleation rate. At that stage, the number of the nuclei per unit volume as a function of

annealing time t is given by:

0 0

t t

N I dt I dtgb dis (3)

The time, t*, taken to precipitate a certain number *N of particles of the new phase for

nucleation to be detected can be determined from Eq. (3).

Application of the above nucleation theory to predict the precipitation-time-temperature

relationship requires the physical parameters, vN , *G and Deff under two different

nucleation mechanisms. In the following we will formulate a model which takes into

account the effect of deformation and softening. The determination of the relevant

parameters will be specified in section 5.2.1.

7.2.1. Activation energy for nucleation

For the case of grain boundary nucleation, the nucleation barrier for a grain boundary

nucleus with two abutted spherical caps is [12]:

* *hom ( )G G Sgb (4)

where 2( ) (2 cos )(1 cos ) / 2S and cos / 2 with being the specific

energy of the interface, , matrix grain boundary energy, *homG , the nucleation

barrier for homogenous nucleation.

The ratio of the nucleation energy on a dislocation to the homogenous nucleation energy

is [13]:

Chapter 7116

*hom/ ( )G G f*

dis (5)

where 22 /G , with 2 /[4 (1 )]Gb for edge dislocations 2 / 4Gb

for screw dislocations, G is the elastic shear modulus, b the Burgers vector and thePoisson ratio. The value of ( )f , which was numerically shown in Fig. 3 in reference

[13], can be approximately expressed as:

0.58( ) 1f (6)

The nucleation barrier, *homG , for a spherical nucleus in an undeformed matrix, is given

as [11,12]:

3 3*hom 2

16

3 ( )G

G GE

(7)

where G is the chemical driving force, GE , strain energy resulting from lattice

distortion, is a modifier of , a measure of the interaction energy between the

surface of the nucleus with the surrounding dislocations, which falls between 0 and 1.

For the precipitation of an intermetallic phase such as MnAl6 ( ) from aluminium with

manganese in solid solution , the chemical driving force per unit volume of precipitate,G (J/mol), is given by [14]:

0ln( / )1

C C RTG C C

C V m (8)

where R is the universal gas constant, T is the isothermal annealing temperature, V m themolar volume of the -phase, C0, the initial concentration of Mn in the solid solution,

C , the equilibrium concentrations of Mn at the annealing temperature. Cß, thecomposition of the intermetallic (MnAl6).

The interfaces of 6/ MnAl and 12/ MnAl are of a semi-coherent nature, with the

orientation relationship being listed in Table 1 [15]. The misfit between the two latticesis defined as ( ) /a a a , with a and a being the lattice parameters of the

matrix and precipitates respectively. The interfacial energy of a semi-coherent interface

can be written as [14]:

c s (9)

where c is a chemical contribution from the coherent portion of the boundary, s , a

structural part due to the dislocations which depends on the misfit between the two

lattices and for a small :


1s C (10)

where C1 is a constant. In this study c and 12MnAl are assumed to be 0.2 J/m2 and

0.22 J/m2 [12], respectively, and 6MnAl is calculated from Eqs. (9) and (10) by applying

data from Table 1, to be 0.26 J/m2.

The value of strain energy EG associated with the semi-coherent interface can be

estimated directly from G as [16]:

1.50( / ) 1 /EG G (11)

where 0 is the specific energy of an incoherent interface, which is taken as 0.5 J/m2

[12].

Table 1. The orientation relationship between the precipitates and Al matrix and the

misfit between the two lattices

Precipitate Orientation relationship disregistry

12MnAl Al(310) //(111) 1.88%MnAl12

12MnAl Al[111] //[310] 1.88%

6MnAl Al(001) //(315) 5.69%MnAl6

6MnAl Al[100] //[130] 1.68%

7.2.2. Nucleation site density

Assuming that the heterogeneous nucleation sites at grain boundaries are positionedwithin a plate of thickness, b , around the grain boundary, the number of heterogeneous

nucleation sites at grain boundaries, N gb , relative to the total number of nucleation sites

per unit volume for homogenous nucleation within the matrix, homN , is [12]:

hom

N

N d

gb

b

g

(12)

where dg is the mean grain diameter and b is the thickness of the grain boundary.

Assuming that the number of available sites for nucleation along dislocations is equal to

the number of atoms in the core of the dislocations, then heterogeneous nucleation

density on dislocations, i. e. the number of atoms per unit volume, N dis associated with

dislocations can be written as [17]:

Chapter 7118

2

hom

Nr

N

dis

c (13)

where is dislocation density, rc is the core radius of the dislocation.

7.2.3. The evolution of dislocation density

Dislocations in a deformed material are assumed to arrange into cellular structures. Thecell size after deformation, , can be estimated as [18]:

0.35 0.17 / (14)

where is the true strain. The dislocation density in the deformed state, , can be

obtained by applying the principle of similitude [19]:

0.5c (15)

where c is a constant about ~5. On the early stages of the annealing process the

fraction residual strain hardening can be described by [20]:

1 ln(1 )RT t

RA

rec (16)

where 0 expQ

RT

rec , A and 0 , constants, Qrec , the activation energy for recovery.

The softening fraction, which is mainly from recovery decay on the early stage of the

annealing process, can be obtained from the microhardness measurement as:

H HR

H H

rexrec

def rex

(17)

where Hdef is the hardness of the material in the as-deformed state, H , the instantaneous

hardness after annealing time t , Hrex , the hardness of completely recrystallized

material.

By assuming that the value of microhardness is proportional to 1/ 2 , the dislocation

density change during softening can be determined from Eqs. (16) and (17).

7.2.4. Effect of micro-segregation of Mn

The redistribution of the precipitate-forming elements between the dislocation cell walls

and the cell interiors during deformation may result in an increase of local

supersaturation, which will affect the thermodynamics of precipitation. As the

mechanism for the segregation is not completely understood yet, in this paper we apply

the solute atmosphere theory [21,22] to quantitatively describe the solute distribution


around dislocations in order to discuss the effect of the segregation on the precipitation

process.

The extent of solute enrichment around a moving dislocation depends on the velocity of

solute atom diffusion relative to the dislocation velocity at the deformation temperature

and the difference between the molar volume of the solute atom and that of the matrix

atom [21,22]. As an approximation, for a dilute solution the concentration of solutearound a dislocation, Cdis , when the dislocation velocity approaches zero, is given by:

0 exp ( ) /C C p V V RTdis s m (18)

where (1 )sin

3 (1 )

Gbp

r, Vm and Vs are the partial molar volumes of matrix and

solute elements, respectively, G, the shear modulus, b, Burgers vector, , Poisson'sratio, r and are the polar coordinates with respect to the core of the moving

dislocations, 0C , the concentration of the solute far away from a dislocation.

7.3. Experimental

The actual chemical composition of AA3003 has been determined using spectroscopic

analysis, and is given in Table 2. The starting material was provided in the transfer state

condition (26 mm thick hot-rolled plate). Two pieces of the plate were solution treated

at 630° for 15 h and water quenched to room temperature. Samples I were directly

annealed isothermally between 550 and 350°C in a salt bath with the temperature

controlled to within 1°C. Samples II were cold rolled with a 50% reduction in thickness

before applying the same isothermal annealing conditions. These treatments produced

samples with essentially two different microstructures: sample I was highly

supersaturated with Mn (~0.68 wt. %) and sample II was highly supersaturated as well

as being strain hardened. The precipitation kinetics was monitored using conductivity

measurements. Sigmatest D2.068 was used to measure the conductivity at 25±2°C. All

the specimens were carefully prepared to have the same thickness. The surface of the

specimen was polished to make good contact with the conductivity detector. 20

measurements were taken for each experiment to give an averaged value. The accuracy

of the measurement is within 1% MS/m. The distribution of the undissolved dispersoids

and the grain structures of all the samples were examined by quantitative optical

metallography. The recovery and recrystallization processes were followed by

microhardness measurements and optical metallography. The micro-hardness was

determined using 50 g load and 15 s loading time on a Buehler OMNIMET MHT

automatic micro hardness tester. An experimental point represents an average value of

18 measurements.

Chapter 7120

Table 2. The chemical composition of the material studied

Si Fe Cu Mn other Al

AA3003 <0.30 <0.56 <0.25 1.10 <0.15(total) bal

7.4. Results

7.4.1 The evolution of microstructure during homogenization

The optical microstructure of the transfer gauge shows an incompletely recrystallized

structure. A fully recrystallized structure was obtained after homogenization at 630°C

for 10 min. However, the recrystallized grains were still elongated along the rolling

direction with an aspect ratio l/w~3-5. The average grain size (equivalent diameter from

area measurement) was determined as 150 µm. The grain size seems not to increase

with increasing homogenization time up to 15 h owing to the pinning effect of the

second phase particles.

There are two kinds of intermetallic particles in the transfer gauge material,-(Fe,Mn)Al6 or MnAl6 particles (rod-like with a aspect ratio >5, size >2 µm) and

-(FeMn)SiAl (spherical, size between ~1 µm and 4 µm). The rods of (Fe,Mn)Al6 are

oriented in the rolling direction.

The small particles (less than 1.0 µm) dissolve with increasing time during

homogenization. However, the dispersion of insoluble particles of (Fe,Mn)Al6 and

(FeMn)SiAl did not change during annealing at 630°C while the rod-like particles

became more or less spherical. Application of 50% cold rolling deformation to the

homogenized samples did not change the size of the undissolved particles and their

distribution.

The electrical conductivity of AA3003 in the transfer state is measured as 24.8 MS/m. It

decreases rapidly with homogenisation time (t=30 min) and then approaches a constant

value of 19.96 MS/m, which indicates that the solid solutions of (Mn, Fe, and/or Si) in

the matrix reach the equilibrium value. After applying cold rolling to 50% thickness

reduction to the homogenized samples, the final conductivity of the material became

20.15 MS/m.

7.4.2 The evolution of the conductivity and precipitation during

isothermal annealing

Fig. 1 shows the conductivity evolution of the samples for both treatments (no

deformation and with deformation) during isothermal annealing at temperatures

between 350°C and 550°C. As can be seen, the conductivity increases rapidly at the


very beginning and the rate slows down gradually with annealing time. The

conductivity of the cold rolled samples increases faster than that of the non-strained

sample. This indicates that strain promotes the precipitation processes. The effect of the

cold rolling on the precipitation increases as the annealing temperature decreases.

(a) (b)

Fig. 1. The conductivity evolution of the samples after homogenization at 630°C, (a)

without strain; (b) 50% cold rolling, then annealed at various temperatures.

(a) (b)

Fig. 2. The microstructure of the precipitates after annealing at 400°C for 35 min, (a)

without strain; (b) 50% cold rolling. (Coarse particles-undissolved, fine particles-

precipitates)

19

20

21

22

23

24

25

26

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04Time (min)

Co

nd

uctivity (

MS

/m)

350°C

400°C

450°C

500°C

550°C

10-2

104

103

102

101

100

10-1

20

21

22

23

24

25

26

27

28

29

30

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04Time (min)

Co

nd

uctivity (

MS

/m)

350°C

400°C

450°C

500°C

550°C

10-2

10-1 10

110

410

310

210

0

Chapter 7122

The optical microstructures after annealing at 400°C for 30 min are shown in Fig. 2.

The difference between the two types of samples is obvious. In the samples without

strain, the precipitation starts at the original grain boundaries, as shown in Fig. 2(a). As

annealing time increases, more and more precipitation takes place within the original

grains and the number and size of precipitates increase with annealing time. However,

in the deformed samples, fine precipitates are more or less uniformly distributed in the

matrix (Fig. 2(b)). This indicates that the heterogeneous nucleation sites provided by

dislocations now play an important role.

7.4.3. The isothermal precipitation kinetics

The electrical conductivity is the reciprocal of the resistivity. It is well established that,for low solute concentration, the resistivity of an alloy, , is given by the following

relation [23],

' '( )i i i iC CAl (19)

where 62.64 10Al cm is the electrical resistivity of the pure matrix. iC and 'iC

the respective concentration of solute atoms. i and 'i are the specific resistivities of

the ith elements in solid solution and in precipitated form, respectively, which are listed

in Table 3. Metals in solid solution promote the resistivity to a greater extent than when

out of solution.

Using Eq. (19) and Table 3 the contribution that each alloying element makes to the

resistivity can be predicted. Because of the low solid solubility of Fe in aluminium and

the relatively low total concentration of Si, their contribution to the resistivity change

during heat treatment can be neglected in comparison with the one due to the change of

the Mn concentration. Therefore, the analysis of the conductivity curves can be based

on an analysis of the quasi binary Al-Mn system, taking into account that the addition of

the elements Fe and Si to the pure Al-Mn alloy reduces the solid solubility of Mn in Al

[4,24]. The solubility of Mn in the solid solution corresponding to the equilibrium phase

MnAl6 in the current Al-Mn-Si alloy as a function of temperature is shown in Fig. 3

[25,26].

Table 3. Effect of the additions on the resistivity of pure Al at 20°C

Element Mn Fe Si

i (µ cm/wt.%) 2.94 2.56 1.02

'i (µ cm/wt.%) 0.34 0.058 0.088

Max solubility,wt.% 1.55 (658°C) 0.052 (655°C) 1.65 (580°C)


Fig. 3. The solubility of Mn in Al and the saturation conductivity as a function of

temperature.

The precipitation fraction can now be determined using:

0 0

0 0

t t

t

x (20)

where t is the conductivity at annealing time t, 0 , the conductivity of the samples

before annealing and , the saturation level of the conductivity of the fully

precipitated samples at long time, which corresponds to the equilibrium content of Mnin solid solution and increases with decreasing temperature. i is the corresponding

resistivity. can be calculated from equation (19) by inputting the solubility of Mn

from Fig. 3 and other data from Table 2 and Table 3. The graphical presentation of the against temperature is shown in Fig. 3, too. The precipitation fraction as a function

of the annealing time is shown in Fig. 4.

It is supposed that the precipitation process can be described by the JMAK equation, i. e

[12].

1 exp nx kt (21)

By applying Eq. (21) to the experimental data, the parameters, k and n are determined

by iteration on the method of least squares and are listed in Table 4. n is close to 0.5

indicating that the precipitation is diffusion controlled process. Application of the

deformation results in a decrease in n but a increase in k. Increase in k indicates that

0

0.2

0.4

0.6

0.8

300 350 400 450 500 550 600 650 700Temperature (°C)

So

lub

ility

Mn

(w

t%)

19

21

23

25

27

29

31

33

Sa

tura

tio

n c

on

du

ctivity (

MS

/m)

Chapter 7124

cold deformation reduces the nucleation barrier of the precipitation. Decrease in n is due

to the concurrent recovery and recrystallization processes, which weakens the effect ofthe dislocations. The relation between ln k and (1/T) is not linear but shows a kink,

which indicates two precipitation reactions taking place in the temperature range

studied.

