computer modeling of deformation influence on cct-diagram ...austenite decomposition was built for a...

International Journal "Information Content and Processing", Volume 6, Number 1, © 2019

49

COMPUTER MODELING OF DEFORMATION INFLUENCE ON CCT-DIAGRAM

FOR MN-, NI- AND MO-ALLOYED LOW-CARBON STEEL

Vladislav Kaverinsky

Abstract: A physically based mathematical model was developed to describe thermodynamics and kinetics of phase transformations in solid state. Specific software has been written for the model implementation. The model developed is primary aimed on austenite decomposition process description. It is used to arbitrary cooling regimes including isothermal stops and thermo-cycles and could be applied to complex-alloyed steels. Using the model CCT-diagram austenite decomposition was built for a Mn, Ni and Mo-alloyed low-carbon steel. Then an investigation was carried out to study an effect of deformation degree and deformation velocity on the CCT-diagram. When simulation the diagram, it was assumed that the cooling rates are constant over the entire temperature range. The effect of high-temperature deformation was considered, which occurs at the temperature above the temperature of the initial moment of the cooling simulation. The deformation degrees considered are 0 % (no deformation), 30 % and 50 %. The deformation velocity was considered in terms of time the certain deformation degree is reached. For 30 % deformation two levels of deformation velocity were under consideration: deformation in 30 seconds (slow) and in 0.3 seconds (fast). For 50 % deformation only slow regime (in 50 seconds) has been studied in the article. The results obtained by the model show a significant impact of the deformation and its degree on acceleration of diffusion controlled phase transformation processes, which consequently affect shear and semi-shear transformations.

Keywords: computer model, phase transformation, austenite decomposition, CCT-diagram.

ITHEA Keywords: I.6.6 Simulation Output Analysis.


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Introduction

Modern steels and alloys are complicated multi-component systems. In steels micro-structure formation a significant role play phase transformation especially austenite decomposition [Sokolov, 2013]. It is the variety of overcooled austenite transformation products, the composition and structure of which depend on the temperature conditions of the process, steels owe their unique technological properties. Mainly built on the austenite decomposition processes control are the technologies of thermal and thermo-deformation processing of steel [Novikov, 1979].

Continuous cooling transformation diagram (or CCT-diagram) is a graphic representation of austenite transformation kinetics of at different cooling rates built for a certain material [Bhadeshia, 1992]. There also exist time temperature transformation (TTT) diagrams built for isothermal transformation conditions which are not under consideration in this article. CCT-diagrams are rather more useful because most of the heat processing conditions are actually cooling but not isothermal.

Experimental building of a CCT-diagram is a quite hard task. It needs special equipment that allows temperature change control in a very wide range, indirect indication of a transformation occurrence by measurement of several physical properties changing (dilatation, magnetic properties, thermal analysis etc.), microstructure control using optical and electronic microscopes, X-ray phase analysis. To build a CCT-diagram a number of long and expensive experiments should be carried out. Computer simulation is not a full alternative to an experimental but it allows rather fast and not expensive estimation of austenite decomposition kinetic processes. Basing on modeled results a CCT-diagram could be build.

In spite of difficultness, there were built a grate number of CCT and TTT diagrams. Hundreds of kinetic diagrams for steels and titanium alloys are given in a popular manual by L.E. Popova and A.A. Popov [Popova, 1991], but this handbook is rather old, and certainly not complete. It becomes clear when you start dealing with the heat processing, rolling, forging, welding and others


51

technologies including high temperature heat and cooling. There are many combinations of alloying elements and impurities. If we take, for example, 10 basic chemical elements that can be contained in steel, and for each one we take 10 levels from 0 to the characteristic maximum content, we get billions of possible combinations. No handbook can contain so much data.

There is an influence on kinetic of phase transformation on only of chemical composition. Impact of initial structure characteristics (especially austenite grain size), heating temperature and previous deformation could be significant [Sokolov, 2013], [Kwon, 1992]. Despite of its essential importance in thermo-deformation processing, influence of pre-deformation factors were out of consideration in most of investigations. There is no information on deformation influences in popular manuals. However, this effect exists and it could be significant [Zolotorevskiy, 2011].

