Modeling the Microstructural Evolution during Hot Deformation of Microalloyed Steels

Modeling the Microstructural Evolution during Hot Deformation of

Microalloyed Steels

Linda Bäcke

Doctoral thesis

Royal Institute of Technology

School of Industrial Engineering and Management Material Science and Engineering

Division of Mechanical Metallurgy

SE-100 44 Stockholm, Sweden

Modeling the Microstructural Evolution during Hot Deformation of Microalloyed Steels LINDA BÄCKE

ISRN KTH/MSE--09/12--SE+MEK/AVH ISBN 978-91-7415-267-8

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan framlägges till offentlig granskning för avläggande av Teknologie doktorsexamen måndagen den 20 april 2009. Fakultetsopponent är Dr. Eric J. Palmiere vid The University of Sheffield, England.

© Linda Bäcke, 2009

Tryckt av: Universitetsservice US-AB, Stockholm, Sverige, 2009

Modeling the Microstructural Evolution during Hot Deformation of Microalloyed Steels LINDA BÄCKE

Abstract

Recrystallization kinetics, during and after hot deformation, has been investigated for decades. From these investigations several equations have been derived for describing it. The equations are often empirical or semi-empirical, i.e. they are derived for certain steel grades and are consequently only applicable to steel grades similar to these. To be able to describe the recrystallization kinetics for a variety of steel grades, models which are more physically- based are necessary.

This thesis contains the development of a physically-based model describing the microstructural evolution during hot deformation of microalloyed steels. The work is mainly focused on the recrystallization kinetics. During hot rolling, the repeated deformation and recrystallization provides progressively refined recrystallized grains. Also, recrystallization enables the material to be deformed more easily and knowledge of the recrystallization kinetics is important in order to predict the required roll forces. Hot strip rolling is generally conducted in a reversing roughing mill followed by a continuous finishing mill. During rolling in the roughing mill the temperature is high and complete recrystallization should occur between passes. In the finishing mill the temperature is lower which means slower recrystallization kinetics and partial or no recrystallization often occurs. If microalloying elements such as Nb, Ti or V are present, the recrystallization can be further retarded by either solute drag or particle pinning. When recrystallization is completely retarded and strain is accumulated between passes, the austenite grains will be severely deformed, i.e. pancaking occurs. Pancaking of the grains provides larger amount of nucleation sites for ferrite grains upon transformation and hence a finer ferrite grain size is achieved.

In this work a physically-based model has been used to describe the microstructural evolution of austenite. The model is built-up by several sub-models describing dislocation density evolution, recrystallization, grain growth and precipitation. It is based on dislocation density theory where the generated dislocations during deformation provide the driving force for recrystallization. In the model, subgrains act as nuclei for recrystallization and the condition for recrystallization to start is that the subgrains reach a critical size and configuration. The retarding effect due to elements in solution and as precipitated particles is accounted for in the model.

To verify and validate the model axisymmetric compression tests combined with relaxation were modeled and the results were compared with experimental data. The precipitation sub- model was verified by the use of literature data. In addition, rolling in the hot strip mill was modeled using process data from the hot strip mill at SSAB Strip Products Division. The materials investigated were plain C-Mn steels and Nb microalloyed steels. The results from the model show good agreement with measured data.

Keywords: modeling, austenite, microalloyed steels, hot deformation, microstructure

evolution, static recrystallization, dynamic recrystallization, metadynamic recrystallization.

Preface

This work was carried out at Dalarna University and at SSAB Strip Products in Borlänge and was one of the projects in the Post Graduate School in Metal Forming at Dalarna University, Borlänge. I wish to thank the sponsors of this project, the Swedish Steel Producer’s Association (Jernkontoret), The Swedish knowledge foundation, SSAB and Dalarna University.

I am very grateful for the discussions and contributions from:

My assistant supervisor, Professor Göran Engberg at Dalarna University for supervising me, for encouraging me in my work, for sharing his great experience in microstructure evolution with me and for lots of stimulating discussions.

My colleague, Ulrika Borggren for co-authorship and fruitful discussions.

My supervisor, Professor Stefan Jonsson, at KTH, for his valuable feedback.

All my colleagues at both Dalarna University and at SSAB Strip Products for general support and for stimulating discussions.

Finally, I want to thank my family for believing in me and especially my husband Fredrik for his endless support and love.

Linda Bäcke

Borlänge, March 2009

List of papers and author contribution in each paper

During the work with this thesis, the author changed her surname from Lissel to Bäcke. This thesis consists of an introduction and the following seven papers, referred to by roman numerals:

I. Prediction of microstructural behavior during hot rolling L. Lissel and G. Engberg

Conference proceeding of the 8

International Conference on Technology of Plasticity (ICTP), Verona, Italy 2005.

II. A physically based microstructure model for predicting the microstructural evolution of a C-Mn steel during and after hot deformation

G. Engberg and L. Lissel

Published in Steel Research International, Vol. 79 (2008), pp. 47-58

III. Prediction of the microstructural evolution during hot strip rolling of Nb microalloyed steels

L. Lissel and G. Engberg

Conference proceeding of the 3

International Conference on Recrystallization and grain growth, ReX & GG III; Jeju Island; Korea. Published in Materials Science Forum, Vol. 558-559 (2007), pp. 1127-1132

IV. Modeling precipitation and its effect on recrystallization during hot strip rolling of niobium steels

L. Lissel, G. Engberg and U. Borggren

Conference proceeding of the 3

International Conference on TMP; Padua; Italy, 2008

V. Physical based model for predicting the microstructural evolution at hot rolling L. Bäcke and G. Engberg

Invited paper to the International Conference MEFORM 2009, Freiberg, Germany VI. Modeling the effect of solute drag on recovery and recrystallization during hot

deformation of Nb microalloyed steels

L. Bäcke

Manuscript submitted to ISIJ international

VII. Modeling the kinetics of strain induced precipitation and its effect on recovery and recrystallization in Nb microalloyed steels

L. Bäcke, U. Borggren and G. Engberg

Manuscript to be submitted to an international journal

Author contribution Paper I

Evaluation of experimental results from literature, collecting and evaluating relevant process data, model parameter-fitting and calculations and writing in close co-operation with the co- author

Paper II

Determining experimental setup along with evaluation of experimental data, model parameter-fitting and calculations and writing in close co-operation with the co-author

Paper III

Collecting and evaluating relevant process data, model development, model parameter-fitting and calculations and writing in close co-operation with the co-author

Paper IV

Collecting and evaluating relevant process data, model development, model parameter-fitting and calculations and writing in close co-operation with the co-authors