Table 4. Fitting parameters n and k for the precipitation kinetics using JMAK equation

undeformed 50% Cold rollingTemperature

n k n k

350 0.40 0.008 0.39 0.058

400 0.42 0.018 0.35 0.091

450 0.39 0.036 0.37 0.065

500 0.42 0.030 0.34 0.107

550 0.37 0.055 0.30 0.159

(a) (b)

Fig. 4. The precipitation fraction as a function of the time, annealed at various

temperatures, (a) without strain; (b) 50% cold rolling.

If the beginning of the precipitation is defined as 0.05x transformed, the time-

temperature-transformation representing nucleation kinetics of the precipitation reactioncan be constructed by solving Eq. (21) at 0.05x for each temperature. The resulting

0

0.2

0.4

0.6

0.8

1

0.01 0.1 1 10 100 1000 10000

Time (min)

Fra

ctio

n

550°C

500°C

450°C

400°C

350°C

102

10-1

10-2

100

101

104

103

0

0.2

0.4

0.6

0.8

1

0.01 0.1 1 10 100 1000 10000Time (min)

Fra

ctio

n

550°C

500°C

450°C

400°C

350°C

10-2

10-1

104

103

100

102

101


C-curves are shown in Fig. 5. In the TTT diagram of the non-strained samples, there is a

nose toward 450°C and the other at higher temperatures, indicating two precipitation

reactions, which is in a good agreement with the literature [4,15]. At temperatures

higher than 500°C, the equilibrium MnAl6 precipitate forms directly. At lower

temperatures (T<500°C), initially the metastable phase G1 (MnAl12) is precipitated

which then transforms into MnAl6 for long annealing times. In the cold deformed

material the precipitation starting time is appreciably shorter and MnAl6 phase forms

even at lower temperatures. The character of the double nose is more noticeable.

Fig. 5. The temperature dependence of the start (t0.05) of the precipitation (C-curve).

7.4.4. The softening kinetics during isothermal annealing

The microhardness evolution after annealing at various temperatures is shown in Fig. 6.

In theory, the microhardness evolution should include the contributions from both

softening and precipitation. The microhardness is measured to be 35±2 HV for the

sample after 630°C solution treatment for 15 h and quenching. After annealing at 400°C

for various times such as 30 min, 15 h, 96 h, the microhardness values are 36±2, 35±2

and 35±2 respectively. They indicate that the microhardness is not sensitive to the

precipitation process. Therefore the microhardness measurement can be considered only

to reflect the softening behavior. The best fit of the equation (16) and (17) to the

experimental data yields the activation energy for recovery Qrec=220 kJ/mol, which is

approximately the activation energy of Mn diffusion in Al [24]. The other twoparameters, A and 0 , are listed in Table 5.

300

350

400

450

500

550

600

0.1 1 10 100 1000Time (min)

Te

mp

era

ture

(°C

)

50%CR No strain

MnAl6

MnAl12

101

103

102

100

10-1

Chapter 7126

Fig. 6. The microhardness as a function of annealing time in the samples with 50% cold

rolling.

Table 5. The fitting parameters A and 0 for recovery

350°C 400°C 450° 500°C

A (kJ/mol) 210 204 198 195

0 (1/s) 219.86 10 251.05 10 261.20 10 271.96 10

7.5. Discussion

7.5.1. Analysis of the experimental results

The enhancement of precipitation kinetics due to prior plastic deformation can be

attributed to the following reasons. Deformation increases the dislocation density, which

increases both the nucleation site density and diffusivities of precipitate-forming

elements in the material. Furthermore, the nucleation barrier on dislocations is much

lower than that for homogenous nucleation. In addition, the solute atmosphere around

dislocations enhances the rate of precipitation via an increase in the local chemical

driving force. The last point can be indirectly seen from the current experimental

results, and forms the basis of the new analysis of the precipitation reaction in AA3003.

As it is well known, applying deformation to the uniform solution will increase its

resistivity, i.e. decrease electric conductivity. However, in the present experiment the

30

35

40

45

50

55

60

65

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04Time (min)

Ha

rdn

ess (

VH

N)

450

400

350°C

500

10-2

102

101

100

10-1

104

103


conductivity of the deformed material shows a slight increase (compare Fig. 3a and

Fig. 3b), which corresponds to a decrease of 0.025 wt. % Mn in solute solution. This

indicates that solute atom redistribution has occurred during cold rolling. The Mn

segregation at dislocations results in a decrease of Mn concentration in the matrix,

leading to an increase of the conductivity.

In the deformed samples, the precipitation of MnAl6 or MnAl12 and recovery are

coupled phenomena during annealing at the temperature range studied here. Obviously,

when annealing at higher temperatures, some of dislocations formed during cold rolling

can be eliminated before the precipitation starts as a result of concurrent restoration

process. Therefore, the effect of the cold rolling on the precipitation kinetics is likely to

decrease with annealing temperature, as is indeed observed. In the sample annealed at

550°C, the deformed samples fully recrystallize within about 12 s, while the

precipitation starts at 15 s. As the time to the complete recrystallization is less than the

start time of precipitation, dislocations formed during cold rolling have been eliminated

completely before precipitation starts. It would be expected that there would be no

enhancing effect of strain-induced precipitation in this case in the point of view that the

presence of the dislocation will increase the nucleation sites and diffusivites of the

precipitation-forming elements. However, it is of interest that the precipitation kinetics

is still promoted (Fig. 5). This, again, can be explained as being a result of the Mn

segregation effect. Precipitate-forming elements are enriched at dislocation cell walls

during deformation or after deformation as a result of segregation. While the dislocation

cell walls are swept out by moving boundaries of new recrystallized grains, the already

established concentration distribution during deformation is unlikely to be altered. Thus,

precipitation may still take place at regions of prior dislocation cell walls where the

higher local supersaturation of precipitate-forming elements is retained for a sufficiently

long time.

In the following section the effect of the initial and transient dislocation density and Mn

segregation on the nucleation kinetics of precipitation will be further analyzed by means

of the theoretical predictions.

7.5.2. Application the model to the experiment data

1. Parameters determination

The physical parameters used in the prediction from literature and other constants

determined from the tuning are listed in Table 6. The diffusion coefficient of Mn in Al

reported in the literature shows a very large scatter [24,27,28]. Therefore, in this study

the activation energy Qd is firstly evaluated from the experimental C-curve obtained in

this work according to the procedure described by Ryum quoted in [29]. The Qd is

determined from Fig. 5, by dividing the slope of the lower asymptote by gas constant R,

Chapter 7128

respectively, as 223 kJ/mol and 198 kJ/mol from the non-strained C-curve and strained

C-curve. Hence the diffusion coefficient of Mn in Al in the literature closest to thesevalue, d 211Q kJ/mol [27] is applied in the prediction.

The solvus boundary for metastable phase MnAl12 in aluminium needs to be

determined. A mathematical description of the solvus boundaries of the equilibrium

precipitate MnAl6 is given as [29]:

eq 6ln[% ] [% ]

HT

S R Al Mn (22)

where S and H are the standard entropy and enthalpy of the reaction. [%Al] and

[%Mn] are the matrix concentrations of elements Al and Mn in wt%. Teq. is the solvus

temperature of MnAl6. By fitting Eq. (22) to the literature data (solid line in Fig. 3) thevalues of S and H are determined as 25 JK/mol and 65 kJ/mol, respectively.

Assuming that the thermodynamic properties of the metastable MnAl12 phase are

similar to those of the equilibrium MnAl6 phase and taking into account the extra

pressure on the particle from the curvature of the interface of the precipitate interface in

the solvus enthalpy, the solvus temperature of metastable equilibrium, Tmet, can be

expressed as:

met. 6ln[% ] [% ]

HT

S R Al Mn (23)

where is the contribution of the interface curvature to the reaction enthalpy, given by:

m2 /V r (24)

where mV is the molar volume of the precipitate, r the radius of the curvature, the

interfacial energy of the particle. The ratio between Eqs. (22) and (24) is:

met.

eq

T H

T H (25)

which is independent of temperature and composition, and only dependent on the typeand shape of precipitate. By setting the average radius 1r nm and 0.26 Jm-2,

then:

met.

eq

0.90T

T (26)

The temperature dependent chemical driving force, G , and the strain energy term,

EG can be determined from Eqs. (8) and (11), respectively, and from these values the

nucleation barriers for grain boundary nucleation and for dislocation nucleation can then

be calculated from Eqs. (4) to (7). The total nucleation site density takes the sum of the


gbN and disN which could obtained from Eqs. (12) and (13), respectively. All thesevalues can be put into Eq. (3), which could be solved numerically, to obtain the 0.5t .

Table 6. Parameters and variables employed in the model prediction

Symbol value definition

a 0.40496 nm Lattice parameter of Al

V Al510 m3/mol Molar volume of Al

V6MnAl

69.68 10 m3/mol Molar volume of MnAl6

V12MnAl

69.10 10 m3/mol Molar volume of MnAl12

dg 150µm The original grain size

b 0.8 nm [12] Grain boundary thickness

6MnAl 0.26 J/m2 Specific energy of

6/ MnAl interface

12MnAl 0.22 J/m2 [12] Specific energy of

12/ MnAl interface

0.324 J/m2 [14] Matrix grain boundary energy

rc 0.286 nm The core radius of dislocation

Qd 211 kJ/mol [27] Activation energy for Mn diffusion

in Al

D034.934 10 m2/min Pre-exponential factor of Mn

diffusivity in Al

0 106 /mm2 Dislocation density in the fully

recrystallized state

Qrec 220 kJ/mol Activation energy for recovery

G 25.4 GPa Shear modulus

b 0.286 nm Burgers vector

0.33 Possion’s ratio

0.16 modifier

Chapter 7130

The C-curve of the unstrained case is now used as a reference to adjust the constants,

(in Eq. (7)) and 0D in the model. As shown in Fig. 7, in the non-strained case the grain

boundary nucleation is dominant mechanism because of lower nucleation barrier and

relatively higher nucleation site density on the grain boundaries. The nose in the C-

curves of dislocation nucleation appears at lower temperature than that of grain

boundary nucleation. This is because that the nucleation barrier at a grain boundary islower than that on dislocations. On the other hand, since is proportional to G ,

which decreases with increasing temperature, the effectiveness of a dislocation to

catalyze nucleation decreases as temperature increases.

Fig. 7. Application of the model to the experimental C-curve of the case without

deformation to determine parameters, and 0D .

2. Effect of the dislocation density, recovery and Mn segregation on the C-Curve

The effect of the dislocation density on the precipitation kinetics is shown in Fig. 8. As

the dislocation density increases, the nose of the C-curve shifts to the left. This is

because of the strong increase in the nucleation site density produced by dislocations

and as a result, the dislocation nucleation gradually becomes dominant.

The effect of the softening reaction on C-curve can be also seen from the dotted line in

Fig. 8. Here we assume that the softening follows Eq. (16) and when it is fully

recrystallized the dislocation density is fixed at 106 /mm2. As can be expected, softening

reaction reduces the dislocation density, and therefore, shifts the upper part of the C-

300

350

400

450

500

550

600

650

1.E-01 1.E+01 1.E+03 1.E+05 1.E+07

Time (min)

Te

mp

era

ture

(°C

)

MnAl6 MnAl12 Experimental

GB =106/mm

2

10-1

105

103

101

107


curve to the right and this effect decreases as annealing temperature decreases because

of the sluggish recovery process in this alloy.

The segregation increases the Mn concentration near the dislocations and/or grain

boundaries, and therefore, increases the driving force for precipitation. Fig. 9 shows that

the effect of the different segregation level of local Mn content on the C-curves. The

bulk concentration of Mn is 0.68 wt. %. As can be seen, Mn segregation displaces the

nose of the C-curve towards higher temperatures and shorter times. The enhancing

effect of the segregation is stronger at higher temperatures.

Fig. 8. The effect of the dislocation density and softening restoration on the C-Curve of

the precipitation. Solid lines: dislocation effect, dotted lines: dislocation effect and

softening effect, dash-dotted lines: grain boundary nucleation.

250

300

350

400

450

500

550

600

650

1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06

Time (min)

Tem

pera

ture

(°C

)

=1013

/mm2

1011

109 10

6

GB

GB

MnAl6

MnAl12

10-4

104

102

100

10-2 10

6

Chapter 7132

(a) (b)

Fig. 9. The effect of the strain-induced microsegregation of Mn on the C-Curve of the

precipitation.

3. Prediction of the C-Curve after the current deformation

The model was applied to the experimental data after cold rolling. In the as-rolled state

the dislocation density is calculated to be 95 10 /mm2 and the relative

concentration ratio of 0/C Cdis

Mn Mn is ~1.35 from Eq. (18). Fig. 10 shows predictions of C-

Curves together with experimental data. The dotted curve only considers the

contribution from the dislocations. The double dotted curve is the one assuming95 10 /mm2 and with Mn segregation from 0.69 to 0.90 ( 0/C Cdis

Mn Mn =1.35). The

solid black curve is the one in which all the effects, the dislocation density, Mn

segregation and softening are taken into account. As can be seen, the presence of

dislocations promotes precipitation reaction. However, only considering the increase in

the nucleation site density from dislocations does not lead to a satisfactory description

of the experimental data. The effect of the microsegregation of Mn is significant. The

prediction of the present model gives a good agreement with the experiments when all

the effects from dislocation density, Mn microsegregation and recovery are considered.

300

350

400

450

500

550

600

650

1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03

Time (min)

Te

mp

era

ture

(°C

)

=1011

/mm2

MnAl6

0.7wt.%Mn

0.8

0.91.0

1.2

GB

10-3 10

310

110

-1300

350

400

450

500

550

600

0.001 0.01 0.1 1 10 100 1000

Time (min)

Te

mpe

ratu

re (

°C)

MnAl12

=1011

/mm2

0.7wt.%Mn

0.9

1.2

0.81.0

GB

10-3

101

100

10-1

10-2

103

102


Fig. 10. The comparison of the prediction with the experimental C-curve of the case

with 50% cold rolling (CR). Dotted curve: 95 10 /mm2, double dotted curve:

95 10 /mm2 and a segregation of Mn from 0.69 to 0.90 %wt. Solid curve: including

the effect of dislocation density, Mn segregation and recovery. Circle points and square

points: experimental C-curves of 50% CR and without strain, respectively.

7.6. Conclusions

1. Micro-segregation of the solute atom Mn at the dislocation network during cold

deformation has an apparent effect on the nucleation kinetics of the precipitation of

MnAl6 and MnAl12 in AA3003 alloy.

2. The present model can account for the effects of Mn micro-segregation on the

nucleation kinetics of the precipitation in addition to the contributions from the

dislocations and concurrent softening reaction.