For investigation of phase transformation thermodynamics and kinetics in multi-component alloyed steels for many of which there is a deficiency of accessible CCT-diagrams experimentally built a mathematical model is crucial. Inspire of existing of several theoretical studies of phase transformation kinetics and theoretically based models development [Sokolov, 2013], [Zolotorevskiy, 2011], [Vasilyev, 2012] for the beginning of our research there was a lack of available and accessible program implementation of the physically grounded mathematic models for austenite decomposition. Though, building of such software becomes our essential task.

Empirical models, which have been actively developed since 70-th, could be precise but only in a restricted range of cases. Physically based models are much more complicated and need modern computational equipment for calculation, but they are flexible and have greater prediction potential [Sokolov, 2013].

Kinetics of austenite transformation in extra low-carbon steels alloyed by Mn, Ni and some amount of Mo is not studied enough, at least we do not have such information. And there is a certain lack of information on deformation influence on this kinetics. According to the work [Zolotorevskiy, 2011] such factors as


52

degree, velocity and temperature of a previous deformation could affect austenite decomposition. Thus, their influence on such kind of steels is of quite interest.

Related Work

For today, a number of models have been created that describe the thermodynamics and the kinetics of decomposition of overcooled austenite in carbon and alloyed steels. But so far the most widespread are purely empirical models [Kwon, 1992], [Lee, 2012], [Suehiro, 1987], [Kern, 1992], [Sun, 2002]. They are easy to develop and, in some cases, can provide sufficient precision. One of the most accurate [Sokolov, 2013] empirical model for the diffusion type austenite transformation was developed in works [Sarkar, 2007]. The basic kinetic equation of this model for low-alloyed steels has the following form:

n

T

TM

eqPFPF

S

TdTn

bTAb

dff

)(

)(exp1exp1

231

(1)

where: )(T - instantaneous cooling speed, n = 0.9, M, b1 and b2 - empirical parameters

meff

eqPFPF d

tbff)(

exp1

(2)

PF

C

fw

bTAbbb1

)(exp 02310 (3)

where: m, b0, b1 and b2 - empirical parameters

Equations (1) and (2) give as an output actually the same thing but are suitable for different types of steels and different initial data. The model above is the basis of the HSMM computer program developed since the middle of 1990s by


53

the University of British Columbia (Canada) in conjunction with the American Institute of Iron and Steel (AISI). The model is not very sophisticated and demanding for computing compared to physically grounded models and has a satisfactory accuracy for many practical cases. Nevertheless, in work [Sokolov, 2013] it is noted that the disadvantage of this model is the limited ability to predict the effects of changing the chemical composition of steel on a simulated processes. For different grades of steel, the values of empirical parameters and the type of equations itself are quite differ.

A rather precise empirical model describing the kinetics of martensitic transformation proposed in work [Lee, 2012]. Its equations are as follows:

nSM TMmf )(exp1 (4)

where fM - the part of martensite at temperature T; MS – temperature of the martensitic transformation start, K; T – current temperature, K; m and n – empirical parameters depending on the chemical composition of steel.

][3,11][4,2][9,8][6,16][5,14][6,30][6,3022,764 CuMoCrNiSiMnCMS (5)

][0193,0][0074,0][0017,0][0105,00231,0 MoCrNiCm (6)

][3108,0][0739,0][0258,0][7527,0][1836,14304,1 2 MoCrNiCCn (7)

Since the martensitic transformation goes very rapidly and under cooling conditions only, the kinetic formulas like (4) describing the process as a temperature dependent (temperature varies over time), and not directly time dependent, it is quite acceptable for use.

More complicated are physically grounded models, which use the equations of mathematical physics to describe the processes. At the same time, the most reliable experimental data is used.

The model of phase transformation in steels can be divided into following parts: ferrite transformation, pearlite transformation and bainite transformation.


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The formation of the ferrite becomes thermodynamically advantageous when the thermodynamic force of ferrite formation becomes equal to the difference in the Gibbs energies of ferrite and austenite and less than zero. The thermodynamic calculations in multi-component systems are usually carried out at some degree of approximation. There are three degrees of approximation: complete equilibrium (equilibrium) – the atoms of all substitution elements have time to be redistributed between austenite and ferrite; ortoequilibrium – there is only a redistribution of intrusion elements (usually carbon and any other intrusion element); paraequilibrium – the atoms of only one intrusion element (usually carbon) are redistributing [Golod, 2010]. In most of physical based models the approach of paraequilibrium is used for austenite transformation describing. Such approach is justified, because the atoms of substitution alloying elements have much less diffusion mobility compared to carbon atoms.