Paper V

Evaluating results and writing in close co-operation with the co-author Paper VI

Single author Paper VII

Evaluation of experimental results from literature, model development, model parameter-

fitting and calculations and writing in close co-operation with the co-authors

Table of contents

1 Introduction ___________________________________________________________ 1 1.1 Hot strip rolling _________________________________________________________ 1 1.2 Thermomechanical controlled rolling _______________________________________ 4 1.3 Background to the work __________________________________________________ 7 1.4 Aim of the work _________________________________________________________ 8 2 Microstructure evolution during hot deformation _____________________________ 9

2.1 The grain structure _____________________________________________________ 10 2.2 Dislocation generation and recovery _______________________________________ 11 2.3 Recrystallization and grain growth ________________________________________ 12 2.4 The effect of microalloying elements on recovery and recrystallization___________ 16 3 Modeling recrystallization _______________________________________________ 21

3.1 Physically-based model __________________________________________________ 21

3.2 Empirical models _______________________________________________________ 25

4 Experimental procedure_________________________________________________ 27

5 Summary of appended papers ____________________________________________ 31

5.1 Paper I________________________________________________________________ 31

5.2 Paper II _______________________________________________________________ 32

5.3 Paper III ______________________________________________________________ 34

5.4 Paper IV ______________________________________________________________ 36

5.5 Paper V _______________________________________________________________ 38

5.6 Paper VI ______________________________________________________________ 38

5.7 Paper VII _____________________________________________________________ 40

6 Conclusions and future work_____________________________________________ 43

7 References____________________________________________________________ 45

APPENDED PAPERS

1 Introduction

The microstructural evolution during hot deformation of microalloyed steels involves accumulation, annihilation and rearrangement of dislocations, recrystallization and grain growth. Microalloying elements, such as Nb, Ti and V can form carbides, nitrides and/or carbonitrides affecting the course of the microstructural evolution. For hot deformation, the microstructural evolution that occurs in the material is dependent on the size of the reductions, the strain rate, the temperature and the length of the holding times between reductions. For hot strip rolling of steel it is the desired final mechanical and geometrical properties of the strip that determines the so called rolling schedule, which is the setup for the rolling process, i.e.

the amount of reduction, the rolling velocity and the temperature.

High strength low alloy (HSLA) steels are steels microalloyed with Nb, Ti and/or V.

Microalloying elements added give a contribution to both grain size refinement and precipitation hardening. Before hot rolling of microalloyed steels, the steels are soaked at high temperature to dissolve the particles to get enough microalloying elements in solution. At low temperatures the microalloying elements start to precipitate causing a retardation of recrystallization. For conventional controlled rolling the finishing passes are carried out below the temperature where the material recrystallizes. When no recrystallization occurs between passes, the strain is accumulated and the austenite grains become severely deformed. This provides for more preferential nucleation sites for ferrite during subsequent transformation and a fine ferrite grain size is obtained.

1.1 Hot strip rolling

At SSAB Strip Products Division in Borlänge, Sweden, the hot rolled steel strips produced are of a wide range of strength, from mild steels to ultra high strength steels. The wide range in strength is mainly achieved by adding different amounts of alloying elements. High strength is mainly obtained by microalloying with Ti and/or Nb. In Figure 1 a schematic picture of the hot strip mill at SSAB Strip Products Division is shown.

In the hot strip mill slabs with height of ~220 mm, length of 5500 - 11000 mm and width of 650 - 1650 mm are reduced to thin strips with thickness 1.6 - 16 mm. The hot strip mill consists of, from start to finish, reheating furnaces, descaler, reversing roughing mill, coilbox, crop shear, descaler, finishing mill with six stands, run-out table with accelerated cooling and finally down-coilers.

In the reheating furnaces the slabs are heated up to ~1200°C and a fully annealed structure is

obtained. The heating temperature and the time in the furnaces are dependent on the steel

grade and for microalloyed steel they are chosen in order to dissolve the right amount of

particles. During heating, some of the particles formed during the continuous casting of the

slab are dissolved provided that they are small enough and that the temperature is high enough for dissolution. Particles that are stable at higher temperatures than the chosen reheating temperature are of course not dissolved.

Figure 1. The hot strip mill at SSAB Strip Products Division in Borlänge.

When the slab is withdrawn from the furnace, the surface is covered with oxide scale which is removed in the descaler after the furnace. Descaling is also conducted before some of the passes in the roughing mill. In the reversing roughing mill five to seven passes are conducted and the total reduction is ~85%. The inter-pass times in the roughing mill are long, which depends on the fact that the bar is rolled back and forth. After rolling in the roughing mill, the transfer bar is usually transported down to the coilbox where it is coiled, but sometimes it is directly transferred down to the finishing mill. When the transfer bar is coiled in the coilbox the temperature along it gets more evenly distributed.

During rolling in the roughing mill and during transportation, the temperature of the transfer bar decreases and before entering the finishing mill it is ~1000°C. Before entering the finishing mill, the new layer of oxide scale is removed and the irregular head and tail end are generally cut off, leaving an even edge to enter the first pass. In the finishing mill, the bar is reduced to the desired thickness. Since the finishing mill is a continuous process, the inter- pass times are shorter than in the roughing mill. After rolling in the finishing mill, the strip is cooled to the desired temperature at the run-out table and subsequently coiled.

The temperature in the hot strip mill is only measured at a few places in the process: after the roughing mill, after the finishing mill and at the down coiler. It is only possible to measure the temperature at the surface of the strip and the temperature gradients in the thickness direction of the strip therefore have to be calculated. Also, the temperature between the measuring points has to be calculated. An example of the calculated temperature during rolling, using STEELTEMP® [1], is presented in Figure 2. STEELTEMP is a finite-difference program for temperature and heat-transfer analysis during, for example heating and flat rolling.

The temperatures at the surface and at the centre of the workpiece are shown in the figure, as well as the measured surface temperatures after the roughing mill and after the finishing mill.

The figure shows rolling of a 220 mm slab to a 4 mm thick strip. In the first passes there are

large temperature gradients in the thickness direction but these are reduced during rolling and at the end of the finishing mill, there are practically no remaining temperature gradients.

Figure 2. Calculated temperature during rolling in the roughing mill and in the finishing mill.

As mentioned above, after heating in the furnaces the material is fully annealed. The dislocations are arranged in configurations of lower energy and the dislocation density is relatively low. Furthermore, the energy of the material is lowered by grain growth and the grains are quite coarse after the heat treatment.