300

350

400

450

500

550

600

650

0.01 0.1 1 10 100 1000

Time (min)

Tem

pera

ture

(°C

) MnAl6

MnAl12

=0.67, =5x109/mm

2

10-1 10

310

210

110

010

-2

Chapter 7134

References

1. E. Nes, J.D. Embury, Z. Metallkde. 66 (1975) 589-593.

2. A.K. Jena, D.P. Lahiri, T.R. Ramachandran, M.C. Chaturvedi, J. Mat. Sci. 16 (1981)

2544-2550.

3. G. Hausch, P. Furrer, H. Warlimont, Z. Metallkde. 69 (1978) 174-180.

4. D.B. Goel, U.P. Roorkee, P. Furrer, Neuhausen, H. Warlimont, Aluminium 50

(1974) 511-516.

5. N.J. Luiggi, Metall. Mater. Trans. B 28 (1997) 125-133.

6. B. Dutta, E.J. Pamiere, C.M. Sellars, Acta Mater. 49 (2001) 785-794.

7. E.J. Pamiere, C.I. Garcia, A.J. DeArdo, Metall. Mater. Trans. A 25 (1994) 277-287.

8. T. Abe, H. Onodera, J. of Phase Equilibria 22 (2001) 491-497.

9. A.H. Cottrell, B.A. Bilby, Proc. Phys. Soc. (London) A62 (1949) 49-61.

10. H. Yoshinaga, S. Morozumi, Phil. Mag. A23 (1971) 1351-1356.

11. K.C. Russell, Adv. Colloid Interface Sci. 13 (1980) 205-318.

12. D.A. Porter, K.E. Easterling, Phase Transformations in Metals and Alloys, Stanley

Thornes (Publishers) Ltd, 1992.

13. J.W. Cahn, Acta Metall. 5 (1957) 169-172.

14. A.K. Jena, M.C. Chaturvedi, Phase Transformations in Materials, Englewood Cliffs,

New Jersey 07632, Prentice Hall, 1992.

15. L.F. Mondolfo, Aluminium Alloys: Structure and Properties, Butterworth and Co.,

London, 1976.

16. W.J. Liu, J.J. Jonas, Mater. Sci. Techn. 5 (1989) 8-12.

17. R. Gomez-Ramirez, G.M. Pound, Metall. Trans. 4 (1973) 1563-1570.

18. J. Gil Sevillano, P. van Houtte, E.A.D. Aernoudt, Pro. Mat. Sci. 25 (1980) 69-412.

19. C.M. Sellars, in: B. Hutchinson et.al (Eds), Thermomechanical Processing in

Theory, Modeling & Practice [TMP]2 (1996) 35-51.

20. E. Nes, Acta Metall. Mater. 43 (1995) 2189-2207.

21. R. Fuentes-Samaniego, R. Gasca-Neri, J.P. Hirth, Phil. Mag. A49 (1984) 31-43.

22. H. Yoshinaga, S. Morozumi, Phil. Mag. A23 (1971) 1367-1385.

23. J.E. Hatch, Aluminium: Properties and Physical Metallurgy, American Society for

Metals, Metals Park, Ohio, 1988.

24. J.J. Theler, P. Furrer, Aluminium 50 (1974) 467-472.

25. M. Hasen, Constitution of Binary Alloys, Mc Graw Hill, New York, 1958.

26. F.A. Shunk, Costitution of Binary Alloys, Second supplement, Mc Graw Hill, New

York, 1969.

27. G.M. Hood, R.J. Schultz, Phil. Mag. A23 (1971) 1479-1489.

28. T.S. Lundy, J.F. Murdock, J. Appl. Phys. 33 (1962) 1671-1673.

29. O. Grong, Metallurgical Modeling of Welding, The Institute of Materials, 1994.

S.P. Chen

Chapter 8

On the precipitation and recrystallization

behavior in an AA3003 alloy following hot

deformation

An AA3003 alloy, pre-heat treated to produce two different starting microstructures,

was deformed in plane strain compression at strain rates in the range of 0.1 to 10 /s at

temperatures varying between 350°C to 550°C. As-deformed specimens were

subsequently annealed at different temperatures to study the effect of the hot

deformation parameters on the precipitation and recrystallization behavior. The solute

content of Mn in the matrix was found to have a large effect on the deformation,

decomposition and softening behavior of the alloy. The precipitation kinetics was

enhanced by deformation but weakened by recovery and recrystallization. Dynamic

precipitation during concurrent deformation depends on the strain rate and deformation

temperature. The softening kinetics is either fast or sluggish depending on the pre-heat

treatments, deformation conditions and the annealing temperature. When

recrystallization is slow the contribution of the recovery to the softening can be as high

as 70%. When recrystallization is fast recovery is responsible for only ~25% of the

softening. Recrystallization kinetics is not simply proportional to the value of the Zener-

Hollomon parameter. The fully recrystallized grain size increases as annealing

temperature decreases as a result of precipitation effect. Although particle stimulated

nucleation (PSN) is observed in the cold deformed samples annealed at higher

temperatures (>500°C), nucleation by subgrain growth and strain induced boundary

migration (SIBM) are the dominant nucleation mechanisms in the hot deformed samples

annealed in the temperature range 450-540°C.

Chapter 8136

8.1. Introduction

It has been recognized that Mn is one of the most efficient elements to affect the

deformation and recrystallization behavior of Al [1]. This influence strongly depends on

the supersaturation of the alloy and on the size and distribution of the precipitate

particles present in the matrix [2-4].

In AA3003 alloy the precipitation of MnAl6 and the recrystallization reaction during

thermo-mechanical treatments are coupled phenomena [5,6]. The competition of these

two processes determines the final microstructure, and therefore, the properties. The

presence of the second phase particles is well known to affect recrystallization kinetics

and grain size and shape. The coarse particles speed up recrystallization while fine

particles slow it down.

Numerous investigators have studied the precipitation kinetics and recrystallization

behavior in 3 alloys after cold working [2,7,8]. However, relatively few workers

have studied the concurrent Mn precipitation, or examined the potential interaction

between this precipitation and the matrix recrystallization during and after hot

deformation such as break down rolling. Hence, systematical studies on the

recrystallization kinetics in AA3003 after hot deformation are limited in number. As the

deformation temperature and strain rate clearly affect both the dislocation substructure

and precipitation kinetics, microstructure evolution due to warm deformation may be

quite different from that due to cold deformation. As concurrent precipitation strongly

affects the kinetics of the softening process, the content of the Mn in the matrix and

particle size distribution just prior to the warm deformation will, for a given

deformation condition, play an important, but as yet unclear, role.

The objectives of the present investigation are therefore: (1) to elucidate the effect of the

thermo-mechanical processing on precipitation and softening behavior of AA3003

following hot deformation, with particular attention being paid to the roles of: pre-heat

treatment; deformation temperature; strain rate; and annealing temperature. (2) To

provide basic experimental data for further modeling study on how the hot deformation

affects MnAl6 precipitation and how the latter, in turn, influences the recrystallization

process.

8.2. Experimental

The chemical composition of AA3003 (in wt.%), determined using spectroscopic

analysis, was 0.30 Si, 0.56 Fe, 0.25 Cu, 1.10 Mn and bal. Al. The starting material was

provided in the transfer gauge state (26 mm thick hot-rolled plate) that had been DC

cast, homogenized, break-down-rolled and annealed. The transfer gauge material

On the precipitation and recrystallization behavior in an AA3003 137

(AA3003) was thermally pre-treated to produce two different initial levels of

manganese supersaturation and dispersoid distribution. One piece of the plate was

solutionised at 630° for 16h and water quenched to room temperature (designated by A:

630°C 16h). The other piece was directly annealed at 450°C in a salt bath for 24 h to

produce a fully recrystallized structure and to reduce the Mn content in the matrix

(designated by B: 450°C 24h). These treatments produced samples with essentially two

different microstructures: sample A was highly supersaturated with Mn and sample B

was highly precipitated with a low level of Mn dissolved in the matrix.

To simulate the deformation that occurs during conventional industrial processing such

as hot rolling, specimens of both treatments were deformed in plane strain compression

using a Gleeble 3500 over the temperature range 350 to 550°C. Rectangular PSC

specimens, dimensions 5 30 10 mm3, were machined from the middle thickness of the

sheet. The elongation direction of the deformation in PSC samples is parallel to the

original rolling direction. A tool with width of 10 mm is used so that the ratio of the tool

width to the sample thickness is 2 to ensure almost uniform deformation in the sample.

Each sample was heated to the test temperature at a heating rate of 5 /s, held for 30 s at

the test temperature, deformed at a given strain rate to a strain of 0.70, then quenched by

water within 1 s to room temperature. The deformation temperatures were 350°C,

450°C and 550°C. The strain rates used were 0.1 /s, 1 /s and 10 /s. All tests were

conducted in vacuum. A water based graphite lubricant was used to reduce the friction

between the specimen and the tools. As the Gleeble uses resistance heating, the contact

between the two platens and the specimen becomes crucial for obtaining a uniform

distribution of the temperature within the specimen. Hence, it is very important to

machine specimens precisely to the exact shape. The specimens were ground on

sandpaper to ensure the difference in thickness at both ends being less than 0.02 mm.

The temperature difference between the two ends of a sample is less than 5°C. To show

the effect of the deformation temperature material heat treated according to schedule A

was also cold rolled to the same amount of strain of 0.7. Samples are labeled by

referring preheat treatment and deformation conditions. For example, A350°C 0.1/s

refers a case of treatment A, deformed at 350°C with a strain rate of 0.1/s, CR refers

samples after cold rolling deformation.

Following the hot or cold deformation, specimens were isothermally annealed in a salt

bath for times ranging from 6 s to 500 h at temperatures of 450, 500 and 550°C to study

the softening kinetics and decomposition kinetics.

For all samples the distribution of the undissolved dispersoids and the grain structures

were determined by quantitative optical metallography. The recovery and

recrystallization processes were followed by microhardness measurements and optical

Chapter 8138

metallography. Specimens for optical metallography were anodized with an aqueous

solution 3% Barker agent at a potential of 15 V for about 100 s. the microstructure was

studied using polarized light. The micro-hardness was determined on a Buehler

OMNIMET MHT automatic micro hardness tester using 300 g load and 15 s loading

time. An experimental point in the graphs represents the average value of 18

measurements. Sigmatest D2.068 eddy current conductivity tester was used to measure

the conductivity at 20±2°C. All specimens were carefully prepared to have the same

thickness. The surface of the specimen was polished to make a good contact with the

conductivity detector. 20 measurements were taken for each experiment. The accuracyof the measurement is within 0.05 MS/m.

8.3. Results

8.3.1. The microstructural evolution during preheat treatment

The optical microstructure of the starting material showed an incompletely

recrystallized structure. A fully recrystallized structure was obtained after

homogenization at 630°C for 2 min. In the fully recrystallized material, the grains are

still elongated along the rolling direction with an aspect ratio l/w~4-6. The average grain

size (equivalent diameter from area measurement) was determined as 125 µm. The grain

size seems not to increase with increasing homogenization time up to 168 h due to the

pinning effect of the second phase particles.

There are two kinds of intermetallic particles present in the transfer gauge material,-(Fe,Mn)Al6 or MnAl6 particles (rod-like with a aspect ratio >5, size >2 µm) and

-(FeMn)SiAl (spherical, size between ~1 µm and 4 µm). The (Fe,Mn)Al6 particles are

oriented in the rolling direction.

The small particles dissolve with increasing time during homogenization at 630°C.

However, the size distribution of insoluble particles of (Fe,Mn)Al6 and (FeMn)SiAl did

not change during annealing while the rod-like particles became more or less

spherical. Therefore material with treatment A has a distribution of coarse particles.

Table 1 shows the effect of homogenization time on the number of particles, their size

and aspect ratio and the area fraction. Particles with a size smaller than 1µm are not

included in these statistics.

The average grain size and the aspect ratio of the grain after treatment B are almost the

same as those after treatment A. There are two kinds of particles in the samples with

treatment B. The coarse particles (>0.5 µm) are those already present in the transfer

state which have undergone a further growth process during subsequent treatments. The

fine dispersoids are those produced by precipitation and growth during further


isothermal annealing at 450°C. The average size of the coarse particles is 1.95 µm. The

average size of the fine dispersoids after a similar treatment is about 0.12 µm according

to Hansen at al [9].

For the modeling purpose of the PSN effect it is necessary to characterize the large

particle size distribution. The larger particle size distribution observed in a two-

dimensional section is generally in the form of [10]

( ) exp( )f H L (1)

where is the particle diameter measured in 2-D, H and L are experimental constants

which can be determined from particle size distribution on a plane. 22 / dL ,2

0H L N . Where 2d is the average diameter of the particles with diameter larger than

c in the 2-D measurement, 0N the total number of particles in the plane. The number

of particles per unit volume, which is larger than in 3-dimensions, is given by

Sandstrom’s analysis [10]:

2 3( ) exp( ) ( )

4v

H LN L erfc L

L (2)

Application of cold rolling and hot deformation to a strain of 0.7 does not change the

size of the undissolved particles and their distribution in the samples A. However, the

fine dispersoids in the samples B tend to form bands parallel to the rolling direction, as

shown in Fig. 2d.

Table 1. Particle evolution during homogenization at 630°C and annealing at 450°C

Treatment

conditions

Average size

(µm)

Aspect ratio Number/mm2

103

%Area

transfer 1.82 1.78 23.24 9.9

450°C 24h 1.95 1.76 25.12 10.5

630°C 16h 2.25 1.72 13.96 6.2

630°C 78h 2.94 1.70 9.46 6.0

630°C 168h 3.42 1.63 5.06 5.6

8.3.2. High temperature mechanical behavior

The flow stress-strain curves under different deformation conditions are shown in

Fig. 1. The steady state stress of the sample A is always higher than that of the sample B

Chapter 8140

under same deformation conditions. This indicates that the level of Mn in solute

solution and the presence of the fine second phase particles have a complicated effect on

the flow stress. The mechanical behavior is well described by Sellars-Tagart constitutive

equation [11]:

(sinh ) exp( / )m

gA Q R T (3)

where A, , m are material constants, , true strain rate, Rg, the universal gas constant,

T, absolute temperature, Q, the activation energy of deformation and , the flow stress.

Conditions of and T are often incorporated into a single parameter, the Zener-

Hollomon parameter, Z [11]

exp( / )gZ Q R T (4)

Fig. 1. The equivalent stress-equivalent strain curves of samples A and B, a). Deformed

at three temperatures for a fixed strain rate of 1 /s, b) deformed at 450°C for three

strain rates.