For the rate of ferrite nucleation calculating modifications of an equation of the form (8), which goes back to work [Lange, 1988], are used.

TkG

K

B

NPFPF

BPF

PF

eTka

DAI

2

3

4 32

(8)

where: DN – coefficient of diffusion related to the process limiting the nucleation (usually it is the coefficient of carbon diffusion in austenite); KPF, APF – parameters depending on the origin and shape of the new ferrite crystals; σ – the effective value of specific energy of the phase separation boundary.

Some variations of the equation (8) are given in works [Sokolov, 2013], [Vasilyev, 2012], [Zolotorevskiy, 2011]. To describe the growth rate of ferrite, which is controlled by carbon diffusion, the equation of parabolic growth (9) [Cristian, 1978] is used:

dtDdR C22 (9)

where: DC – the coefficient of volumetric carbon diffusion; λ – parameters of parabolic growth.


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For the velocity of displacement of the boundary between phases controlled by the restructuring of the crystalline lattice, use the equations (10, 11) from work [Liu, 2004]:

GMV / (10)

TRQ

eMM

*

0/ (11)

where: Q * - an activation energy of the crystalline lattice rearrangement (is equal to about 140000 J/mol)

There exist equations for more precise evaluation of the activation energy of the crystalline lattice rearrangement depending on chemical composition of solid solution. Some information on this can be found in work [Vasilyev, 2012].

Now modes that simultaneously deal with both limiting processes become popular. Such models are called mixed mode models. In the above models, the concept of the effective value of carbon concentration at the boundary between phases is introduced, which ensures the equality of the speeds of this boundary displacement due to both factors: the diffusion outflow of carbon and lattice rearrangement. The indicated concentration of carbon (interface concentration) is in the range from the current concentration of carbon in the middle of austenitic grain to the equilibrium concentration of carbon at a given temperature.

The origin of pearlite in eutectic steels occurs at the boundaries of austenite grains. In low and medium carbon steels it begins in the interphase regions that are enriched in carbon. In [Schaslivzew, 2006], an expression is used to describe the pearlite colony generation rate:

TRG

TRQ

PE eeNI

*

(12)


56

where: N – density of nucleation places; β – frequency factor; Q – an activation energy of the diffusion of the corresponding atoms across the phase boundary; ∆G * - thermodynamic barrier of nucleation.

The limiting process of a perlite colony growth is volume diffusion of carbon in austenite [Sokolov, 2013]. Hence, the equation for the pearlite growth rate has the form (13) presented in [Hillert, 1957]:

SS

Sxxxx

ffD

VCC

CCCPE

0112

(13)

where: S – the distance between the lamellae, equal to the sum of the thickness of a plate of ferrite Sα and a plate of cementite Sθ; S0 – a critical distance for which, according to the work [Mecozzi, 2011], an equation (14) can be writen:

)(2

0 TGS

PE

(14)

In works [Capdevila, 2002] and [Shapiro, 1968] an approach was developed in which as a limiting process diffusion along the interphase boundary is considered. The equation for a pearlite colony growth speed in this case (for a Mn alloyed steel) is written as (15):

SS

SSxxx

DKVMn

MnMnMnPPE

01112

(15)

where KP – coefficient of grain boundary segregation; MnD – coefficient of grain

boundary diffusion of Mn atoms; δ – thickness of the interphase boundary; Mnx

and Mnx – equilibrium concentrations of Mn at the corresponding boundaries of

the phase separation; Mnx – average content of Mn in austenite.

In some models, the description of pearlite nucleation is neglected, because it is believed that it occurs quite quickly [Sokolov, 2013]. Pearlite colonies grows rate is controlled by diffusion of carbon. In this case, the leading phase is


57

cementite [Hillert, 1957]. Therefore, the simulation of the pearlite colony growth could be reduced to the description of the growth of a cementite plate into depth of an austenite grain.

The most difficult to simulate is bainitic transformation. Its physical nature is not completely understood and is a subject of discussion. It is diffusive and sheared simultaneously. Work [Unemoto, 1992] considers it as primarily diffusion, whereas in work [Gaude-Fugarolas, 2006] it is assumed mainly as sheared one.