During deformation, the stored energy of the material is raised by the presence of dislocations and boundaries. Simultaneously, recovery occurs because of the high temperature. During recovery, rearrangement and annihilation of dislocations occurs which lowers the stored energy. The remaining stored energy is the driving force for subsequent recrystallization. In the roughing mill, the temperature is high and the inter-pass times are long, allowing for full recrystallization to occur between passes. In the finishing mill the temperature is lower and the inter-pass times are shorter than in the roughing mill which means that no or partial recrystallization could occur between passes.

Recovery and recrystallization are important processes during hot rolling. They lower the flow stress of the material enabling it to be deformed more easily. After complete recrystallization the energy of the material may be further reduced by normal grain growth.

These softening mechanisms: recovery, recrystallization and grain growth are described more

thoroughly in section 2.

1.2 Thermomechanical controlled rolling

To achieve the desired final mechanical properties of the strip such as a good combination of strength, fracture toughness and weldability, thermomechanical processing (TMP) is utilized [2, 3]. TMP means control of the reheating temperature, the rolling schedule, the cooling rate and the coiling temperature, i.e. control of the entire processing sequence. The purpose of TMP is to obtain optimum ferrite refinement and it is therefore necessary to “maximize the area of austenite grain boundary per unit volume at the onset of phase transformation” [4].

There are different ways of achieving this and the main types of controlled rolling are recrystallization controlled rolling, conventional controlled rolling and dynamic recrystallization controlled rolling [5]. The type of controlled rolling refers to the conditions in the final rolling passes, i.e. in the finishing mill for conventional hot strip rolling. First, the as-reheated microstructure is eliminated in the roughing mill where complete recrystallization occurs between passes.

During hot strip rolling, the material generally stops recrystallizing below a certain temperature. This temperature is often called the recrystallization stop temperature (RST) or the no-recrystallization temperature, T

. The T

denotes the temperature above which static recrystallization occurs between passes [6]. It depends on the deformation, the cooling rate and the inter-pass times in the rolling process. Below this temperature recrystallization is retarded due to strain-induced precipitation of second-phase particles.

Recrystallization is retarded by adding small amounts of certain elements to mild steels (microalloying). High strength low alloy (HSLA) steels is the commonly used name for microalloyed steels. Microalloying generally means the addition of less then 0.1% of elements to mild steel compositions and refers to the addition of Ti, Nb or V [7]. The effect of the concentration of Ti, Nb, V and Al in a 0.07C-1.40Mn-0.25Si (wt.%) steel on the recrystallization stop temperature was determined by Cuddy [8], see Figure 3.

Figure 3. Effect of the initial solute content on the recrystallization stop temperature,

from Cuddy [8].

In Cuddy’s investigation [8] multi-pass deformation was conducted with a total reduction of about 65%. As seen in Figure 3, the most effective raise of the RST is provided by Nb. Also, recrystallization is impeded at higher temperatures with smaller additions of Nb than with Ti, Al and V.

When heating the slabs of Nb microalloyed steels before rolling, the temperature is usually high enough so that Nb is in solution. During rolling, Nb forms nitrides, carbides and/or carbonitrides which effectively retards recovery and recrystallization. The highly work hardened (pancaked) austenite provides numerous nucleation sites for the ferrite during the subsequent transformation which give very fine ferrite grains [9].

Recrystallization controlled rolling means that rolling is conducted above the no- recrystallization temperature, T

, and that full recrystallization is obtained between passes.

The repeated recrystallization of austenite provides progressively refined recrystallized grains.

A schematic time-temperature diagram of recrystallization controlled rolling is shown in Figure 4.

Figure 4. Schematic time-temperature diagram of recrystallization controlled rolling showing the recrystallized microstructure between passes. T

and A

denote the no- recrystallization temperature and the phase transformation temperature, respectively.

From Siciliano et al. [5].

Conventional controlled rolling is illustrated in Figure 5. It is conducted above the phase transformation temperature, A

, and below T

so that the strain is accumulated between passes and the austenite grains get severely deformed, i.e. pancaking of the austenite occur.

The severely deformed austenite grains provide numerous nucleation sites during subsequent transformation.

Dynamic recrystallization controlled rolling, Figure 6, means that dynamic recrystallization,

DRX, is triggered in one or more passes during rolling and this in turn triggers the fast

metadynamic recrystallization, MDRX. DRX is induced either by applying large single

deformations or by accumulation of strain from pass to pass. Siciliano and Jonas [5] state that

one of the benefits with this approach is the intense grain refinement caused by dynamic

recrystallization.

Figure 5. Schematic time-temperature diagram of conventional controlled rolling, showing the deformed microstructure between passes. T

and A

denote the no- recrystallization temperature and the phase transformation temperature, respectively.

From Siciliano et al. [5].

Figure 6. Schematic time-temperature diagram of dynamic recrystallization controlled rolling, showing the very fine recrystallized microstructure after dynamic (DRX) and metadynamic recrystallization (MDRX). T

and A

denote the no-recrystallization temperature and the phase transformation temperature, respectively. From Siciliano et al. [5].

To be able to practice controlled rolling and thereby get the desired final mechanical

properties of the product, knowledge of the microstructural evolution of the austenite during

rolling is essential. In order to design optimized rolling schedules that fit the needs and

constraints of each case, models that describe the microstructural evolution are extremely

useful.

1.3 Background to the work

For several decades the microstructural evolution of austenite during hot deformation of steels has been investigated [4, 5, 8-15]. Over the last few years, more and more attention has been paid to the development of mathematical models that can predict the microstructural evolution during hot rolling [4, 5, 10, 11, 13-18]. The goal is usually to be able to predict the microstructure of the materials and to optimize the different process conditions to obtain the best combination of strength, ductility and weldability in the as-rolled product.

For the hot rolling process, a setup (rolling schedule) determines the temperature and reduction that has to be met in each pass in order to obtain the desired final product. The setup is determined on the basis of the desired final mechanical and geometrical properties of the strip.

With an on-line model predicting the mechanical properties, the hot rolling process could be controlled towards the mechanical properties. If for example a deviation from the predicted target occurs during rolling, a new setup could be determined in order to meet the final mechanical properties. This would require a model with very short calculation times in order to calculate a new setup and adjust the process in time. Therefore, for this purpose, empirical or semi-empirical models with short calculation times are often used. Semi-empirical models refer to models that are derived for certain steel grades and are generally only applicable to these.

With an off-line model, rolling schedules for different steel grades can be optimized. For example, the INTEG hot strip mill model (HSMM) [16] predicts the final mechanical properties of as-rolled products. The microstructural evolution is described by semi-empirical models and one setup of parameters is required for each steel grade. Another model predicting the final mechanical properties is presented by Andorfer et al. [17]. The model is used in an on-line control system, CAQC (Computer Aided Quality Control), in order to calculate a new setup during rolling. The model is used for the seven-stand hot strip mill of voestalpine’s steel plant in Linz, Austria. The description of recrystallization and grain growth is based on semi- empirical equations.