The parameters obtained by fitting Eq. (3) to the steady state flow stresses are listed in

Table 2 in the first two rows. Although the exact physical significance of each constant

in Eq. (3) is rather vague, it is interesting to compare the obtained values with those of

commercial purity aluminum and with those of conventional Al-Mn series alloys [12-

14]. A very wide variation has been reported on these parameters of Al-Mn alloys in

differing states, a few examples are listed in Table 2. If the parameter A is regarded as a

frequency factor (proportional to the density of activateable sites), the higher value in

the treatment A reflects that more dislocations overcome obstacles to move in unit time.If the parameter is a measure of activation volume, the lower value in treatment B

0

10

20

30

40

50

60

70

0 0.2 0.4 0.6 0.8Strain

Str

ess (

MP

a)

A350°C

B550°C

A550°C

B450°C

A450°C

B350°C

1/ s a

0

10

20

30

40

50

0 0.2 0.4 0.6 0.8Strain

Str

ess (

MP

a)

Td=450°C

A1/s

A0.1/s

A10/s

B1/s

B10/s

b


suggests a smaller volume of material participating in a single deformation event. And if

the parameter m is associated with climb-controlled processes, irrespective of dispersed

particles, the lower value in the treatment A indicates a larger solute drag effect during

deformation. The activation energy basically reflects the energy barriers over which

deformation can take place. For pure aluminum the activation energy is an indicator of

the energy barriers for self-diffusion, which largely governs dislocation motion. For

aluminum alloys, the activation energy is affected by solute concentration and the

dispersion of second-phase particles in a complicated way, which usually act as addition

barriers to the free motion of dislocations. Thus, as the high temperature mechanical

behavior of the materials depends largely on the content of Mn in the matrix and the

amount of dispersoids, the effect of the pre-deformation state should not be ignored.

Table 2. The fitting parameters of the material constants in the constitutive equations for

treatments A and B and some related data in literature.

A sec-1 MPa-1m Q (kJ/mol) deformation

state

Treatment A 111.0 10 0.056 4.0 186

Treatment B 101.2 10 0.052 4.5 160

Al-1Mn [12] 101.21 10 0.039 5.0 156 transfer

Al-1Mn [14] ~ 0.052 2.6 152 homogenized

AA3003 [13] ~ ~ 3.1 180 homogenized

AA1050 [12] 111.96 10 0.036 5.0 155 transfer

The subgrain size, , directly after hot deformation was estimated from the following

relationship [15]:

1 /( )av s iM Gb (5)

where 1 is a constant of a order 3, avM , the average value of Taylor factor, G, the

shear modulus, b, the burgers vector and i , s , the initial flow stress and the steady

state flow stress, respectively.

As the stored energy is inversely proportional to the subgrain size, the driving force for

recrystallization in the samples of the treatment A having a higher flow stress is higher

than that of the treatment B under the same deformation conditions.

Chapter 8142

8.3.3. The decomposition kinetics of the supersaturated matrix during

deformation and isothermal annealing

The electrical conductivity of AA3003 in the transfer state is measured as 24.8 MS/m

which corresponds to 0.34 wt.% Mn in solution. For the samples with treatment A,

during homogenizing at 630°C the conductivity decreases rapidly in the first (within 30

minutes) and then approaches a constant value of 19.96 MS/m, which indicates that the

solid solution (of Mn, Fe, and/or Si) in the matrix reaches the equilibrium level. The

solubility of Mn in solute solution in this temperature is 0.68 wt.% and this value is

maintained during down quenching. For the samples after treatment B, the conductivity

is 26.98 MS/m. The corresponding Mn content in the matrix is 0.22 wt.%.

Table 3. The conductivity measurement in the samples A after deformation and the

fraction decomposed during deformation with respect to the equilibrium concentration

at different annealing temperatures

The fraction decomposed during

deformation in the annealing temperature

Deformation

conditions

LnZ Conductivity

(MS/m) 450°C 500°C 540°C

As-

homogenized

19.96 0.0 0.0 0.0

CR 20.15 3.6 4.3 5.7

550°C 0.1/s 24.9 20.46 8.0 9.5 12.6

550°C 1/s 27.2 20.35 6.5 7.6 10.1

450°C 0.1/s 28.6 20.36 6.6 7.8 10.4

550°C 10/s 29.5 20.27 5.3 6.3 8.4

450°C 1/s 30.9 20.23 4.8 5.6 7.5

450°C 10/s 33.2 20.18 4.1 4.8 6.4

350°C 0.1/s 33.6 20.42 7.5 8.8 11.7

350°C 1/s 35.9 20.20 4.4 5.1 6.8

It is interesting to note that the conductivity measured in the deformed samples of the

treatment A is higher than that before deformation, as listed in Table 3. Fig. 2 shows the

distribution of the second phase particles in the as-deformed samples for 4 different

deformation conditions. In Fig. 2a, for material of treatment A after cold rolling, the

matrix is clean except for coarse primary particles. However, fine precipitates are found

along the grain boundaries and subgrain boundaries in the samples after hot deformation

(Fig. 2b and Fig. 2c). More precipitates can be observed in the sample A350°C 0.1/s

than in A350°C 1/s. Fig. 2d shows the particle distribution in B450°C 1/s in which the

smaller particles are formed before deformation. The conductivity in the deformed


samples B also increases when the deformation temperature is below 450°C. The

increment in conductivity is related to the deformation conditions in the same manner as

in the samples A but is smaller. However, when the sample B is deformed at 550°C the

conductivity decreases, in line with resolutionising of Mn.

Fig. 2. The distribution of the precipitates in the as deformed samples, (a) Cold Rolling,

(b) A350°C 0.1/s, (c) A350°C 1/s and (d) B450°C 1/s

That the conductivity of the as-deformed samples is higher than that in the undeformed

samples indicates that the solute redistribution of the solute atoms (Mn, Fe, Si) has

occurred during deformation. The fine dispersoids observed along grain boundaries or

subgrain boundaries in some of the hot deformed samples give a further indication that

strain-induced precipitation could happen during hot deformation.

Fig. 3 shows the conductivity variation with annealing time in the samples of treatment

A after different deformation conditions. In Fig. 3a, material annealed at 450°C, the rate

of the precipitation in the sample after cold deformation is highest at the very beginning

of the annealing (~1 min.) and then decreases. The effect of the hot deformation

Chapter 8144

conditions on the precipitation kinetics depends on the Zener-Hollomon parameter, Z

(lnZ is listed in Table 4) and deformation temperature. In Fig. 3b, material annealed at

500°C, the enhancing effect of the deformation on the precipitation at first sight bears

no relation with Z. However, when the effect of the recrystallization on the precipitation

kinetics is considered these results could be fully understood. This point will be

elaborated in detail in the discussion.

Fig. 3. The decomposition kinetics in the samples A measured by means of conductivity

( 0.70 for both cold and hot deformation), (a) annealed at 450°C, (b) annealed at

500°C.

19

20

21

22

23

24

25

26

27

28

29

30

0.01 0.1 1 10 100 1000 10000

Annealing time (min)

Co

nd

uctivity (

MS

/m)

no strain 50% CR

350°Cx0.1/s 350°Cx1/s

450°Cx0.1/s 450°Cx1/s

450°Cx10/s 550°Cx0.1/s

550°Cx1/s 550°Cx10/s

a

19

20

21

22

23

24

25

26

27

28

0.01 0.1 1 10 100 1000 10000Annealing time (min)

Conductivity (

MS

/m)

no strain 50% CR

350°Cx0.1/s 350°Cx1/s

450°Cx0.1/s 450°Cx1/s

450°Cx10/s 550°Cx0.1/s

550°Cx1/s 550°Cx10/s

b


Fig. 4 is the evolution of the conductivity measured in the samples of B450°C 1/s

annealed at different temperatures. During annealing at 540°C the conductivity

decreases rapidly and then approaches a constant value. This indicates that the second

phase particles are dissolving during early stage of annealing. Deformation not only

promotes the precipitation process but also enhances the particle dissolution process.

Fig. 4. The conductivity in the samples of B450°C 1/s as a function of annealing time

at different temperatures.

8.3.4. The softening kinetics

The softening kinetics is determined from the microhardness measurement and it is

given by:

def t

def rex

H Hy

H H (6)

where defH , rexH and tH are the values of the microhardness measured in the as

deformed state, fully recrystallized state and at annealing time t respectively. The

softening kinetics after different deformation conditions is listed in Table 4 and is

shown in Fig. 5 to Fig. 7 for materials after hot deformation and Fig. 8 for material after

cold rolling. OPM in Fig. 5a labels the recrystallization kinetics measured from optical

microscopy. As can be seen, the softening kinetics is either fast or slow, strongly

25

26

27

28

29

30

31

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04Annealing time (min)

Co

nd

uctivity (

MS

/m)

450°C

500°C

540°C

10-2 10

410

310

110

-110

210

0

Chapter 8146

depending on the pre-heat treatment, deformation temperature, strain rate and annealing

temperature.

For a given deformation condition, the rate of softening increases as the annealing

temperature increases. For a given strain rate, an increase in the deformation

temperature results in a decrease in the rate of recrystallization. For a given deformation

temperature, an increase in the strain rate leads to an increase in the rate of

recrystallization kinetics. Generally, for a given annealing temperature, the softening

kinetics increases as the Z value increases. However, attention should paid to special

cases. Although the Z values for the deformation conditions of A350°C 0.1/s and

A450°C 10/s are almost the same, the softening kinetics in the samples of

A450°C 10/s is much faster than that in A350°C 0.1/s (see Table 4, Fig. 6 and Fig. 7).

This seemingly unexpected result was verified in a new set of experiments.

When recrystallization is fast, recovery is responsible for only ~25% of the softening.

However, when the softening kinetics is sluggish the contribution of the recovery to the

softening can be as high as 70%, as shown in Fig. 5 to Fig. 7, in which arrows indicate

the start of recrystallization.

The softening kinetics in the cold rolling samples is faster than that in the hot deformed

samples under the same annealing temperatures. This is in line with a finer dislocation

network due to cold deformation. The contribution of the recovery to the softening is

about 60% and it slightly increases as at annealing temperature increases.

It should be noted that when the recrystallization is fast, the softening rates in samples

of the treatment B is higher than that in samples of the treatment A (see Fig. 5a, Fig. 6a

and Fig. 7a). Contrarily, when the recrystallization is slow, the softening rates in

samples of the treatment A is higher than that in samples of the treatment B (see Fig. 5b,

Fig. 6b and Fig. 7d). This indicates that the precipitates existing prior to the plastic

deformation play a less effective role in retarding recrystallization even though the

particle size may be in the range of 50~150 nm.

The times required for 5% and 95% recrystallization are now plotted against reciprocal

temperature for several deformation conditions, as shown in Fig. 9. A striking feature of

Fig. 9 is the pronounced upward curvature of the data at low temperature for both the

start and finish of the recrystallization except for the case of B350°C 1/s where there is

no curvature for the start of the recrystallization. The recrystallization time is longer

than would be predicted by extrapolation from high temperature using Arrhenius

relation. The departure of the curves from the extrapolation to low temperature suggests


that both the nucleation and growth processes are retarded by the concurrent

decomposition process.

Fig. 5. The softening kinetics annealed at 450°C after different deformation conditions.


0

0.2

0.4

0.6

0.8

1

1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04


Fra

ctio

n

A350°Cx1/s,OPM

B350°Cx1/s,OPM

A350°Cx1/s

B350°Cx1/s

Ta=450°C

a

10-3

104

103

102

101

100

10-1

10-2

0

0.2

0.4

0.6

0.8

1

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05Annealing time (min)

Fra

ctio

n

A450°Cx1/s

A450°Cx10/s

B450°Cx1/s

B450°Cx10/s

Ta=450°C

b

10-1 10

510

4103

102

101

100

0

0.2

0.4

0.6

0.8

1

1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04


Fra

ction

A450°Cx0.1/s

A450°Cx1/s

B450°Cx1/s

A450°Cx10/s

B450°Cx10/s

Ta=500°C

b

10-3

104

103

102

101

100

10-1

10-2

0

0.2

0.4

0.6

0.8

1

1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03


Fra

ctio

n

A350°Cx0.1/s

A350°Cx1/s

B350°Cx1/s

Ta=500°C

a

10-3

103

102

101

100

10-1

10-2

Chapter 8148


0

0.2

0.4

0.6

0.8

1

0.01 0.1 1Annealing time (min)

Fra

ction

B350°Cx1/s

A350°Cx1/s

A350°Cx0.1/s

Ta=540°C

a

10-2

100

10-1

0

0.2

0.4

0.6

0.8

1

0.01 0.1 1 10 100Annealing time (min)

Fra

ction

A450°Cx0.1/s

A450°Cx1/s

A450°Cx10/s

Ta=540°Cb

10-2

102

101

100

10-1

0

0.2

0.4

0.6

0.8

1

0.01 0.1 1 10Annealing time (min)

Fra

ction

B450°Cx1/s

B450°Cx10/s

A450°Cx1/s

A450°Cx10/s

Ta=540°C

c

10-2

100

101

10-1 0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1 10 100 1000 10000Annealing time (min)

Fra

ction

B550°Cx1/s

A550°Cx0.1/s

A550°Cx1/s

A550°Cx10/s

Ta=540°C

d

10-3

10-2

104

103

102

101

100

101


Fig. 8. The softening kinetics annealed at different temperatures after cold deformation.

Fig. 9. Times to 5% and 95% recrystallization for several deformation conditions.

0

0.2

0.4

0.6

0.8

1

0.01 0.1 1 10 100 1000 10000Annealing time (min)

So

fte

nin

g f

ractio

n

350°C

500

450

400

10-2

101

100

103

104

102

10-1

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.2 1.25 1.3 1.35 1.4 1.45 1.51000/T (K)

t0.05 A350°Cx1/s

t0.95 A350°Cx1/s

t0.05 B350°Cx1/s

t0.95 B350°Cx1/s

t0.05 CR

t0.95 CR

Tim

es t

o 5

% o

r 9

5%

re

x (

min

)

105

100

101

102

103

104

10-2

10-1

a

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.2 1.25 1.3 1.35 1.4 1.45 1.51000/T (K)

t0.05 A450°Cx10/s

t0.95 A450°Cx10/s

t0.05 B450°Cx10/s

t0.95 B450°Cx10/s

t0.05 CR

t0.95 CR

106

10-1

100

101

102

103

104

105

10-2

Tim

es to 5

% o

r 95

% r

ex (

min

)

b

Chapter 8150

Table 4. The start and finish times and recrystallized grain size of the two treatments

after different deformations and different annealing temperatures

Defor-

mation

T°C

Strain

rate

(1/s)

LnZ Anneal

T °C

Starting

time

(min)

Complete

time

(min)

Grain size

(µm) RD/ND

Aspect

ratio

540 0.15 1 1438/362 4

500 60 103 2857/180 16

0.1 33.6

450 No recrystallized within 7620 min

540 0.2 496/175 2.8

500 0.3 2 562/150 3.3

A350

1 35.9

450 1080 104 1088/145 7.5

540 12 40 1875/358 5.2


0.1 28.6


540 0.2 2 1312/375 3.5

500 960 4200 2000/313 6.5

1 30.9


540 0.1 0.5 637/262 2.4

500 0.5 10 725/212 3.4

A450

10 33.2

450 4800 5 104 1375/195 7.2

0.1 24.9 540 No recrystallized within 5040 min

1 27.2 540 240 4320 6500/900 7.2

A550

10 29.5 540 0.2 2 1687/412 4

540 0.15 536/112 4.5

500 0.12 1 612/124 4.9

B350 1 30.9

450 0.3 60 1466/156 9.4

540 0.5 5 3875/295 13-18

500 7620 ~

1 26.6


540 0.1 0.4 1125/150 7.5

500 0.3 5 1450/175 8.2

B450 10 28.9

450 23280 4 105 2140/196 11

B550 1 23.4 540 No recrystallized within 5040 min

500 0.5 26.8 1.36

450 0.2 30 28.5 1.42

400 30 6000 280/123 2.28

A

Cold rolling



8.3.5. Nucleation mechanisms and recrystallized grain structure

After hot PSC deformation the grains are further elongated along the rolling direction.