In diffusion based models the rate of bainitic plates origin is calculated using the equation (16) given by [Sokolov, 2013]:

TRG

TRQ

B eeKI

*

(16)

where: K – empirical parameter; Q – activation energy of a diffusion controlled process; ∆G* – thermodynamic barrier of nucleation.

The following expressions (17 – 20) are used in work [Trivedi, 1970] to describe the growth of bainite elements having the shape of the plates (by the diffusion mechanism):

C

CB

DV

25627 3* (17)

TRxxV

xxx

CC

mol

CC

CС

)(

(18)

200

0*

2121

(19)

CC

CC

xxxx

0 (20)


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where С – the radius of a bainite plate peak, * – a parameter depending on the concentration of carbon in the phases.

In shear models of bainite transformation, the process of bainite elements growth is neglected. The transformation process is described as the origin of subelements, of which the bainite plates are composed. In this case, different types of subelements are distinguished – primary (originated on the boundaries of grains and fragments) and autocatalytic (originating on the surface of existing bainitic elements).

There is a rather small number works that physically substantiate describe the effect of the previous deformation on austenite transformation. As experimental studies of the influence of plastic deformation of austenite on the formation of ferrite, carried out in [Hanlon, 2001], has showed that in the deformed austenite the origin of ferrite grains is substantially accelerated at the initial austenitic boundaries. The analysis carried out in [Lacroix, 2003] showed that this effect could not be explained simply by an increase in the number of potential nucleation places at the borders. The shape of the transformation curves could be reproduced only when significant reduction of the nucleation barrier. In this case, it appears that the increase in the driving force of the transformation (changes in the energy of Gibbs) due to the stored energy of the deformation substructure of austenite is too small to ensure acceleration of the nucleation, to the extent observed in the experiment [Hanlon, 2001].

In the paper [Zolotorevskiy, 2011], the authors believe that the reduction of the barrier of origin is due to the increase in the energy of the borders themselves due to an increase in their degree of defect and the formation of the boundary deformation substructure. This effect is described by the introduction of a form factor, which depends on a deformation degree.

Thus, there is enough data to build a semi-empirical but physically ground model of austenite transformation. For today, the model of thermodynamics and kinetics of austenite transformation in complex alloyed steels is realized and implemented as software by the author of this paper. The compiled program and its source code is available by the following link:


59

https://sourceforge.net/projects/aus-trans-sim/files/. A description and some others examples of the model implementation are given in our works [Kaverinsky, 2017], [Kaverinsky and Sukhenko, 2017]. The most detailed description of the model is given in my monograph [Kaverinsky, 2019] (in Ukrainian).

Task and challenges

The main purposes of the article are as follows: using the developed computer model of kineticas of austenite transformation to build and describe a CCT-diagram for a low-carbon steel of certain chemical composition; using an ability of the model to take into account impact of deformation, to study the influence of degree and velocity of hot previous deformation on the shape of the CCT-diagram.

This type of steels is used to welding because of low content of carbon, but alloyed to increase its strength and ability to hardening during thermo-deformation processing. For technology of such processing development a CCT-diagram including one for deformed state in essential. Unfortunately we do not have this information and experimental building of the CCT-diagram is a changing and long term work. The easer and faster way is using a simulation with our computer model, which was verified and proofed to be rather precise for similar steels [Kaverinsky, 2017], [Kaverinsky and Suchenko, 2017].

Brief description of the model

A description of the model developed was given in our several works, for instance [Kaverinsky and Suchenko, 2017]. The most detailed description and assaying of it can be found in my monograph [Kaverinsky, 2019]. Here is just a brief review of the model where many implementation details are hidden.

The thermodynamics modeling is based on CALPHAD-method [Golod, 2010], [Sokolov, 2013]. The original special software modules were developed for this model realization. The state function of a system under isobaric-isothermal conditions is the Gibbs energy [Golod, 2010], [Dinsdale, 1994]. According to the methods used, Gibbs's total molar energy for a solid solution of substitution is represented as a following sum [Golod, 2010]:


60

GGGGxG mgexidiK

ii

1

0 (21)

where: xi – an atomic proportion of component i in a solution, iG

0 – Gibbs

energy of a pure component i in the modification in a hypothetical nonmagnetic state, Gex – an ideal component of the mixing energy, Gex – the

excessive Gibbs energy, Gmg – the correction for the magnetic ordering

(essential only for ferromagnetic and antiferromagnetic phases).