Semi-empirical models (described in section 3.2) give a good description for the steel grades they are derived for. In order to describe the microstructural evolution for a variety of steel grades there is a need for models which are more physically-based. Such a model would provide the possibility to design optimized rolling schedules for both existing and new steel grades.

The model used in this work is based on a physical description of dislocation density evolution. Only mean values of the microstructural variables are considered in the model and this approach was chosen in order to keep the calculation times fairly short. The model describes the microstructural evolution during hot deformation and is based on the work by Siwecki and Engberg [4] and Wang et al. [18]. It is implemented in a toolbox for process simulations [19].

The model is intended to be able to describe the microstructural evolution during the whole

rolling process, from heating the slabs in the furnace to cooling of the strips when coiled, and

also predict the final mechanical properties. This work is concentrated on the deformation part

in hot rolling, i.e. rolling in the roughing mill and in the finishing mill. The following steps in the rolling process, cooling and coiling of the strip, are not considered here. No phase transformation is yet described in the model, only the austenite region is considered.

1.4 Aim of the work

The aim of this thesis is the verification and validation of a newly developed microstructure model. The model is intended to contribute to the understanding of the microstructural evolution during hot deformation. The work is done for SSAB Strip Products Division who produces steel strips of a variety of low-carbon steel grades. The calculations are primarily intended for the hot strip mill process at SSAB Strip Products Division and for SSAB’s products. At the start time of the work, the existing model contained a description of work hardening and recovery [4] and a recrystallization model developed by Wang et al. [18]. A new simplified recrystallization model was under development and the work in this thesis is the further development of the entire model with emphasis on the recrystallization description.

This was done by:

• Simulating hot strip rolling of C-Mn steels (Paper I)

• Verifying and validating the description of recovery and recrystallization of C-Mn steels by simulating stress-relaxation tests (Paper II)

• Simulating hot strip rolling of Nb microalloyed steels by adding a semi-empirical equation for the description of precipitation (Paper III)

• Simulating hot strip rolling of Nb microalloyed steels by adding a physically-based model for the description of precipitation (Paper IV-V)

• Verifying and validating the description of solute drag on recovery and recrystallization in Nb microalloyed steels by simulating stress-relaxation tests (Paper VI)

• Verifying and validating the precipitation model for Nb(C,N) precipitation by

simulating previously reported experimental data (Paper VII)

2 Microstructure evolution during hot deformation

The stored energy due to the accumulated dislocations during deformation is generally lowered by three processes: recovery, recrystallization and grain growth [20]. Recovery is a process where annihilation and rearrangement of the dislocations occurs, see section 2.2.

Recrystallization involves the formation of new dislocation free grains. The grains grow on the expense of the old deformed grains, leaving a new structure with low dislocation density as a result, see section 2.3. Grain growth is the process when the grains coarsen and the grain boundary area is reduced. Recovery and recrystallization can take place during and after deformation and to distinguish them they are called dynamic and static, respectively. If the recrystallization after deformation is preceded by dynamic recrystallization it is called metadynamic [21]. Metadynamic recrystallization is the continuation of the dynamic recrystallization at the end of deformation and the nuclei for recrystallization are thus already present.

The characteristic effects of work hardening, dynamic recovery and dynamic recrystallization on the stress-strain curve are illustrated in Figure 7.

Work Hardening Dynamic Recovery Dynamic Recrystallization

Z High

True stres s Low

True strain

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

RT Q dt

Z d ε exp

Figure 7. Schematic illustration of work hardening, dynamic recovery and dynamic recrystallization during hot deformation where Z denotes the Zener-Hollomon parameter, see text.

The flow curve is dependent on the conditions of the deformation, such as temperature, T, and strain rate, d / ε dt , which can be expressed by the Zener-Hollomon parameter [3]

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

RT Q dt

Z d ε exp

(1)

where Q

is the activation energy for deformation and R is the gas constant, 8.314 Jmol

K

.

The model used in this work describes the microstructural evolution through several sub- models, which are outlined in Figure 8.

Dislocation generation

Section 2.2

- dislocation generation due to deformation

Vacancies - creation and annihilation Dislocation recovery

Section 2.2

- dynamic and static recovery through climb and glide Subgrains

Section 2.3

- formation, growth, boundary energy and misorientation

Recrystallization and grain growth

Section 2.3

- driving force for recrystallization, number of potent sites for “nucleation”, number of

recrystallized grains, grain growth and coarsening main

model

Precipitation

Section 2.4

- nucleation and growth of precipitates

Flow stress

Section 2.2

- flow stress of deformed and recrystallized material

Solute drag

Section 2.4

- concentration profile over the grain boundary

Figure 8. Outline of the sub-models that together represents the microstructural model.

2.1 The grain structure

The kinetics of recrystallization and grain growth depend on the migration of grain boundaries. Grain boundaries are regions of considerable atomic misfit and act as strong barriers to dislocation motion [22]. A grain boundary is a boundary that separates regions of different crystallographic orientations and the misorientation between two crystals (grains) is an angle (θ) which is the smallest rotation required to make the two crystals coincide [23]. A schematic picture of a grain boundary is shown in Figure 9. Five macroscopic degrees of freedom are needed to define the geometry of a boundary. In Figure 9, AB represents a boundary plane and the overall geometry of the boundary is defined by the orientation of this plane with respect to one of the two crystals (two degrees of freedom) and by the angle θ (three degrees of freedom).

Figure 9. Schematic presentation of a grain boundary between two crystals misoriented

Grain boundaries are usually separated into the categories of low and high angle grain boundaries which are dependent on the size of the misorientation [23]. Low angle grain boundaries (LAGB) or subgrain boundaries are boundaries misoriented by an angle less than 10-15°. High angle grain boundaries (HAGB) are boundaries misoriented by an angle greater than 10-15°. The definition of grains with low angle boundaries and high angle boundaries play an important role in the description of recrystallization as will be discussed in section 2.3 and section 3.1.

2.2 Dislocation generation and recovery

Most of the applied work during rolling turns into heat and only a small part remains as stored energy (~1%) [23]. The increase in stored energy is mainly due to the accumulation of dislocations which is caused by both tangling of existing dislocations and the generation of new. The energy is also raised by the increase of grain boundary area. The stored energy in the material is the driving force for recovery and recrystallization.