The average aspect ratio of the grains now is 14-18. The grain structure after the same

amount of cold rolling is comparable to that after hot deformation. In addition to the

pancaking of the grains, deformation bands can be found in the as-deformed

microstructure. These deformation bands are not uniformly distributed. In the heavily

dynamic precipitated samples, the subgrains are observed to be elongated along the

deformation direction because precipitation occurs on the subgrain boundaries (as

shown in Fig. 3b).

Fig. 10. The partially recrystallized structures in the samples (a) A450°C 10/s,

annealed at 450°C for 128 h, (b) B350°C 1/s, annealed at 450°C for 40 s, (c)

A450°C 1/s, annealed at 540°C for 18 s and (d) CR, annealed at 400°C for 1 h.

Particle stimulated nucleation (PSN) was only found in the cold rolled samples

annealed at temperature above 500°C. In this circumstance, the initial stage of

recrystallization is more or less uniform and the early site saturation can be inferred.

However, in all other cases, the initial stage of the recrystallization appears to be highly

Chapter 8152

localized. Nucleation by subgrain growth (Fig. 10a) and strain induced boundary

migration (SIBM) (Fig. 10b) are often observed. The growing recrystallized grains have

an irregular shape and jagged grain boundaries which is typical for particle impeded

grain boundary migration. No PSN effect was found in the current hot deformed

samples annealed in the temperature range studied under optical microscopy. Evidently,

hot deformation leads to a reduced tendency for nucleation. The recrystallized grain

size in the sample after hot deformation are typically 5~10 times larger than those at the

same stage of transformation after cold deformation (compare Fig. 10c and Fig. 10d).

The deformation inhomogenity, e.g. near original grain boundaries, is prerequisite for

the nucleation of recrystallization (Fig. 10c).

The final recrystallized grain size resulting from annealing at different temperatures

were measured by means of optical light microscopy. The final grain size varies

considerably within the specimens. The average values are listed in Table 4, where the

grain size is given by two dimensions, RD (grain dimension in the elongation direction

in PSC specimens) and ND (grain dimension in the compress direction). Measured grain

dimensions for fully recrystallized material for several cases are given as a function of

annealing temperature in Fig. 11 for both RD and ND. It is clearly shown that for a

given deformation condition, the fully recrystallized grain size and its aspect ratio

decrease as annealing temperature increases. This relation of grain size with annealing

temperature is opposite to that commonly cited as “recrystallization law” which does

not involve dispersoids effects. The size increase at lower temperature is primarily in

the RD direction.

Fig. 11. Recrystallized grain size as a function of annealing temperature for several

deformation conditions.

0

200

400

600

800

1000

1200

1400

1600

440 460 480 500 520 540 560

Annealing temperature (°C)

Re

cry

sta

llize

d g

rain

siz

e

(µm

)

A350°Cx1/s RD

A350°Cx1/s ND

B350°Cx1/s RD

B350°Cx1/s ND

a0

500

1000

1500

2000

2500

440 460 480 500 520 540 560

Annealing temperature (°C)

Re

cry

sta

llize

d g

rain

siz

e (

µm

)

A450°Cx10/s RD

A450°Cx10/s ND

B450°Cx10/s RD

B450°Cx10/s ND

b


For a given deformation temperature, an increase in the strain rate leads to a decrease in

the recrystallized grain size. The grain size is larger for the B preheat practice. The

aspect ratio of the recrystallized grains in the samples of the treatment B remains larger

than that in the samples of the treatment A. The fully recrystallized grain size and its

aspect ratio are much smaller in the cold deformed samples.

The lower the annealing temperature, the larger the recrystallized grain size and the

larger aspect ratio of the grain and these phenomena can be associated with the sluggish

recrystallization process.

8.4. Discussion

8.4.1. Effect of the deformation, recovery and recrystallization on the

precipitation kinetics of supersaturated alloy

The microstructural evolution during annealing is complex and needs to be interpreted

in terms of the combination of precipitation and recrystallization. In Al-Mn alloys,

precipitation occurs only on high or low angle grain boundaries. Thus, the precipitation

kinetics is influenced by the deformed, recovered and recrystallized microstructure.

That deformation of the supersaturated solution increases the precipitation rate is

attributed to the following reasons [16]: deformation increases the dislocation density,

which increases both the nucleation sites and diffusivities of precipitate-forming

elements in the material. Furthermore, the nucleation barrier for precipitation on

dislocations is much lower than that for homogenous nucleation. In addition, the solute

segregation to the dislocation network during deformation enhances the rate of

precipitation via an increase in the local chemical driving force. Therefore, the more

dislocations introduced by deformation, the larger effect of the enhancement. Any

recovery or recrystallization will result in a decrease in the dislocation density, and

therefore, weaken the effect of deformation.

Fig. 12 shows schematically a time dependence of the dislocation density for various

experimental conditions for samples of treatment A deformed at 450°C. The hot

deformation is done at a temperature range where precipitation and dynamic recovery

occur concurrently. The dislocation density reaches a saturation value at a strain of 0.7.

The higher the strain rate, the higher the dislocation density. However, the loading time

increases as strain rate decreases. In the following annealing processes, the dislocation

density decreases because of static recovery and recrystallization. The reduction of the

dislocation density is faster in the samples deformed at a higher strain rate and annealed

at a higher temperature. Here we need to analyze the effect of the deformation on the

Chapter 8154

dynamic precipitation (during deformation) and static precipitation (during annealing

after deformation) separately.

Fig. 12. The dislocation density as a function of time for various experimental

conditions deformed at 450°C.

The solute segregation or dynamic precipitation during hot deformation is a complicated

problem. Data in Table 3 show that the conductivity increases after deformation, which

indicates that the decomposition of the supersaturated matrix has happened during

deformation. Solute atmosphere formation around a moving dislocation leads to the

redistribution of solute elements between the dislocation cell walls and the cell interiors.

When the concentration of the solute atoms at dislocations reaches the composition of

the precipitate nucleus, a precipitate can form. The concentration of the solute atoms

around a moving dislocation is determined by the velocity of solute atom diffusion

relative to the dislocation velocity which is proportional to the macro strain rate. The

solute atom diffusion may be enhanced by vacancy production during hot deformation.

Therefore, this process depends strongly on the strain rate and the deformation

temperature. However, no apparent correlation between dynamic precipitation and the Z

value can be deduced.

The contribution of the dynamic precipitation during deformation to the overall

precipitation kinetics is different at different annealing temperatures thereafter. The

fraction decomposed during deformation is given by:

time

Dis

loca

tion

den

sity

deformation annealing

0.1/ s

10 / s

1/ s

450°C

540°C

500°C

Td=450°C


1h d

h

x (7)

where h and d are the resistivities in the as-homogenized state (before deformation)

and in the as-deformed state, respectively. is the saturation level of the resistivity of

the fully precipitated sample at long time, which corresponds to the equilibrium contentof Mn in solid solution at a given annealing temperature. is calculated in Chapter 7.

The fraction decomposed during deformation can be up to 12% of overall precipitation,

as listed in Table 3.

By referring to the recrystallization kinetics listed in Table 4 the experimental data in

Fig. 3 can be understood as follows. When annealed at 450°C, the recrystallization start

and finish time after cold deformation is 0.1 and 10 min, respectively. The precipitation

rate in the cold deformed sample is relatively high at the very beginning of annealing

(Fig. 3a) and then decreases because of the annihilation of the dislocation density as a

result of recovery and recrystallization. On the contrary, softening in all the other hot

deformed samples is still in the slow recovery stage and the effect of deformation

conditions on the precipitation rate will follow the rule of “the higher the Z value, the

higher dislocation density, the higher precipitation rate”. However, loading time also

plays an important role. For example, although the Z value of the sample 350°C 1/s is

higher than that of sample 350°C 0.1/s, the precipitation rates of the two samples are

almost identical. This is due to the stronger segregation effect caused during

deformation in the sample 350°C 0.1/s. In the case of Fig. 3b, material annealed at

500°C, precipitation rate at deformation conditions of higher Z becomes lower because

of the early recrystallization. Thus, the experimental data in Fig. 3 show, in fact, a

combined effect of deformation and softening on the precipitation.

The JMAK equation is now applied to describe the static precipitation kinetics during

annealing at a given temperature. The precipitation kinetics is written as:

2 1 exp( )nd t

d

x kt (8)

where t is the resistivity at annealing time t. d and are the same as above. k and

n are constants. The best-fitting parameters for k and n under different deformation

conditions are listed in Table 5. The value of n decreases as Z value increases but k

varies in the opposite manner except the cases where the deformation is carried out at

lower strain rate (0.1 /s).

Chapter 8156

Table 5. Fitting parameters for k and n under different deformation conditions using

JMAK equation

Annealing temperature °C

450 500 540

Deformation

conditions

lnZ

n k n k n k

No strain 0.39 0.037 0.42 0.031 0.37 0.057

CR 0.37 0.066 0.33 0.108 0.30 0.160

A550 0.1/s 24.9 0.54 0.016 0.43 0.056 0.34 0.109

A550 1/s 27.2 0.51 0.025 0.42 0.067 0.34 0.125

A450 0.1/s 28.6 0.49 0.032 0.38 0.104 0.33 0.132

A550 10/s 29.5 0.50 0.029 0.41 0.082 0.33 0.135

A450 1/s 30.9 0.43 0.070 0.36 0.135 0.32 0.146

A450 10/s 33.2 0.43 0.076 0.34 0.106 0.30 0.162

A350 0.1/s 33.6 0.38 0.122 0.29 0.212 0.29 0.175

A350 1/s 35.9 0.38 0.132 0.29 0.161 0.29 0.178

The overall precipitation kinetics includes dynamic precipitation and static precipitation

and is given by:

1 ( ) exp( )nd

h

x kt (9)

The times to 15% decomposition and 95% decomposition at different annealing

temperatures are calculated and represented by the C-curves, as shown in Fig. 13. The

introduction of deformation has the effect of shifting the C-curve to shorter times. The

start-up of precipitation is strongly affected by deformation condition, i.e. the strain rate

and deformation temperature. The complete of precipitation is not only promoted by

deformation but also affected by recovery and recrystallization. For example, in

Fig. 13e, the times to 95% precipitation for both cases of A450 10/s and cold

deformation are almost the same when the samples are annealed at 540°C. However,

when materials are annealed at 450°C the time to 95% precipitation of A450 10/s is

much shorter than that of samples after cold deformation. This is due to the fact that the

enhancing effect of deformation is weakened by different degree of recovery and

recrystallization at different annealing temperatures. The recrystallization process in

both cases completes within 0.5 minutes during annealing at 540°C. When dislocations

are eliminated the enhancing effect of the deformation is only from the solute

segregation formed during deformation. When the materials are annealed at 450°C, in


the sample of cold deformation, the recrystallization completes in 10 minutes while in

the sample of A450 10/s the recrystallization takes 7000 minutes to start, at which time

the precipitation is complete.

8.4.2. Effect of precipitation on the recrystallization kinetic

As we have observed in this experiment, the basic nucleation mechanisms in

recrystallization after the hot deformation is subgrain growth and strain-induced

boundary migration. The fine precipitates retard both nucleation and growth of the

recrystallized grains. It can be argued that the large recrystallized grain size results from

a reduction in the number of successful nuclei as annealing temperature decreases. In

the following we will develop a model to explain the effect of the precipitation on the

softening kinetics qualitatively. We simplify the analysis by considering the

substructure to be adequately described using two components only. The major

component is considered as an assembly of equiaxed subgrains of mean equivalentradius, R , mean misorientation, , and with boundaries of mean energy and mobility,

and M respectively. The minor component we consider as “particular” subgrains

(effectively sub-critically sized recrystallization nuclei) which have a larger size ( nR )

and different boundary characteristics ( , ,n n nM ) with respect to that of the “average”

subgrain assembly [17,18]. According to Cahn and Humphreys the presence of the local

misorientation gradient (covering a number of average subgrain diameters) is necessary

for the formation of the nuclei. During “particular” subgrain growth its boundarymisorientation n is related to the size of the “particular” subgrain as [5]

0 0( )n nR R (10)

Where 0 and 0R are initial values of n and nR , and is the orientation gradient

( /n nd dR ).

When a deformed supersaturated solution is annealed, the softening and the

precipitation processes occur simultaneously. The growth rate of the particular subgrain

is given by:

( )nn D Z

dRV M P P

dt (11)

where PD and PZ are the net driving pressure and Zener pinning pressure respectively.

Chapter 8158

Fig. 13. The precipitation-temperature-time (PTT) diagram for the samples of

treatment A after different deformation conditions. (a), (b) and (c) the time to 15%

precipitation; (d), (e) and (f), the time to 95% precipitation.

440

460

480

500

520

540

560

0.001 0.01 0.1 1 10 100Annealing time (min)

Tem

pera

ture

(°C

)

A350°Cx1/s

A350°Cx0.1/s

CR

No strain

a

440

460

480

500

520

540

560


Tem

pera

ture

(°C

)

A450°Cx10/s

A450°Cx1/s

A450°Cx0.1/s

CR

No strain

b

440

460

480

500

520

540

560


Te

mp

era

ture

(°C

)

A550°Cx10/s

A550°Cx1/s

A550°Cx0.1/s

CR

No strain

c

440

460

480

500

520

540

560

0 2 4 6 8 10Annealing time x10

-4 (min)

Te

mpera

ture

(°C

)

A350°Cx1/s

A350°Cx0.1/s

CR

No strain

d

440

460

480

500

520

540

560


-4(min)

Tem

pera

ture

(°C

)

A450°Cx10/s

A450°Cx1/s

A450°Cx0.1/s

CR

No strain

e

440

460

480

500

520

540

560


-4(min)

Tem

pera

ture

(°C

)

A550°Cx10/s

A550°Cx1/s

A550°Cx0.1/s

CR

No strain

f


The net driving pressure acting on the front of a viable nucleus is given by:

( )nD

d AP E

dW (12)

where W and A are the volume and the surface area of the viable nucleus, respectively.is the specific boundary energy of the viable nucleus. E is the energy difference

between the two sides of the recrystallized front, which is approximately the stored

energy in the average assembly and for hot deformation it can be estimated by:

/E R (13)

where is a geometrical constant and has a value of ~1.5 [5]. The boundary energy

depends on the misorientation and is given by [5]:

( ) (1 ln( ))m

m m

(14)

where m and m are, respectively, the values of boundary energy and misorientation

for high-angle boundaries, which is commonly taken as 15° and m is 0.324 J/m².