A model for an intrusion element containing in a solid solution is more complex. These are solid solutions including elements with a small radius of the atom, for example C or N in Fe. For these cases, a sub-lattice model [Vasilyev, 2007] is used in which a solid solution is considered consisting of substitution and intrusion sub-lattices, in which elements interact with each other, both within one sub-lattice and between different sub-lattices [Golod, 2010]. Another component is included in the system: the vacancy nodes of the sub-lattices not occupied by atoms.

In the FCC (face centered cubic lattice) solid solution there is one "pore" per one node of substitution sub-lattice, i.e., the position in an intrusion sub-lattice a = b = 1. For BCC (base centered cubic lattice) solid solution per one node of substitution sub-lattice there are 3 “pores” a = 1, b = 3. Thus, the components of the solution are atoms of an intrusion chemical elements and vacancies. For calculation in the framework of the sub-lattice model, the atomic parts of components xi are replaced by the parts of nodes yi occupied by the components in its subgroups [Vasilyev, 2007], [Golod, 2010].

The sub-lattice model of Gibbs's total energy for solid solutions has the following form:

j

mmg

mex

jii

iijiii j

im GGyybyyaRTGyyG ]lnln[2211

:021 (22)


61

j i

элвнvaiijjэлзамi j

jimex JyyJyyG ][][ .).,(:

12:.).(

21 (23)

where: Ji,j – interaction parameters between components i and j; r – number (order) of the interaction parameter; the upper indices 1 and 2 for yi and yj indicate the concentration of a component atoms in a substituent sub-lattice and in a intrusion one, respectively.

The parameters of interaction are linearly dependent on temperature and are reference values, in the calculations only the interaction parameters from zero to second order are used. The contribution of magnetic ordering to the Gibbs energy was taken into account by the method from work [Dinsdale, 1994]. According to this, the correction to the value of the total Gibbs energy on the magnetic ordering is following:

)()1ln( 0 gBTRGmag (24)

where: τ = T/T*, where T* is the critical temperature (Curie temperature for ferromagnets), T – current temperature, K; B0 – a mean magnetic moment per one atom.

The parameter g (τ) from equation (24) is calculated in following way:

111597511692

1125518

600135611

497474

14079

1)(

15931

p

ppg

when τ 1 (25)

111597511692

1125518

150031510)(

25155

p

g

when τ > 1 (26)

where: p – a proportion of magnetic enthalpy absorbed above the critical temperature (depends on the lattice structure).


62

For all phases of an equilibrium heterogeneous system, the chemical potentials of each component in all phases are equal to each other. On this basis, we can formulate independent equations, since the chemical potential of the same component in different phases is described by different functions of concentrations and temperature. The geometrical representation of the equality of the chemical potentials in a two-component system means that the common tangency of the potential curves corresponding to each phase (concentration dependence of the Gibbs energy) is found. Points of the touch of this tangency line to the potential curves determine the composition of the phases, which corresponds to the state of equilibrium.

In multi-component systems, it is necessary to find the position of a hyperplane of corresponding dimension, which is tangent to multidimensional surfaces describing the thermodynamic phase potentials. The task of finding the chemical potential of a component in a multicomponent system using the above complex models of temperature-concentration dependence of Gibbs energy can be solved only numerically [Golod, 2010]. Mathematical methods of numerical solution of nonlinear systems of equations of this type are given in [Golod, 2008]. In the developed by the author computer program for this task a special module is implemented connecting the library ALGLIB for multiparametric minimization problems solving. At the same time, first using the empirical formulas from work [Sokolov, 2012], the first approximation is calculated, and then the refinement of values is performed. Practice shows that this significantly reduces the computing time.

Ferrite transformation kinetics modeling is based on works [Zolotorevskiy, 2011], [Vasilyev, 2012]. Ferrite nucleation is given by the equation (27):

),()()()(

0 2

3

2

1

1)( AEBeff

AENYTGTkxK

xK

RTYQ

iii eeTtNCJ

(27)

where: Ji – rate of ferrite nucleation by i-th process mode, 1/s m2; Сi - empirical factor, which takes into account the influence of a crystal lattice defects rate in

the certain place on the atoms moving rate, K1/2/sm3; (t)Ni0 - the amount of


63

potential nucleation places by the moment; T – temperature, K; QN(YAE) – the

activation energy of the α lattice rebuilding, J/mol; R – universal gas

constant, J/molK; K1(x), K2(x) – empirical coefficients; σeff – the effective

surface energy of a ferrite, J/m2; kB – Boltzmann constant, J/K; Gγα(T, YAE) – change of the volumetric Gibbs energy in γα transformation as a function of the chemical composition and temperature, J/m3.