Recovery of the material is a process that occurs prior to recrystallization and is primarily due to changes in the dislocation structure. During recovery, the dislocations rearrange in configurations of lower energy. Recovery is actually a series of events: formation of cells, annihilation of dislocations within cells, formation of low-angle subgrains and subgrain growth [23], see Figure 10.

Figure 10. The stages of recovery where first cells are formed due to the rearrangement

of dislocations and then subgrains are formed and grow due to annihilation of

dislocations, from Humphreys and Hatherly [23].

During subsequent recrystallization the subgrains function as the nuclei for recrystallization, see section 2.3. During hot deformation, dislocation accumulation due to deformation and annihilation and rearrangement of dislocations due to dynamic recovery occurs simultaneously. The evolution of dislocations during hot deformation can be separated in two parts – dislocation generation and dislocation recovery

dt d dt d dt

d ρ ₌ ρ

₋ ρ

(2)

where ρ

is the generated dislocation density and ρ

is the annihilated dislocation density. If dynamic recovery is the only form of restoration that occurs in the material, the flow-stress in a stress-strain curve reaches a plateau and then holds a steady-state flow-stress, see Figure 7. This depends on the fact that the rate of recovery and work hardening reaches a dynamic equilibrium.

The flow stress during deformation is dependent on the dislocation density ρ and is usually described by

ρ b G α m σ

σ =

+ ⋅ ⋅ ⋅ ⋅ (3)

where σ

is a constant, mainly due to the strengthening contribution of precipitation, m is the Taylor factor which is dependent on the deformation due to the development of a deformation texture [4], see Figure 11, α is a proportionality constant (~0.15), G is the shear modulus and b is the Burgers vector. Equation (3) gives a good description of the flow-stress, see for example the work by Bergström [24].

Figure 11. Taylor factor as a function of applied strain for uniaxial compression of a fcc metal, from Siwecki and Engberg [4].

2.3 Recrystallization and grain growth

The definition of recrystallization according to Doherty et al. [20] is “the formation of a new grain structure in a deformed material by the formation and migration of high angle grain boundaries driven by the stored energy of deformation”. Consequently, recrystallization is a process where new dislocation-free grains grow on the expense of the old deformed grains, leaving a new structure with low dislocation density as a result. The process is divided into nucleation of new grains and growth of the same. It is nowadays acknowledged that

“nucleation” of recrystallization can not be described with classical nucleation theory [23].

Recrystallization is a much more rapid process and the new grains grow from small regions already present in the structure, like subgrains or cells. The subgrains must have an energy advantage, a larger size, to be able to grow, rather than shrink and vanish. Hence, for the recrystallization to take place, a critical subgrain size has to be reached. This critical size is large in comparison with the mean subgrain size, which means that it can only be reached by abnormal growth.

According to Humphreys [25], no evidence is found that rotation plays an important role in subgrain coarsening. It is a reasonable assumption that both subgrain coarsening and nucleation of recrystallization from pre-existing subgrains are controlled by the migration of low angle grain boundaries. Subgrain coarsening in regions of large orientation gradient is believed to be a mechanism by which recrystallization originates [25]. When an orientation gradient is present in the deformed and recovered material, the subgrains will grow more rapidly than those in an environment where there is no orientation gradient. Faster growing subgrains obtain more misorientation, which results in the creation of a high angle grain boundary and hence a nucleus.

Consequently, for recrystallization to take place a critical subgrain size has to be reached and the misorientation has to be high enough, i.e. it has to be a high angle grain boundary.

Preferable nucleation sites for recrystallization are grain boundaries [23, 26] and the new grains form a “necklace” around the deformed grain, see Figure 12 and Figure 13 (a) and (b).

When the grains are very large, intragranular nucleation occurs where new grains nucleate at the interior of the grains as well and not only at the prior deformed austenite grain boundaries [23], see Figure 13.

prior grain boundary

Figure 12. A schematic picture of the development of recrystallizing grains, where the initial grain size is small in comparison with the new recrystallized grains. The dotted lines illustrate prior grain boundaries and the continuous lines show new recrystallized grains. From Humphreys and Hatherly [23].

Figure 13. A schematic picture of the development of recrystallizing grains, where the

initial grain size is large in comparison with the new recrystallized grains. The dotted

lines illustrate prior grain boundaries and the continuous lines show new recrystallized

grains. From Humphreys and Hatherly [23].

The recrystallized grain size is reduced when larger deformations are applied. Larger deformations give smaller subgrains and thereby increases the number of nuclei for recrystallization [13, 27]. An example is shown in Figure 13 compared to Figure 12. If the initial grain size is the same, more nuclei, as in Figure 13, provide a finer recrystallized grain size. The effect of smaller recrystallized grains with larger strains is most evident at relatively small strains. It becomes less evident with increasing strain and ceases at some critical strain.

A smaller initial grain size also reduces the recrystallized grain size due to the increase of potential nucleation sites caused by the increased grain boundary area [13, 27].

The recrystallized grain size is also dependent on temperature. At lower temperatures, finer recrystallized grains are obtained due to the lower mobility of the grain boundaries and consequently slower growth rate of the recrystallizing grains [27]. The slower growth rate promotes additional nucleation during recrystallization which consequently gives smaller recrystallized grains. As the initial grain size is decreased, this effect becomes less significant.

Recrystallization is a thermally activated process and the temperature dependence for static recrystallization is illustrated in Figure 14. The recrystallization curves are obtained from analyzing stress-relaxation tests performed on a Gleeble 1500 thermomechanical simulator.

As seen in the figure, a sigmoidal shape of the recrystallization curve is obtained when the recrystallized fraction is plotted as a function of log(time). This is due to the nucleation and growth of recrystallizing grains followed by impingement of the growing grains.

Impingement of growing grains

Nucleation

Figure 14. Recrystallization curves from analyzing stress-relaxation tests for a plain C- Mn steel pre-strained to 0.2 true strain at a strain rate of 1 s

deformed and annealed at 850°C-1200°C. The material is the same as that studied in Paper II.

Recrystallization during deformation is called dynamic recrystallization. If dynamic

recrystallization occurs, the flow-stress raises to a peak value followed by a lower steady-state

flow-stress, see Figure 7. Dynamic recrystallization is promoted by low strain rates and high

temperature. If the strain rate is low enough and the temperature high enough the flow stress

does not reach a steady-state flow-stress after the peak, but oscillates around a certain value

due to successive cycles of recrystallization occurring concurrently with the deformation [21].

At higher strain rates there is a single peak behavior of dynamic recrystallization. According to Sakai and Jonas [28], the single peak in the flow-stress curve obtained when the strain rate is increased and the temperature is decreased involve grain refinement. Decreasing the strain rate and increasing the temperature gives multiple peaks in the flow-stress curve and grain coarsening takes place.