The net driving pressure for this particular subgrain to grow is now given by:

2 ( )lnn m n

D

n m m

PR R

(15)

The first term of the right hand of Eq. (15) is the driving pressure from stored energy

(Pv), the second term is retarding pressure from boundary curvature (Pc), and the third

term is a pressure from the deformation gradient.

If the main mechanism of recovery in the subgrain assembly is subgrain coarsening, the

growth rate of a uniform subgrain assembly may be expressed in the form [5]:

( )4

Z

dRM P

dt R (16)

For a random dispersion of spherical particles (average radius r and volume fraction fp)

the drag pressure is given by [19]:

Chapter 8160

6 /z pP f r (17)

A more realistic assumption for the precipitate distribution is that the particles lie on

sub-boundaries in the hot worked structure in AA3003. The precipitate-retarding force

corresponding to this case is [19]:

2

3

2p

z

fP

r (18)

where is the average subgrain diameter, which is given by Eq. (5).

The condition for instability, leading to discontinuous growth of the particular subgrain

is written as [17,20]:

0nn

dR dRR R

dt dt (19)

Setting the left-hand side of inequality (19) equals to zero and assuming that the

misorientation between a viable nucleus and surrounding is 15°, the critical radius of a

viable nucleus can be obtained as:

2

8 ( )4

2( )4

n n n n nn z z

cn n

z z

M M M MR P P

M M M M RR

MP P

M R

(20)

This equation shows that the critical size for a viable nucleus is a function of zP , R ,

as well as the ratio of the mobilities between the high angle grain boundary and low

angle sub-boundary. In general, zP will vary during annealing because of the time

dependence of the nucleation and growth of the precipitates. R will increase due to theconcurrent recovery. In addition, nM and M will increase as the decrease in the solute

concentration of the matrix. Therefore, Rc will vary with annealing time and

temperature. An essential condition for the presence of the solution of the Eq. (20) is the

term under the root sign being not less zero. Taking it to be zero, one obtains:

2( ) 2

8 ( 1)

nn

zCriticaln n

n

M

MPM

RM

(21)


Thus if the value of zP given by Eq. (18) is larger than that by Eq. (21) there will be no

nucleation event.

In order to elucidate the evolution of the Rc with annealing time and annealing

temperature we need to consider the precipitate properties and the boundary properties

in more detail. As derived in the appendix, the volume fraction of the precipitate is

expressed as:

0

0

p

c cf

c c (22)

where 0.738 for Mn3SiAl12, 0.816 for MnAl6, c and 0c are the Mn

concentrations in the precipitate and in the matrix before precipitation occurs,respectively. c is the Mn concentration in the matrix at annealing time t for a given

temperature T, which can be obtained on the basis of the decomposition rate as

determined from the resistivity measurement, i.e.

1 exp( )nd d t

d d

c ckt

c c (23)

where dc , c and c , are the concentrations of Mn in the matrix in the as-deformed

state, after an annealing time t and in equilibrium, respectively. d , t and , are the

corresponding resistivities at the start of annealing, after an annealing time t and inequilibrium, respectively. Values of c can be calculated from the solvus line as in

Chapter 7. Combining Eqs. (22) and (23), one gets:

0

0

( ) exp( )n

dp

c c c c ktf

c c (24)

where k and n are fitting parameters as listed in Table 5.

Assuming that the precipitates have a simple spherical growth morphology, the growth

rate during the growth stage is:

MnDdr c c

dt r c c (25)

By solving Eq. (25) against annealing time we obtain:

Chapter 8162

2 20

0

2 ( )exp( )

( )

t

nMn dD c cr r kt dt

c c (26)

The diffusion coefficient of Mn in aluminum at a given temperature is given by:

0 exp( )dif

Mn

g

QD D

R T (27)

where DMn is the diffusion coefficient, c , the instantaneous concentration of Mn in thematrix, c , the concentration in the particle, c , the concentration of Mn in matrix in

equilibrium with precipitate assuming a planar interface, gR , the gas constant, T,

absolute temperature, 0r , the average radius of the precipitates in the as-deformed state,

r , the particle radius at annealing time t.

Considering the solute drag effect, the grain boundary mobility in a solute solution is

given by Cahn’s solution for low driving force [21]:

11

'n

Pure

M cM

(28)

where

2

b gb m

Pure

g

D VM

b R T (29)

and

2( )' sinhv B b

B B

N k T E E

ED k T k T (30)

where Mpure and c refers to the intrinsic grain boundary mobility and the concentrationof Mn in the matrix, b is the grain boundary width (~1nm), 34 /vN a the number of

atoms per unit volume, a, the lattice parameter, b, the value of the Burger’s vector, Dgb

is the grain boundary self-diffusion coefficient and D is the average value of the

diffusion coefficient in the vicinity of the grain boundary, Vm is the molar volume ofaluminum, Bk , Boltzmann’s constant, E, the interaction energy between a solute atom

and the grain boundary, which can be estimated by [22]:


34 mE Gr (31)

where ( ) /a m mr r r is the misfit parameter, ar , the radius of a solute atom, mr , the

atomic radius of a solvent atom, G, the shear modulus and it is given by [23]:

0

1 ( 300)1

2 933

TG G (32)

where 40 2.54 10G MN/m2, T is annealing temperature.

The mobility of a sub boundary is related to the average misorientation angle of the

boundary. It has a form of [17]:

41 exp( 5( ) )n

m

M M (33)

There appears to be no published value of the grain boundary diffusion coefficient. In

this analysis we simply assumed that the activation energy of the grain boundary

diffusion is one half of that for bulk diffusion. The parameters used in the model for the

diffusion coefficients are listed in Table 6.

Table 6. Model input parameters

0D

ms-1

difQ

kJ/mol

Reference

Bulk diffusion coefficient of Al 10-5 142 [24]

Grain boundary diffusion of Al 10-5 71

Mn bulk diffusion in aluminum 4 10-5 211 [25]

Mn pipe diffusion in aluminum 4 10-5 165

The evolution of the particle fraction and the particle radius is shown in Fig. 14. At

higher annealing temperatures the growth rate of the particle is larger than that at lower

annealing temperature. Therefore at the same amount of the precipitation fraction the

particle size is larger at the high annealing temperature.

Fig. 15 show the driving force evolution during annealing, in which vP is the driving

pressure, cP is the retarding pressure from the boundary curvature of the viable nucleus,

Chapter 8164

zP and zP are the pining drag pressures on the subgrain boundary and on the boundary

of the viable nucleus, respectively. zCriticalP is the pressure calculated from Eq. (21). zP

decreases as the annealing time increases because of the rapid growth of the particle

size. There are no possible nuclei present before zP decreases to zCriticalP (the arrow

point in Fig.15). zP is larger than zP due to the larger size and higher boundary energy

of the viable nucleus.

Fig. 14. The evolution of the particle fraction and particle radius at different

temperatures.

Fig. 15. The driving pressure evolution during annealing at different annealing

temperatures assuming the initial subgrain radius being 1.5 µm (for details see text).

0

0.4

0.8

1.2

1.6

2

2.4

1 10 100 1000 10000

Annealing time (s)

Pre

ssu

rex1

0-5

N/m

2

Pv

Pc

Pz

Pz

Pz critical

T=450°C

100

104

103

102

101

0

0.4

0.8

1.2

1.6

2

2.4

0.1 1 10 100 1000Annealing time (s)

Pre

ssu

rex1

0-5

N/m

2

T=540°C

Pz criticalPz

Pz

Pc

Pv

10-1

103

102

101

100

0

0.04

0.08

0.12

0.16

0.2

0 50 100 150 200 250Annealing time (s)

Ave

rag

e p

art

icle

rad

ius (

µm

)

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

f p

T=540°C

r

f p

0

0.02

0.04

0.06

0.08

0.1

0 1000 2000 3000 4000 5000

Annealing time (s)

Avera

ge

part

icle

rad

ius (

µm

)

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

f p

T=450°C

f p

r


The critical subgrain radius as a function of the annealing time at different temperatures

for 2 initial subgrain sizes is shown in Fig. 16. At a given annealing temperature the

critical nucleus radius first decreases as the pinning pressure decreases and then

increases again due to the increase of the subgrain size. The critical radius decreases as

the annealing temperature increases.

If we assume that the probability for a nucleation event in association with a possible

site is proportional to the number of subgrains able to grow then the smaller the Rc, the

more possible nucleation sites and the smaller the grain size. This explains why the

recrystallized grain size decreases as annealing temperature increases.

Fig. 16. The critical subgrain radius as a function of the annealing time at different

temperatures for 2 initial subgrain sizes.

Another point to be noted here is that, for the precipitate volume fraction encountered,

heterogeneous precipitation on the scale of substructure seems to be essential for the

retarding the recrystallization growth, and so plastic deformation plays an important role

in the geometry as well as the kinetics of precipitation. The precipitates existing prior to

the plastic deformation have no appreciable retarding effect on recrystallization even

though the particles may be small enough. This is why the recrystallization kinetics in

samples with treatment B is faster at high annealing temperature while it is slower at

lower annealing temperature.

2

3

4

5

6

7

8

9

10

11

0.1 1 10 100 1000 10000 100000Annealing time (s)

Critical nucle

us r

adiu

s (

µm

)

2.5µm

500°C2.5µm

540°C

1.5µm

540°C

1.5µm

450°C1.5µm

500°C

R=2.5µm

T=450°C

10-1

105

104

103

102

101

100

Chapter 8166

8.4.3. Interaction between precipitation and recrystallization

In essence, there is a strong two-way interaction between recrystallization and

precipitation after deformation: the precipitation reaction is accelerated by deformation

and weakened by recovery and recrystallization. The recrystallization is retarded by the

precipitation of MnAl6 particles on subgrain boundaries. The inter play of precipitation

and recrystallization phenomena can be studied by referring to Recrystallization-

precipitation-temperature-time diagrams (RPTT). The RPTT diagram is constructed

from experimental data for samples of treatment A, as shown in Fig. 17 for three

different deformation conditions. Here the start-up of recrystallization (taken as 5 %

recrystallized) and complete recrystallization (95 % recrystallized) are plotted, together

with the conditions for 15 % precipitation. The stored energy in the samples of

A350°C 0.1/s is slightly higher than that in the samples of A450°C 10/s. However, the

recrystallization kinetics in the sample A450°C 10/s is much faster than that in the

sample of A350°C 0.1/s. The precipitation kinetics in the sample of A350°C 0.1/s is

faster than that in the sample of A450°C 10/s at early stage of annealing, and therefore,

more precipitates form in the sample of A350°C 0.1/s, which provide a higher

retarding pressure to the recrystallizing front.

Fig. 17. Recrystallization-precipitation-temperature-time (RPTT) diagram for samples

of treatment A after three different deformation conditions. P0.15, Rs and Rf are the time

to 15% precipitation, start-up of recrystallization and completion of recrystallization.

Labels 1, 2, 3 indicate the deformation conditions A450°C 10/s, A350°C 0.1/s and

A350°C 1/s, respectively.

440

460

480

500

520

540

560

1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06 1.E+08


Tem

pera

ture

(°C

)

Rs3 Rs1 Rs2

Rf3 Rf1 Rf2

P0.152 P0.151

P0.153

10-2

108

106

104

102

100

10-4


A complete recrystallization-precipitation-temperature-time diagram in a wider

temperature range is established for material with treatment A after cold deformation,

as shown in Fig. 18. As we have analyzed in Chapter 7, at higher temperature the

precipitate to form is MnAl6 while at lower temperature the precipitate is MnAl12. If the

recrystallization starts before conspicuous precipitates have formed, for example, when

cold rolled material is annealed above 450°C, a fast recrystallization is present and a

fine recrystallized grain size is obtained. However, if conspicuous precipitates have

formed before recrystallization start, such as at annealing temperatures below 450°C, a

retarding effect on recrystallization is due to happen and a larger grain size is obtained.

Fig. 18. Recrystallization-precipitation-temperature-time (RPTT) diagram for samples

of treatment A after cold rolling. CR-50% cold rolling, NS-no strain.

300

350

400

450

500

550

600

1.E-02 1.E+00 1.E+02 1.E+04 1.E+06

Time (min)

Te

mp

era

ture

(°C

)

tr0.05

tp0.15,CR tp0.15,NS

tr0.95

MnAl6

MnAl12

10-2

104

102

100

106

tp0.95,CR tp0.95,NS

Chapter 8168

8.5. Conclusions

(1). For AA3003 alloy, preheat treatments, which changes Mn content in supersaturated

solution and the distribution of the second phase particles, have a large effect on the

mechanical behavior of the alloy, softening kinetics and the fully recrystallized grain

size.

(2). The softening kinetics is either fast or sluggish depending on the preheat treatments,

the deformation conditions and the annealing temperature. When the softening kinetics

is fast, the recrystallization kinetics in the material with a lower supersaturated Mn in

matrix is faster than that in the material with a higher supersaturated Mn in matrix.

When the softening kinetics is slow, an opposite trend is found.

(3). When recrystallization is slow the contribution of the recovery to the softening can

be as high as 70%. When recrystallization is fast recovery is responsible for only ~25%

of the softening.

(4). The recrystallized grain size in AA3003 after cold or hot deformation increases as

the annealing temperature decreases. The size increase at lower temperatures is

primarily in the RD direction.

(5). Recrystallization kinetics is not simply proportional to the value of the Zener-

Holloman parameter as a result of precipitation effect.

(6). The dynamic decomposition, which includes the solute atom redistribution and the

formation of the precipitates, during hot deformation is related strongly to the strain

rate. The static decomposition kinetics of supersaturated alloy during annealing at a

given temperature is affected by the deformation conditions, recovery and

recrystallization processes.

(7). Particle stimulated nucleation (PSN) is only observed in the cold deformed samples

annealed at higher temperatures (>500°C). Nucleation by subgrain growth and strain

induced boundary migration (SIBM) are the dominant nucleation mechanisms in the hot

deformed samples annealed in the temperature range 450-540°C.

(8). A theoretical framework has been established in which precipitation reaction and

recrystallization kinetics has been linked in a semi quantitative manner. The fine

precipitates present prior to deformation have less appreciable retarding effect on

recrystallization.


References

1. L.F. Mondolfo, Aluminium Alloys: Structure and Properties, Butterworth and Co.,

London, 1976.