Ferrite growth rate, controlled by the diffusion of carbon outflow, for a spherical shape ferrite grain is given by equation (28) [Sokolov, 2013], [Vasilyev, 2012]:

)x(x

)x(x

R)Y(T,D

=)Y(T,VαCγC

CγC

α

AECAE

Cαγ

int

int

/ (28)

where: )Y(T,D AEC - the bulk diffusion coefficient of carbon in austenite as a

function of composition and temperature, averaged over the carbon

concentration profile, m2/s; Rα – the radius of a ferrite grain, m; intγC

x - an

interface molar concentration of carbon in the austenite; Cx - a current average

molar concentration of carbon in the austenite; αC

x - an equilibrium molar

concentration of carbon in the ferrite.

The rate of the γ / α - border controlled rearrangement of the lattice described by the equation (29) from [Vasilyev, 2012]:

)Y(T,ΔGRT)(YQ

eM=)Y(T,V AEαγ

AEN

γαAELαγ

0// (29)

where: M0 – the mobility parameter of γ / α-border, M0 ≈ 10 m4/cJ; Y*AE indicates that the values calculated for the concentrations that occur directly in the transition area.

The value of the carbon interface concentration is calculated from the condition of the ferrite growth rates obtained by the equations (28) and (29) equality. Formula (28) was used to compute the effective growth rate.


64

The carbon diffusion coefficient in an alloyed solid solution was calculated by equation (30) from [Vasilyev, 2007]:

RT

yγy+yα+)(yΔU

R

Z

X=S

Z

X=S SySγCy+SySα+)C(yCΔUθ

RT

(T)Ly+(T)Ly)y(yD=)Y(T,D

Z

X=S

Z

X=SSSCSSCC

Z

X=S

SCVaS

FeCVa

Z

X=SS

ccCAEC

exp

exp

12110

(30)

where: DC0 – pre-exponential factor, depends on the crystal lattice and the

average frequency of the atoms thermal vibrations, m2/s; DC0 ≈ a2ν, where a is a lattice parameter, ν – average frequency of the atoms vibrations; yC – sublattice concentration of carbon atoms; yS - sublattice concentration of an

alloying element atoms; (T)LFeCVa - a temperature-dependent interaction energy

of a carbon atom with its nearest vacant place in the implementations sublattice

of Fe lattice, J / mol; (T)LSCVa - a temperature-dependent interaction energy of a

carbon atom with its nearest vacant place in the sublattice of the alloying

element lattice, J/mol; R – universal gas constant, J/molK; T – temperature, K;

- parameter which determines the relationship between entropy and energy of

migrations activation, K-1, )(yU CCΔ - migrations of carbon in Fe activation

energy barrier without alloying as a function of the carbon concentration, J/mol; αS and γS – parameters of of others elements influence on the carbon migration activation barrier, J/mol

The rate of pearlite colonies growth is given by equations (31, 32) from [Vasilyev, 2012], [Sokolov, 2013]:


65

αCθC

γCγC

θα

AECAE

Cpe xx

θxx

S)Y(T,D=)Y(T,V

/int

6.35 (31)

),(4 /

AEYTGS

(32)

where: θ/γC

x - the concentration of carbon in a boundary section

austenite/cementite carbide; θC

x - a concentration of carbon in cementite; intγC

x

- interface molar concentration of carbon in the austenite; )Y(T,D AEC - a bulk

diffusion coefficient of carbon in the austenite as a function of its composition

and temperature, m2/s; θαS - a half-width distance in pearlite interlamellar; θ/ασ

- a specific energy of α/γ - boundary; )Y(T,G AEθγΔ - the Gibbs energy change

of the cementite carbide precipitation from austenite.

Nucleation rate of bainite elements is given by formula (7) from [Bhadeshia, 1992], [Sokolov, 2013]:

TRYTGkYQ

kkAEk

AEkAEB

etFCYTtJ

),()(

)(),,(

(33)

where: Ck - pre-exponential factor, empirical parameter, which depends on the mode of the process; Fk(t) – parameter determining the change over the

process of the number of places of bainitic elements nucleation; )(YQ AEB - an

activation energy of bainitic elements nucleation as a function of the chemical composition of the austenite; kk – an amendment coefficient.