If dynamic recrystallization occurs, metadynamic recrystallization occurs after deformation. If dynamic recrystallization does not occur, static recrystallization may follow the deformation.

Static recrystallization requires an incubation time, when the nuclei forms, while the nuclei for metadynamic recrystallization is already present at the end of deformation [21].

Dynamic recrystallization occurs when a critical strain is reached. The critical strain, ε

, required to initiate dynamic recrystallization is lower than the peak strain, ε

, in the stress- strain curve [13], see Figure 15. Hence, dynamic recrystallization takes place before reaching the maximum flow stress at the peak strain. According to Sellars [13] the relation between the peak strain and critical strain is ε

= 0.8·ε

. For strains below the critical strain there is significant strain dependence on the recrystallization kinetics after deformation, and above there is no strain dependence. This is illustrated in Figure 15 where the critical and peak strains are pointed out in a stress-strain curve in figure (a). Figure (b) show the fraction recrystallized from relaxation curves after pre-strain 0.2, 0.4 and 0.6. The strain of 0.6 is well above the critical and peak strain and dynamic recrystallization followed by metadynamic recrystallization occurs. At strain 0.4, the peak strain is not reached but the critical strain is exceeded. Recrystallization after a strain of 0.4 is as fast as after a strain of 0.6, see figure (b).

Hence, metadynamic recrystallization occurs after a strain of 0.4 which means that dynamic recrystallization had occurred.

0 25 50 75 100 125

0 0.2 0.4 0.6

True strain

True stres [MPa]

εc εp

1000°C dε/dt=1.0 s^-1

0 0.5 1

0.001 0.01 0.1 1 10 100

Time [s]

Fraction recrystallized

ε=0.2 ε=0.4 ε=0.6 1000°C

(a) (b) Figure 15. (a) Stress-strain curve with arrows indicating the critical strain, ε

, and the

peak strain, ε

, and (b) recrystallization curves derived from relaxation tests for the C- Mn steel investigated in Paper II. After a pre-strain of 0.2, lower than ε

, strain- dependent static recrystallization occurs and after pre-strains of 0.4 and 0.6, exceeding ε

, strain-independent metadynamic recrystallization occurs.

After complete recrystallization, the energy in the material may be further reduced by grain

growth. Grain growth is driven by the decrease in energy per unit volume which is

accomplished by the reduction of grain boundary area and by the increase of average grain

boundary curvature [27, 29].

2.4 The effect of microalloying elements on recovery and recrystallization

Microalloying elements retard the progress of recrystallization either as dissolved in the austenite (solute drag) or as precipitates (precipitation pinning). The most common microalloying elements are niobium, vanadium and titanium which are all strong carbide- forming elements.

The effect of Nb in solution is shown in the dynamic recrystallization-time-temperature (RTT) curves in Figure 16, taken from Paper VI. The construction of RTT curves as a way to study recrystallization was proposed by Akben et al. [30]. RTT curves are constructed from the recrystallization start time at a certain temperature taken as the peak strain through the strain rate. The values in Figure 16 are obtained from the flow stress curves at strain rate 0.1 s

and temperature 1000-1200°C. It can be seen that recrystallization is much faster in the plain C-Mn steel compared to the Nb microalloyed steels. The effect is enhanced at lower temperatures.

1000 1100 1200

1 2 3 4 5

Time to peak strain [s]

Temperature [°C]

Nb2 Nb1 CMn

Strain rate 0.1 s^-1

Figure 16. The effect of Nb in solution on dynamic RTT curves at a strain rate of 0.1 s

(from Paper VI). The Nb1 steel contains 0.034 wt.% Nb and the Nb2 steel contains 0.055 wt.% Nb.

Yamamoto et al. [31] investigated the effect on recrystallization of Nb, V and Ti in solution and as precipitated carbides and nitrides. The softening in the material was studied by double compression tests and the fraction softening between deformations was determined by the offset-method (see section 4). The fraction softening was compared to the fraction recrystallized determined for a quenched steel with high hardenability (0.04C-2.0Mn-0.3Mo- 0.10Nb in wt.%) by microscopic observations. The microscopic observation compared to the softening curves indicated that recrystallization starts at the softening ratio around 20%.

To determine the effect of solutes, 0.006C steels were decarburized to less than 0.002C so that

the microalloying elements remained dissolved at 900°C. The softening kinetics in steels with

varying Nb content were studied, see Figure 17 (a). Yamamoto et al. found that the retarding

effect of Nb in solution is very significant in the region below 20% softening, i.e. before the

start of recrystallization. Above 20% softening the effect is smaller which indicates that solute

Nb retards recovery and the onset of recrystallization. The solute drag showed almost no

effect on the progress of recrystallization, only the onset. Also for Ti steels, softening is

retarded, but not as much as for the Nb steel, see Figure 17 (b). For V steels, softening is only

0 20 40 60 80 100

0.01 0.1 1 10 100 1000 10000

Holding time [s]

Softening ratio [%]

0.002C, 900°C Nb free

0.097Nb

0.171Nb 0.05Nb

0 20 40 60 80 100

0.01 0.1 1 10 100 1000 10000

Holding time [s]

Softening ratio [%]

Si-Mn

0.097Nb 0.092V

0.095Ti 0.002C, 900°C

(a) (b) Figure 17. The effect of (a) Nb in solution on softening behavior (b) Nb, V and Ti in

solution, from Yamamoto et al. [31].

Yamamoto et al. also investigated the effect of solutes from other microalloying elements, Ni, Cr, Mn and Mo, see Figure 18, to establish if the effect of solute drag is resulting from lattice distortion by solute atom in the matrix. They found that the degree of the retarding effect may be explained by the degree of lattice distortion. The retarding effect increases in ascending order of Ni, Cr, Mn, V, Mo, Ti, Nb.

0 20 40 60 80 100

0.01 0.1 1 10 100 1000

Holding time [s]

Softening ratio [%]

1.0 % Cr 1.0 % Ni 1.9 % Mn 0.9 % Mo

Si-Mn 0.097Nb

900°C

Figure 18. The effect of Cr, Ni, Mn and Mo in solution on softening behavior compared to a Nb steel, from Yamamoto et al. [31].

In the investigation by Yamamoto et al. [31] precipitates where examined using a transmission electron microscope (TEM) and extraction replicas. The Nb precipitated was determined by chemical analyses of the extracted replicas and is shown in Figure 19 (a). At 1000°C, the ultra low carbon steel (0.002C) showed the same softening behavior as the higher carbon steel (0.019C). This indicates that no precipitation occurs at 1000°C during the time for softening. At lower temperatures the higher carbon steel displays slower softening behavior than the ultra low carbon steel, see Figure 19 (b). With higher C content, more precipitation of NbC particles occurs. This indicates that recrystallization is influenced more by precipitated particles than by solute drag.