2. E. Nes, J.D. Embury, Z. Metallkde. 66 (1975) 589-593.

3. G. Hausch, P. Furrer, H. Warlimont, Z. Metallkde. 69 (1978) 174-180.

4. P. Furrer, N. Rheinfall, H. Warlimout, Hanau, Aluminium 54 (1978) 135-142.

5. F.J. Humphreys, M. Haltherly, Recrystallization and Related Annealing Phenomena,

Pergamon, London, 1996.

6. E. Nes, Aluminium 52 (1976) 560-563.

7. F. Gatto, G. Camona, M. Conserva, P. Fiorini, Mater. Sci. Eng. A3 (1968) 56-61.

8. P.L. Morris, B.J. Duggan, Met. Sci. January (1978) 1-7.

9. V. Hansen, B. Andersson, J.E. Tibballs, J. Gjonnes, Metall. Mater. Trans. B 26

(1995) 839-849.

10. R. Sandström, Z. Metallkde. 71 (1980) 741-751.

11. C.M. Sellars, W.J. McGregor Tegart, Int. Met. Rev. 17 (1972) 1-24.

12. H. Shi, A.J. McLaren, C.M. Sellars, R. Shahani, R. Bolingbroke, Mater. Sci. Techn.

13 (1997) 210-216.

13. R. Akeret, Z. Metallkde. 61 (1970) 3-10.

14. H. Zhang, E.V. Konopleva, H.J. McQueen, Mater. Sci. Eng. A319-321 (2001) 711-

715.

15. O.D. Sherby, P.M. Burke, Pro. Mat. Sci. 13 (1967) 325-332.

16. S.P. Chen, N.C.W. Kuijpers, S.van der Zwaag, Mater. Sci. Eng. A341 (2003) 296-

306.

17. F.J. Humphreys, Acta Mater. 45 (1997) 4231-4240.

18. F.J. Humphreys, Acta Mater. 45 (1997) 5031-5039.

19. S.S. Hansen, J.B. Vander Sande, M. Cohen, Metall. Trans. A 11 (1980) 387-402.

20. C.V. Thompson, H.J. Frost, F. Spaepen, Acta Metall. 35 (1987) 887-890.

21. J.W. Cahn, Acta Metall. 10 (1962) 789-798.

22. N.F. Fiore, C.L. Bauer, Pro. Mat. Sci. 13 (1967) 85-88.

23. H.J. Frost, M.F. Ashby, Deformation Mechanism Maps, Pergamon Press, 1982.

24. R.D. Doherty, J.A. Szpunar, Acta Mater. 32 (1984) 1789-1798.

25. E.A. Brandes, G.B. Brook, Smithells Metals Reference Book, 1988.

Chapter 8170

Appendix: The precipitation volume fraction

Assuming a system containing N0 mole atoms, Np in the precipitate, Nm in the matrix,the molar (atom) volume of the precipitate and the matrix is pV , mV , respectively,

pM and mM are the molar atom weight of precipitate and matrix, respectively.

The volume fraction of the precipitate is expressed as:

(1 )p p p p p p

p

m m p p m m p m

N V N V x Vf

N V N V N V x V (1)

0/p px N N is the mole fraction of the precipitate.

We use c and c to represent the Mn concentration in the precipitate and the matrix at

annealing time t for a given temperature T, 0c is the average Mn concentration of the

alloy before precipitation occurs. The weight fraction of precipitate is pW , then

0(1 )p pc W c W c (2)

From (2) we can derive

0p

c cW

c c (3)

The relationship between the mole fraction and the weight fraction is:

1/p p p

p

p p m

W W Wx

M M M (4)

Substitute (3), (4) into (1) we got:

0

0

p mp

m p

V M c cf

V M c c (5)

/p mV V relates to the lattice parameters. The lattice parameter of the precipitate

Al12Mn3Si is ap=12.65 Å, the number of atoms in the unit cell is 138. The lattice

parameter of the precipitate Al6Mn is a=6.5 Å, b=7.55 Å, c=8.87 Å, the number of

atoms in the unit cell is 28. The lattice parameter of pure aluminum is aal=4.0496 Å.


The volume fraction of the precipitate is expressed as:

0

0

p

c cf

c c (6)

where 0.738 for Mn3SiAl12, 0.816 for Mn Al6.

Chapter 8172

S.P. Chen

Summary

Microstructural modeling of thermomechanical processing is well established as a

valuable procedure for optimizing processing conditions in the steel industry, while the

similar work for aluminum industry is still an opening problem. Recovery,

recrystallization and grain growth are core elements of this processing. This thesis

concentrates on the recovery and recrystallization behavior of aluminum alloys with a

focal point on the static softening kinetics after deformation at elevated temperatures.

The aim of the project is to investigate recovery and recrystallization processes in

aluminum alloys as a function of deformation condition, thermal history, composition

and precipitate content, and therefore to develop a validated physical model to predict

the recrystallization kinetics as a result of such processes. We performed experiment

and built models to achieve this objective.

After a general introduction, in Chapter 2 the quantification of the recrystallization

behavior in AA1050 using different techniques, optical microscopy, Electron Back

Scattering Diffraction (EBSD) and microhardness is described. Based on this study, a

new methodology for the determination of the recrystallized volume fraction from

anodically etched aluminum alloys using optical microscopy is proposed. The method

involves the creation of a composite image from multiple micrographs taken at a series

of orientations. The multiple orientation image method is shown to consistently yield a

recrystallized volume fraction which is significantly higher than that determined from a

single image while multiple orientation imaging and Orientation Imaging Microscopy

(OIM) results are found to be in good agreement. Furthermore it is shown that, after the

subtraction of the effect of concurrent recovery using the rule of mixtures,

microhardness indentation can also be used to determine the recrystallized volume

fraction. The multiple orientation image method in combination with microhardness test

is employed throughout this work to determine the recrystallization kinetics.

In Chapter 3, a physical model to predict the recrystallization kinetics of single phase

polycrystalline metals, based on a single grain representation of deformed

microstructure (characterized by a mean subgrain size and mean misorientation of

subgrain boundaries), is presented. The model takes into account the grain geometry,

the position and the density of the nucleation sites. The selected geometry is a regular

Summary174

tetrakaidecahedron, combining topological features of a random Voronoi distribution

characteristic for polycrystalline material with the advantages of a single-grain

calculation. The model employs empirically determined relationships from existing

literature to describe the deformed microstructure and in so doing, enables the

prediction of the recrystallization behavior when only the deformation strain and the

recrystallization temperature are known. The boundary mobility and the driving force as

well as the nucleation density are related to the true plastic strain of deformation

through the microstructure. The model also describes the effects of concurrent recovery

on the overall recrystallization kinetics. The model predictions show that the grain

geometry change during deformation has a large effect on the recrystallization kinetics.

Chapter 3 lays the foundation for the kinetic model of the single grain approach. It is

further modified to incorporate the textural components and then applied to predict the

recrystallization kinetics in the AA1050 after cold deformation in Chapter 4. After

introducing the deformation inhomogenity from grain to grain due to difference in

Taylor factor, the single grain approach could predict the JMAK exponent of the

recrystallization kinetics correctly. The simulation shows that the JMAK exponent

depends on the grain geometry, nucleation site density, the initial grain size prior to

deformation and the main textural components after deformation. The effect of grain

size on recrystallization kinetics is predicted to depend on the amount of the prior strain

applied. This prediction is well supported by literature data.

In order to explain the difference between the laboratory simulation results and real

industrial processing, in Chapter 5 FEM is applied to calculate and compare the strain,

strain rate and stress distributions during plane strain compression in Gleeble 3500

stimulator and in the industrial hot rolling operation. It is shown that the evolution of

the equivalent strain rate and temperature during hot rolling operation and in constant

nominal strain rate PSC testing is quite different. In the hot rolling operation, the

highest strain rate distribution is concentrated just beneath the surface of the slab around

the entry point. The strain rate in the center of the rolling slab experiences a peak

variation, i.e. first increases and then decreases as the slab passes the roll gap. The

equivalent stress, which is a combined effect of the strain rate, strain and the

temperature, reaches a maximum values at about one third to two third of the arc contact

length from the entry, the position of the peak corresponding to that of the equivalent

strain rate. While in the PSC experiment the distribution of the equivalent strain, strain

rate and stress depend on the ratio of the tool width to the thickness of the sample in the

compression direction and the friction at the surfaces between the tool and the

specimen. In Chapter 6, the actual distribution of the strain, strain rate and stress in the

PSC samples, as determined by using FEM simulation were used as input parameters

for the recrystallization kinetic model.

Summary 175

The physical model based on the single grain approach is refined and extended to apply

to the prediction of the recrystallization kinetics of a single-phase metal following hot

deformation in Chapter 6. The model accounts properly for the effect of the concurrent

recovery and textural components in the deformed microstructure on the

recrystallization kinetics. As a physical model, all the input parameters are related to the

microstructure and can be determined from experiment only leaving the pre-exponent

factor of the mobility as an adjustable parameter. Experimental work is conducted on

the recovery and recrystallization kinetics in AA1050 following plastic deformation at

elevated temperatures as encountered during break down rolling to extract the physical

input parameters and to validate the model. The Plane strain compression (PSC) test is

used to simulate the hot rolling deformation. The predictions are in good agreement

with the experimental results.

As for most of industrial alloys such as AA3003, decomposition, recovery and

recrystallization take place in the same temperature range. Therefore, the interaction

between precipitation and recrystallization has profound implications for the control of

the microstructure. When recrystallization starts earlier than precipitation, the

recrystallization kinetics is fast and the finial recrystallized grain size is usually small.

However when precipitation starts earlier than recrystallization, recrystallization process

is sluggish and coarse-grained finial structure is due to obtain. In Chapter 7, nucleation

kinetics of the precipitation of MnAl6 and MnAl12 in the aluminum alloy AA3003 alloy

have been investigated experimentally and theoretically. The results show that cold

rolling enhances the rate of precipitation and this effect increases as the annealing

temperature decreases. Micro-segregation of the Mn solute atom at the dislocation

network during cold deformation is found to have a significant effect on the nucleation

kinetics of the precipitation of MnAl6 and MnAl12 in AA3003 in addition to the effect of

dislocations, which increase the nucleation site density and reduce the nucleation

barrier. A model to predict the start times of the precipitation during isothermal holding

is constructed by considering the effects of dislocations and recovery as well as micro-

segregation of Mn on the nucleation kinetics of precipitates.

In Chapter 8, experiments on AA3003 are described which were performed to study the

effect of the hot deformation parameters on the precipitation and recrystallization

behavior. It is shown that the solute content of Mn in the matrix has a large effect on the

deformation, decomposition and softening behavior of the alloy. The decomposition

process retards the softening processes and the later, in turn, reduce the rate for the

formation of the precipitates. Dynamic precipitation during concurrent deformation

depends on the strain rate and deformation temperature. The static precipitation during

annealing after deformation is enhanced by deformation but weakened by recovery and

Summary176

recrystallization. The softening kinetics is either fast or sluggish depending on the pre-

heat treatments, deformation conditions and the annealing temperature. When

recrystallization is slow the contribution of the recovery to the softening can be as high

as 70%. When recrystallization is fast recovery is responsible for only ~25% of the

softening. Recrystallization kinetics is not simply proportional to the value of the Zener-

Hollomon parameter. Particle stimulated nucleation (PSN) is believed to be an

important nucleation mechanism in the cold deformed samples annealed at higher

temperatures (>500°C). However, nucleation by subgrain growth and strain induced

boundary migration (SIBM) are the dominant nucleation mechanisms in the hot

deformed samples annealed in the temperature range 450-540°C.

In this thesis it has been shown that the use of the single grain approach provides a new

insight into the recrystallization kinetics in a single phase metals after hot or cold

deformation. When combined with a precipitation model it has the potential to predict

the recrystallization kinetics in supersaturated alloys. The models form a strong

foundation for larger scale modeling of the thermomechanical behavior of industrial

aluminum alloys. However, in order to use the model for quantitative predictions, more

experimental data on the grain boundary mobility are required.

S.P. Chen

Samenvatting

Het modelleren van de microstructuurontwikkeling tijdens thermomechanische

processen wordt erkend als een waardevolle procedure voor het optimaliseren van de

procesparameters in de staalindustrie, terwijl dergelijke modelvorming voor aluminium

momenteel een belangrijk onderwerp van onderzoek is. Herstel, rekristallisatie en

korrelgroei zijn de voornaamste elementen tijdens het proces van aluminiumproductie.

In dit proefschrift wordt onderzoek beschreven dat is gericht op het herstel- en

rekristallisatiegedrag van aluminiumlegeringen, met bijzondere aandacht voor de

kinetiek van statisch herstel en rekristallisatie na deformatie op hoge temperaturen.

Het doel van het onderzoek is het bestuderen van herstel- en rekristallisatieprocessen in

aluminiumlegeringen als functie van de deformatiecondities, de thermische

geschiedenis, de chemische samenstelling en de aanwezigheid van precipitaten, hetgeen

beoogd wordt te leiden tot een fysisch model dat de rekristallisatiekinetiek voorspelt

tijdens thermomechanische behandeling. In het onderzoek zijn zowel experimenten

uitgevoerd als modellen ontwikkeld om dit doel te bereiken.

Na een algemene inleiding wordt in hoofdstuk 2 een quantitatieve studie beschreven van

het rekristallisatiegedrag in AA1050, met behulp van verschillende experimentele

technieken: optische microscopie, Electron Back Scattering Diffraction (EBSD) en

microhardheidsmetingen. Gebaseerd op dit onderzoek wordt een nieuwe methodologie

voorgesteld voor de bepaling van de volumefractie gerekristalliseerd materiaal met

behulp van optische microscopie aan anodisch geëtste aluminiumlegeringen. De

methode maakt gebruik van een samengesteld beeld, verkregen uit verschillende

microstructuurfoto’s genomen bij een reeks van oriëntaties. Deze multiple orientation

imaging methode resulteert consistent in een volumefractie gerekristalliseerd materiaal

die significant hoger is dan de volumefractie bepaald aan de hand van een enkele

opname. De resultaten van de multiple orientation imaging methode en de Orientation

Imaging Microscopy (OIM) vertonen een goede overeenkomst. Bovendien wordt

aangetoond dat ook microhardheidsmetingen, gebruik makend van een mengregel en na

aftrek van de effecten van gelijktijdig herstel, ook gebruikt kunnen worden om de

fractie gerekristalliseerd materiaal te bepalen. De multiple orientation imaging methode

wordt, in combinatie met microhardheidsmetingen, in de hele dissertatie gebruikt om de

rekristallisatiekinetiek te bepalen.