Let’s consider on deformation influence discretion in our model. The origin of ferrite in the body of a grain occurs mainly after previous defamation. In our model, it was believed that the places of ferrite nucleation in the middle of grains are mainly boundaries sub-grains, and the rate of its origin depends on the size of the sub-grain and the proportion of large-angle boundaries. The


66

subzero dimension was estimated by the formula (34) obtained in work [Zolotorevskiy, 2011]:

01

)(

bT

kd

(34)

where: k1 – an empirical parameter taking into account stress relaxation at the

moment of the transformation; – Poisson coefficient; μ(T) – shear modulus at the deformation temperature; b – Burgers vector module: σ0 – internal stress after deformation.

The part of high-frontier boundaries was calculated according to the formulas (35, 36) from work [Zolotorevsky, 2010]:

3

)(1

Tbe

(35)

400850

5,0)(

T

eTb (36)

where: T is temperature of deformation.

A level of internal stress after deformation was estimated by the empirical formula (37) from work [Zolotorevsky 2011].

Ted2880

07.0048,0223,00 7,22

(37)

where: – deformation degree; έ – strain velocity, dγ – austenite grain size, T – the deformation temperature.

For the rate of origin of ferrite within the austenite grains, the following formulas have been used. The expression (38) is for the sub-grain and (39) for the sympathetic nucleation:


67

RTYTGkYQ

AU

AEPAEAF

etfd

CJ),()(

1.33 )(

(38)

RTYTGkYQ

AU

AESAEAF

etNtfCJ),()(

32.33 )()(

(38)

where: fAU(t) – austenite volume fraction to the current time, N3(t) – number of ferrite grains originated to the current time point, QAF(YAE) – an energy of the austenite lattice rebuilding into ferrite lattice when transformed inside a grain, as

a function of austenite chemical composition, J/mol, QAF(YAE) QN(YAE), kP and kS – correction coefficients, Gγα(T, YAE) – change of the molar Gibbs energy when γα transformation as a function of chemical composition and temperature, J/mol.

In spite of the model considers only the effect of deformation on ferrite transformation it could have an indirect influence on others transformations. This effect is manifested in the fact that the ferrite transformation begins and ends at different time and, therefore, because of cooling conditions, at different temperatures. This means that the equilibrium and actual carbon concentrations in the phases present will be different. Subsequent transformations also begin at another time and temperature and so they take place under differing conditions.

Application of the developed model to estimate an effect of deformation degree and strain velocity on CCT-diagram of a low-carbon steel

The chemical composition of the material investigated is given in table 1.

Table 1. The chemical composition of the material investigated (wt. %)

Fe C Mn Si Cr Ni Mo Cu

base 0.07 1.75 0.25 0.03 1.00 0.18 0.02


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It is a low-carbon steel alloyed with Mn and Ni. It has also a higher concentration of Mo. Level of Si is ordinary. It is just a deoxidizer. Cr and Cu are on the levels of impurity. Calculated temperatures of the critical points are as follows: A1 = 692 ºC and A3 = 813 ºC.

A CCT-diagram was built using our model. For this purpose a number of transformation processes were simulated for conditions of different cooling temperatures from 2000 ºC/sec to 0.01 ºC/sec. Because of large time range and high interest to fast cooling rates the time scale is made logarithmic.

A job file contains the source data for the calculation. For different aims the content of this file could be other. Detailed instruction on task files could be found in wiki for the program. It is accessible by the following address: https://sourceforge.net/p/aus-trans-sim/wiki/Work%20with%20the%20solver%20module/ For the considered task this file includes chemical composition of the material (in wt. %) excluding base component, which is clear from material type field, initial austenite grain size (in microns), deformation degree (in parts of 1), deformation speed (in % per second), Poisson ratio, rigidity modulus, module of Burgers vector, deformation temperature (in K), cooling conditions (for constant cooling rate only start temperature (in K) and cooling rate in deg/sec), time step for simulation (0.001 second was assumed in this investigation), maximum time (in seconds) and temperature ranges, a devisor for numerical differentiation and maximum number of iteration for numerical solutions and precision for thermodynamic calculation of critical temperatures (in K).