It was demonstrated that the Ti steel also shows an effect of higher C content, see Figure 20

(a), but the softening behavior in V steel is not influenced by an increase of C or N, see

Figure 20 (b). Recrystallization is most likely complete before the start of VC or VN

precipitation. The reason for the small effect of V on softening was determined to be the weak solute drag effect that does not retard the recovery and the onset of recrystallization.

0 0.02 0.04 0.06 0.08 0.1

1 10 100 1000 10000

Holding time [s]

Precipitated Nb [wt%]

850°C 900°C 950°C 1000°C 1050°C 0.019C-0.095Nb

0 20 40 60 80 100

0.01 0.1 1 10 100 1000 10000

Holding time [s]

Softening ratio [%]

0.1Nb, 900°C 0.002C(Nb free)

0.002C

0.019C 0.006C

(a) (b) Figure 19. (a) The progress of strain induced precipitation of Nb(C,N) in a 0.095 Nb steel

at temperatures varying between 850-1050°C. (b) The effect of C content on the softening behavior in 0.1 Nb steels at 900°C, from Yamamoto et al. [31].

0 20 40 60 80 100

0.01 0.1 1 10 100 1000 10000

Holding time [s]

Softening ratio [%]

0.002C(Ti free)

0.029C 0.006C 0.002C

0.10Ti, 900°C

0.042C

0 20 40 60 80 100

0.01 0.1 1 10 100 1000 10000

Holding Time [s]

Softening ratio [%]

0.002C-0.003N 0.014C-0.002N 0.034C-0.002N 0.002C-0.0134N 0.002C(V free)

0.10V, 900°C

(a) (b) Figure 20. The effect of C content on the softening behavior in (a) 0.1 Ti steels and (b)

0.1 V steels at 900°C, from Yamamoto et al. [31].

Precipitation is a diffusion controlled process and is dependent on temperature. Weiss and Jonas [32] used hot compression tests and analyzed the strain-to-peak stress, ε

, to determine the precipitation kinetics. By determining changes in the dynamic recrystallization, the start (P

) and finish (P

) times of precipitation could be evaluated. They used this method to develop Precipitation-Time-Temperature (PTT) diagrams, see Figure 21. The PTT curves have the classical C-shape.

Investigations have shown that deformation accelerates the precipitation kinetics [32, 33], see Figure 21 to Figure 23. Weiss and Jonas found that after a pre-strain of 5% the rate of precipitation is about ten times faster than in a dislocation-free material. Watanabe et al. [33]

also found that deformation increased the precipitation kinetics by a factor of more than ten in

terms of isothermal holding time. In their investigation, they used a hot rolling simulator and

determined precipitation kinetics by electrolytic extraction.

800 850 900 950 1000 1050

1 100 10000 1000000

Time [s]

Temperature [°C]

Undeformed Deformed 5%

P_s P_f

0.05%C, 0.035% Nb

Figure 21. Precipitation-time-temperature diagram for undeformed and deformed austenite, where P

and P

are the start and finish times of precipitation respectively, from Weiss and Jonas [32].

Figure 22. Precipitation kinetics of Nb(C,N) particles in recrystallized austenite of a 0.062%C-0.074%Nb steel, from Watanabe et al. [33].

Figure 23. Precipitation kinetics of Nb(C,N) particles in deformed austenite (30%

reduction at 950°C) of a 0.063%C-0.084%Nb steel, from Watanabe et al. [33].

Precipitation and recrystallization are two interacting processes. If precipitation occurs before or in the initial stage of recrystallization, recrystallization is retarded by the precipitates. If recrystallization is complete before the onset of precipitation, precipitation kinetics is reduced due to fewer nucleation sites.

Kang et al. [34] investigated the individual effect on recrystallization of each stage of precipitation, i.e. nucleation, growth and coarsening. In their investigation, they followed the softening kinetics of a 0.03wt.% Nb steel by double compression tests. Precipitate sizes were determined using TEM and carbon extraction replicas. At temperature 1000°C and above, the softening curve has the typical sigmoidal shape and below 1000°C, the softening curve exhibited a “hump” with initial softening followed by hardening due to precipitation, see Figure 24.

Figure 25 shows the mean particle size as a function of time. In the figure, the time dependence of the particle size could be divided into two regions: one with a steep and one with a week slope. The transition point was regarded as the end of the growth stage and the start of the coarsening stage. A Comparison of the softening curve, Figure 24, with the mean sizes of the precipitates, Figure 25, showed that during nucleation and growth, softening is impeded by particle pinning. During the coarsening stage of precipitates, softening is again resumed. At 1000°C, recrystallization was completed before precipitation started and the precipitation hence occurred in undeformed material.

0 0.2 0.4 0.6 0.8 1

0.01 0.1 1 10 100 1000 10000 10000

Time [s] 0

Fraction softening

850°C 900°C 950°C 1000°C 1050°C

Figure 24. Softening curves at various temperatures after a pre-strain of 0.3 at strain rate 10 s

, from Kang et al. [34].

1 10 100 1000

0.01 0.1 1 10 100 1000 10000 1E+0

Time [s] 5 Mean particle size [nm] 850°C

900°C 950°C 1000°C

Figure 25. Mean particle size of Nb(C,N) precipitates as a function of holding time after

deformation, from Kang et al. [34].

3 Modeling recrystallization

3.1 Physically-based model

The recrystallization model used in this work is based on Humphreys’ model [35] and is described in the appended papers. The following section will give a brief summary of the reasoning behind the model development.

In the model, subgrains act as nuclei for recrystallization. The condition for recrystallization to be triggered in the model is that the subgrains are larger than a critical size and that they are misoriented enough, the subgrain boundaries have to be of high angle type (see section 2.1 and 2.3). The condition for the misorientation to be high enough is that the subgrains have a boundary energy close to that of grains.

The subgrain size, R

, depends on the dislocation density, ρ , [36]

s

ρ

sub

k

R = (4)

where k

is a constant. The boundary energy of subgrains, γ

, is assumed to be dependent on the misorientation and the boundary energy of a high angle grain boundary, γ

, [35]

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛ −

⋅

=

g sub g

sub gb

sub

1 ln

θ θ θ

γ θ

γ ₍₅₎

where θ

and θ

are the misorientation of low and high angle grain boundaries, respectively.