Samenvatting178

In hoofdstuk 3 wordt een fysisch model gepresenteerd, dat de rekristallisatiekinetiek

voor éénfasig polykristallijn materiaal voorspelt. Het model is gebaseerd op een

weergave van de gedeformeerde microstructuur door een enkele korrel, gekarakteriseerd

door de gemiddelde afmeting van de subkorrels en de gemiddelde misoriëntatie over

een subkorrelgrens. Het model neemt de korrelgeometrie en de kiemposities en -

dichtheid in rekening. De gekozen geometrie is een regelmatig tetrakaidecahedron,

hetgeen de topologische eigenschappen van een willekeurige Voronoi distributie voor

polykristallijn materiaal combineert met de voordelen van berekeningen aan een enkele

korrel. Het model gebruikt empirisch bepaalde verbanden uit de literatuur om de

gedeformeerde microstructuur te beschrijven, en is zo in staat om het

rekristallisatiegedrag te voorspellen op basis van slechts de plastische rek en de

rekristallisatietemperatuur. De grensvlakmobiliteit, de drijvende kracht en de

kiemdichtheid zijn gerelateerd aan de ware plastische rek via de microstructuur. Het

model beschrijft ook de effecten van gelijktijdig herstel op de rekristallisatiekinetiek.

Het model laat zien dat de verandering van de korrelgeometrie tijdens de deformatie een

grote invloed heeft op de rekristallisatiekinetiek.

Hoofdstuk 3 legt de basis voor het kinetiekmodel gebaseerd op de single-grain

benadering, dat toegepast wordt in de volgende hoofdstukken. Het model wordt verder

ontwikkeld door textuurcomponenten in rekening te brengen, en vervolgens toegepast in

hoofdstuk 4 voor de voorspelling van de rekristallisatiekinetiek in AA1050 na koude

deformatie. Variaties in de deformatie per korrel worden in rekening gebracht via

verschillen in de Taylor-factor, waarna de single-grain benadering een goede

voorspelling levert van de exponent in de Johnson-Mehl-Avrami-Kolmogorov (JMAK)

beschrijving van de rekristallisatiekinetiek. De simulaties laten zien dat de JMAK-

exponent afhangt van de korrelgeometrie, de dichtheid van kiemplaatsen, de

oorspronkelijke korrelgrootte (vóór deformatie), en de belangrijkste textuur

componenten na deformatie. Er wordt door het model voorspeld dat de invloed van de

korrelgrootte op de rekristallisatiekinetiek afhangt van de hoeveelheid toegepaste

deformatie. Deze voorspelling is in overeenstemming met gegevens in de literatuur.

Om de verschillen tussen laboratoriumexperimenten en industriële procescondities te

bestuderen wordt in hoofdstuk 5 Eindige-Elementen Modellering (Finite-Element

Modelling, FEM) toegepast om de rek, de reksnelheid en de spanningsverdelingen

gedurende plane-strain compressie (PSC) uit te rekenen en te vergelijken in een Gleeble

3500 simulator en onder de condities van het industriële warmwalsproces. Er wordt

aangetoond dat de ontwikkeling van de equivalente reksnelheid en de temperatuur heel

verschillend is gedurende het warmwalsproces en gedurende PSC tests bij nominaal

constante reksnelheid. Gedurende het warmwalsproces treedt de maximale reksnelheid

Samenvatting 179

op vlak onder het oppervlak van de plaat, dicht bij het punt waar de plaat de wals ingaat.

De maximale variatie in reksnelheid wordt gevonden in het midden van de plaat: eerst

neemt de reksnelheid toe; later vindt een afname plaats als de plaat de wals passeert. De

equivalente spanning, die afhangt van een combinatie van rek, reksnelheid en

temperatuur, bereikt een maximum op ongeveer een derde tot twee-derde van het

walscontact vanaf het beginpunt, waarbij het maximum overeenkomt met het maximum

in equivalente reksnelheid. In een PSC experiment hangt de verdeling van de

equivalente rek, reksnelheid en spanning af van de verhouding van de walshoogte tot de

dikte van de plaat, en van de wrijving tussen het instrument en het preparaat. In

hoofdstuk 6 wordt de eigenlijke verdeling van rek, reksnelheid en spanning in PSC

preparaten, bepaald met FEM simulaties, gebruikt als invoergegevens voor het model

voor de rekristallisatiekinetiek.

Het fysische model gebaseerd op de single-grain benadering is verder verfijnd en

ontwikkeld in de toepassing voor de rekristallisatiekinetiek voor een éénfasig metaal na

warmdeformatie in hoofdstuk 6. Het model geeft een goede representatie van de

effecten van gelijktijdig herstel en de textuurcomponenten in de gedeformeerde

microstructuur op de rekristallisatiekinetiek. Omdat het model een fysische basis heeft

kunnen alle invoergegevens gerelateerd worden aan de microstructuur, en verkregen

worden uit experimenten, waarbij alleen de pre-exponentiële factor van de mobiliteit

een aan te passen parameter is. Om de fysische invoergegevens te bepalen en het model

te valideren, zijn experimenten uitgevoerd aan de rekristallisatiekinetiek in AA1050 na

deformatie op hoge temperatuur, zoals toegepast wordt gedurende break-down walsen.

De plane-strain compressie test wordt gebruikt om warmwalsen te simuleren. De

modelvoorspellingen zijn goed in overeenstemming met de experimentele resultaten.

Voor de meeste industriële legeringen, zoals AA3003, vinden decompositie, herstel en

rekristallisatie plaats in hetzelfde temperatuurstraject. De interactie tussen precipitatie

en rekristallisatie heeft daarom vergaande consequenties voor de controle van de

microstructuur. Als rekristallisatie eerder start dan precipitatie is de

rekristallisatiekinetiek snel en vormt zich een fijne gerekristalliseerde microstructuur.

Als daarentegen de precipitatie eerder start dan de rekristallisatie is het

rekristallisatieproces relatief traag en wordt een vrij grove microstructuur verkregen. In

hoofdstuk 7 wordt de nucleatiekinetiek van de precipitatie van MnAl6 en MnAl12 in de

aluminiumlegering AA3003 experimenteel en theoretisch bestudeerd. De resultaten

laten zien dat koudwalsen de precipitatiesnelheid verhoogt, welk effect toeneemt bij

afnemende gloeitemperatuur. Microsegregatie van Mn in oplossing op het

dislokatienetwerk gedurende koude deformatie heeft een significante invloed op de

nucleatiekinetiek voor de precipitatie van MnAl6 en MnAl12 in AA3003. Daarbij komt

nog het effect van de dislokaties, die de dichtheid van kiemplaatsen doen toenemen en

Samenvatting180

de barrière voor kiemvorming doen afnemen. Een model is opgesteld dat de starttijden

voor precipitatie tijdens isotherme warmtebehandeling voorspelt, gebaseerd op de

invloeden van zowel dislokaties en herstel als microsegregatie van Mn op de

nucleatiekinetiek.

In hoofdstuk 8 worden experimenten aan AA3003 beschreven, die uitgevoerd werden

om de invloed van de warmvervormingsparameters op het precipitatie- en

rekristallisatiegedrag te bestuderen. Er wordt aangetoond dat het gehalte van Mn in

oplossing in de matrix een grote invloed heeft op het gedrag van de legering in

deformatie, decompositie, en herstel en rekristallisatie. Het decompositieproces

vertraagt het herstel en rekristallisatie, die op hun beurt de vormingssnelheid van

precipitaten doen afnemen. Het verloop van dynamische precipitatie gedurende

gelijktijdige deformatie hangt af van de reksnelheid en de deformatietemperatuur.

Statische precipitatie, gedurende een gloeibehandeling na de deformatie, neemt toe door

de deformatie, maar neemt af als gevolg van herstel en rekristallisatie. De kinetiek van

herstel en rekristallisatie kan zowel snel als traag zijn, afhankelijk van de voorafgaande

warmtebehandelingen, de vervormingscondities en de gloeitemperatuur. In het geval

van trage rekristallisatie kan het aandeel van herstel wel 70% bedragen; in het geval van

trage rekristallisatie slechts 25%. De rekristallisatiekinetiek is niet simpelweg evenredig

met de Zener-Hollomon parameter. Door deeltjes gestimuleerde kiemvorming (particle-

stimulated nucleation, PSN) is waarschijnlijk een belangrijk kiemvormingsmechanisme

in koudvervormde preparaten die op een hoge temperatuur (hoger dan 500°C) gegloeid

worden. In het temperatuurgebied van 450 tot 540°C zijn kiemvorming door groei van

subkorrels en kiemvorming door rek geïnduceerde grensvlakverplaatsing (strain-

induced boundary migration, SIBM) de belangrijkste mechanismen.

In deze dissertatie wordt aangetoond dat het gebruik van een fysisch model op basis van

de single-grain benadering nieuw inzicht levert in de rekristallisatiekinetiek in éénfasig

metaal na koude of warme deformatie. Als dit model gecombineerd wordt met een

precipitatiemodel ontstaat de mogelijkheid om de rekristallisatiekinetiek in

oververzadigde legeringen te voorspellen. De modellen vormen een sterke basis voor de

modellering van het thermomechanische gedrag van industriële aluminiumlegeringen op

een grotere schaal. Voor het gebruik van de modellen voor quantitatieve voorspellingen

zijn echter meer experimentele gegevens met betrekking tot de grensvlakmobiliteit

nodig.

181

Publications

S.P. Chen, Z.J. Lok and S. van der Zwaag, On the precipitation and recrystallizationbehavior in an AA3003 following hot deformation, in Proceedings of int. conf. onThermomechanical processing: Mechanics, Microstructure and Control, 23-26 June2002, The university of Sheffield, England, 171-180.

S.P. Chen, N.C.W. Kuijpers and S. van der Zwaag, Effect of microsegregation anddislocations on the nucleation kinetics of precipitation in aluminum alloy AA3003,Mater. Sci. Eng. A341, 2003, 296-306.

S.P. Chen, D.N. Hanlon, S. van der Zwaag, Y.T. Pei, and J.TH.M. de Hosson,Quantification of the recrystallization behavior in Al-alloy AA1050, J. Mat. Sci., 37,2002, 989-995.

S.P. Chen, I. Todd, and S. van der Zwaag, Modeling the kinetics of grain boundarynucleated recrystallization processes, Metall. Mater. Trans. 33A, March, 2002, 529-539.

S.P. Chen and S. van der Zwaag, EBSD study on the recrystallization behavior inAA1050, in 'Nederlandse Vereniging Voor Microscopie, Jaarboek 2001', 2001, 40-41.

S.P. Chen, T. Zuidwijk, and S. van der Zwaag, Recovery and recrystallization kineticsin AA 1050 after simulated break down rolling, in 'Recrystallization and Grain Growth,Proceedings of the First Joint International Conference', G. Gottstein and D.A. Molodov(Eds), Springer, Volume 2, 2001, 821-830.

S.P. Chen, Y. Van Leeuwen, S.B. Davenport, D.N. Hanlon, and S. van der Zwaag, in'Microstructures, Mechanical Properties and Processes', The effect of grain geometry onthe kinetics of recrystallization processes, Y. Brechet (Eds), Euromat1999, 3, 1999, 96-101.

S.P. Chen, M.S. Vossenberg, F.J. Vermolen, J. van de Langkruis, and S. van der Zwaag,Dissolution of particles in an Al-Mg-Si alloy during DSC runs, Mater. Sci. Eng. A, 272,1999, 250-256.

S.P. Chen, K.M. Mussert, and S. van der Zwaag, Precipitation Kinetics in Al6061 andan Al6061-alumina Particle Composite, J. Mat. Sci., 33, 1998, 4477-4483.

182

S.P. Chen and S. van der Zwaag, A comparative FE study of hot rolling and PSC testingof AA1050, accepted for publication in J. Test. and Eval., 2003.

S.P. Chen and S. van der Zwaag, A single grain approach applied to modelingrecrystallization kinetics in single-phase metal, submitted to Metall. Mater. Trans. A,2002.

S.P. Chen, A. Miroux and S. van der Zwaag, Modeling recrystallization kinetics inAA1050 following simulated break down rolling, to be submitted to Acta Mater. 2003.

S.P. Chen, Z.J. Lok, A. Miroux and S. van der Zwaag, The effect of dispersoids andconcurrent precipitation on recrystallization kinetics in hot deformed AA3003, inpreparation.

183

Acknowledgements

The research described in this thesis was carried out at the Laboratory for Materials

Science, Delft University of Technology. The contributions of many people to the

development of this thesis are greatly acknowledged.

I am extremely grateful to my promoter Prof. Dr. ir. S. van der Zwaag, it was his kind

acceptance that gave me an opportunity to pursue my Ph. D degree at the Delft

University of Technology. His inspirational guidance, continuing support and

enthusiastic encouragement were tremendous for me. I also appreciated very much for

kind care that he gave to my family.

I wish to express my gratitude to Dr. Yvonne van Leeuwen for her phase transformation

model. The recrystallization model in this thesis is developed from the phase

transformation model. I am grateful to Dr. J. Wang and N.C.W. Kuijpers for beneficial

discussions on some interesting topics, to Dr. J. Sietsma, R. Lok and S. Kruijver for

translation of the summary and propositions from English into Dutch, to Dr. A. Miroux

for critical reading of the manuscript.

I would like to acknowledge all technicians in the Group of Microstructure Control of

Metals (MCM), particularly, T. de Haan, N. Geerlofs, T. Zuidwijk and E. Peekstok.

Their good cooperation was indispensable for the success of the project.

I would like to thank all the present and former colleagues in MCM Group for their help

and their discussions about my research and various other subjects. Also thanks go to

Drs. T. Hurd, M.R. van der Winden, J. van de Langkruis and C. Liu from the Aluminum

group, Corus R&D for their invaluable discussions and suggestions in the project.

Further I would like to thank all the members in the Aluminium Cluster of NIMR for

the beneficial questions and discussions during and after cluster meetings.

I would like to take this opportunity to acknowledge The Netherlands Institute for Metal

Research for offering me this rewarding position.

Special thanks go to my parents for their enlightening: effort, endeavor and endurance.

Last, but surely not least, I am greatly indebted to my wife, Ying, to my daughters, Sijia

and Silu, for their love, patience, understanding, and support in every aspect.

185

About the author

Shangping Chen was born on August 25, 1962 in Shaanxi Province, P. R. China. He

obtained a BSc degree in 1983 and a MSc degree in 1987 in Materials Science and

Engineering in Xi’an Jiaotong university, Shaanxi, China. He was employed as a

research assistant in 1988, and a lecturer in 1990, and an associate professor in 1996, at

Department of Mechanical Engineering, Shaanxi Institute of Technology. In 1997, he

was awarded to work as a Visiting Scientist for one year in Laboratory for Materials

Science, Delft University of Technology, the Netherlands, within a scientific

cooperation project between the Royal Dutch academy of sciences and the Chinese

academy of sciences. Since 1st December 1998, he has been studying for a PhD degree

in the Netherlands Institute for Metal Research (NIMR) under the guidance of Prof.

dr.ir. S.Van der Zwaag. The main research work during these four years is documented

in the thesis entitled “Recovery and Recrystallization kinetics in AA1050 and AA3003

Aluminium Alloys”.

Recovery and Recrystallization Kinetics in AA1050 and AA3003 Aluminium Alloys

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