The evolution of the ratio θ

/ θ

and also the subgrain size during deformation are shown in Figure 26. In the model, the condition of a high angle boundary is determined by the ratio θ

/ θ

which has to be close to one for recrystallization to start.

0 0.2 0.4 0.6 0.8 1

0 0.05 0.1 0.15 0.2

True strain θsub

/θ^g

0 5 10 15 20 25

Rsub [µm]

R_sub θsub/θg

Figure 26. The evolution of θ

/θ

and subgrain size during deformation of a C-Mn steel

at 900°C.

The second condition for recrystallization to be triggered is that the driving force, F

, is larger than zero

0 )

- (

2 d sub sub rec gb

rec

= − + + c ⋅ G ⋅ b ⋅ − P

− P

>

R

F R γ γ ρ ρ

(6)

where R

is the recrystallized grain size, c

is a constant, ρ

and ρ

are the dislocation density in the deformed and recrystallized material, respectively, and P

and P

are the retarding forces due to particle pinning and solute drag, respectively. From this condition the critical subgrain size is derived

SD

Z

P

P b

G R c

−

−

⋅

⋅

⋅

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛ −

−

= ( - )

ln 1 1

rec def 2 d

g sub g

gb sub

crit

sub,

ρ ρ

θ θ θ

γ θ

(7)

This means that for larger deformations (higher dislocation density), providing smaller subgrains, a smaller critical size is required for recrystallization to start.

If recrystallization starts, the number of recrystallized grains is determined. The grains are assumed to be spherical and the potential nucleation sites are subgrains at high angle grain boundaries, i.e. along the old deformed boundary. In the model, the grain boundary area is calculated for deformed grains.

In Figure 27 and Figure 28 a schematic picture of the course of recrystallization is shown.

During recrystallization, growth of recrystallizing grains and coarsening of recrystallized grains take place concurrently. Figure 28 shows the recrystallizing grains growing into the deformed material and the coarsening of recrystallized grains, i.e. normal grain growth.

(a) (b) (c) Figure 27. A schematic picture of (a) a spherical grain, (b) deformed grain and (c)

deformed grain with new recrystallized grains along its periphery.

Old grain

boundary Coarsening of

recrystallized grain, normal grain growth (interior grain)

Recrystallizing grain (surface grain)

”Reduced” grain boundary

Figure 28. A schematic picture of deformed material (light gray) where new grains

In the model, the recrystallizing/recrystallized grains are treated as two classes: surface grains and interior grains, which represent growing and coarsening grains, respectively, see Figure 28. If the grain boundary is covered with new recrystallized grains, a “reduced” old grain boundary is calculated where new grains can start to recrystallize.

Second-phase particles exert a retarding force, P

, on recrystallization and grain growth. Also, microalloying elements dissolved in the austenite provides a retarding force, P

. For the material to recrystallize, the driving force for recrystallization has to be larger than the pinning force exerted by precipitates and the retarding force exerted by solute drag, see equation (6).

The drag on a grain boundary caused by elements in solution depends on the velocity of the boundary relative to the diffusivity of the impurity and also its interaction with the grain boundary. To be able to model the effect of solute drag on recrystallization and grain growth, the interaction energy between the solute and the grain boundary, ΔG, has to be determined.

ΔG is the dissipation of free energy due to diffusion when the boundary passes through a volume containing one mol of atoms [37]. A wedge-shaped or a truncated wedge-shaped function is often chosen for ΔG in order to simplify the calculations, see Figure 29 (a).

The atoms are dragged along the boundary and the concentration profile at the boundary for a certain element at constant temperature depends on the velocity of the boundary. In Figure 29 (b) the concentration, c, of Nb along the boundary at different boundary velocities is shown.

Higher velocity yields lower concentration profile.

-2.5

-1 -0.5 0 0.5 1

y/w

ΔG

Boundary width

-0.005 0 0.005 0.01 0.015 0.02

-1 -0.5 0 0.5 1 1.5 2 2.5

y/w

c-c0 direction

v₁

v₃ v₂

v1<v2<v3

c₀ = 0.01

(a) (b) Figure 29. (a) Wedge-shaped (dashed line) and truncated wedge-shaped (solid line)

function for ΔG. (b) Impurity concentration profile across a grain boundary moving with velocity v calculated using the model by Hillert and Sundman [37] assuming a wedge shaped interaction profile and constant solute diffusivity (c

is the matrix solute concentration). The position along the grain boundary, y, has been normalized through division with the boundary width, w.

A concentration profile dependent on the boundary velocity means a retarding force

dependent on the velocity. In Figure 30 the effect of the retarding force as a function of

velocity is shown. For recrystallization, the driving force is large, i.e. the boundary velocity of

recrystallized grains is high, and the retarding force is in the “high velocity region”. As the

velocity is decreased due to accumulation of atoms in the boundary, the retarding force also

gets higher. Hence, only at the initial stage of recrystallization, the retarding force is in the

high velocity region.

0.E+00 1.E+07 2.E+07

0 10 20 30 40 5

Dimensionless velocity v ·w /D PSD [J/m3 ]

0 High velocity region

Figure 30. The variation of the retarding force with a dimensionless boundary velocity as calculated with Hillert and Sundman’s model [37]. A wedge shaped interaction profile and constant solute diffusivity is assumed.

The retarding force due to second phase particles, P

, with radius r and volume fraction f is, for randomly distributed particles, described by [23]

r P

f 2 3

Z

= γ (8)

commonly known as Zener pinning. The number of particles per unit volume, N

, is given by

( ^{4 r}

³ )

f

N

= π . Assuming that particles within a distance r on either side of the boundary will intersect it gives the number of particles intersecting unit area of the boundary N

=2r·N

. With the maximum restraining effect on the grain boundary given by F

=πrγ

[23] and the retarding force given by P

=N

·F

, equation (8) is obtained.

Randomly distributed particles, equation (8), do not give enough retardation to impede recrystallization completely [38, 39]. A more realistic assumption for the precipitate distribution is that the particles lie on the subgrain boundaries. A schematic picture of randomly distributed particles and particles distributed at the grain boundary is shown in Figure 31. When the particles are distributed at the boundaries, the number of particles per unit area of boundary is N

= δ N

/ S

, where S

is the grain boundary area per unit volume (=3/2R

) and δ is a factor depending on the position of the particles in the boundaries [23].

Consequently, particles on grain boundaries provide a higher retarding force than the one given by equation (8)

2 r

R f F

N

P

δγ

=

= (9)

which gives enough retarding force on the growth of recrystallizing grains.

(a) (b)

v v

Figure 31. Schematic picture of a boundary with velocity v where the particles are (a)

randomly distributed and (b) distributed on the boundary.