Output file includes calculated characteristics of the system for each of calculation step. These data includes, equilibrium and actual carbon concentrations in phases, actual amount of phases and structural components, current time and temperature, number of crystallites per volume unit, grain or other structural parts sizes, values of calculated thermodynamic functions. These files are rather long and complicated. So another program was developed to make selection of certain columns of values to easer consider and save them into an Excel file. Basing on analysis of this data columns and kinetic curves built using them it is possible to build a kinetic diagram.


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The calculated diagram is shown on Figure 1. A feature of the calculated diagram is the allocation of globular (FG) and acicular ferrite (FA), pearlite (P) and fine pearlite (S), as well as upper (BH) and lower (BL) bainite regions. These arias allocation really paper on the most of existing diagrams, especially old ones.

Figure 1. CCT-diagram for low-carbon steel alloyed with Mn and Ni based on modeling (not pre-deformed material)

It can be seen from the diagram that quenching to martensite is possible for this steel at cooling rates of more than 100 ºC/sec. Although martensite in such steels is low carbon and hardening during hardening is not very strong [Novikov, 1979]. Pure bainitic structures, on the basis of the built diagram, do not form when this steel is cooled. But higher bainite can be formed in some quantity along with ferrite-pearlitic structures. Lower bainite is also formed, mainly at cooling rates of 1 to 10 ºC/sec. The amount of the bainite formed is small, about 1 – 5 %. The basis of the structure is ferrite.


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Let’s consider an effect of deformation on the shape of the diagram. Figure 2 shows a CCT-diagram for the steel of the same chemical composition, but previously deformed to 30 % in 30 seconds (competently slowly). It corresponds to slow rolling or extrusion. Deformation temperature assumed here is 927 ºC (1200 K). At this temperature the metal is steel plastic to deform but deformation effects on phase transformation already could be shown quite bright.

A significant difference is seen comparing the diagrams from Figure 1 and Figure 2. Ferrite transformation is displaced to the left. That means that it starts sooner and appears at faster cooling rates. Even cooling rate of 2000 ºC/sec does not leads to completely quenching, but only about 90 % of the microstructure is to be martensite. A significant part of structure at fast cooling of previous deformed steel is acicularis ferrite, which is known as a deformation inducted structure [Sokolov, 2013].

Figure 2. CCT-diagram for low-carbon steel alloyed with Mn and Ni based on modeling (predeformation 30 % at 927 ºC during 30 seconds)


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Following the ferritic transformation, the beginning of pearlitic one is shifted. But the amount of pearlite is small in this steel because of low carbon content. As it known from practice it would be rather a separate carbide phase rather than a “traditional” pearlite. A shape of bainite transformation curve changed. Amount of bainite is also small but in formes in a wider temperature range. The most of the bainte formed is higher bainite. But the curve of lower binate appearance displaced. This curve is actually corresponds the range of temperature where redistribution of carbon doesn’t go. Changes for the low cooling rate area of the diagram are not significant.

Figure 3 shows a CCT-diagram for the same steel, but previously deformed to 50 % in 50 seconds at the same temperature.

Figure 3. CCT-diagram for low-carbon steel alloyed with Mn and Ni based on modeling (predeformation 50 % at 927 ºC during 50 seconds)

The general direction of changes in the diagram corresponds to those in Figure 2 compared to Figure 1, but some more significant. The start temperature of ferrite transformation increases, especially at faster cooling rates. These fast


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cooling rates lead to more amount of acicularis ferrite and less part of martensite in the final structure. The area of pearlite transformation extends, but the amount of pearlite is also small. So it looks on the kinetic curve as an almost plane region. Decreasing of the martensitic transformation start temperature begins at faster cooling rates. The nature of this decreasing is carbon enrichment of ferrite caused by carbon redistribution during ferrite formation, which goes significantly faster in deformed steel. Temperature area of low bainte formation extends even at slow cooling rates.

However, additional changes after extra increasing are significantly less than they were competently with non-deformed material. Additional studies, not presented at the article, show decreasing of extra changes of the diagram shape with more increasing of deformation degree. Thus, this influence has a fading character.

Influence of the strain (deformation) velocity on the austenite transformation kinetics is also studied. Figure 4 shows a CCT-diagram for the same steel, but deformed to 30 % in 0.3 seconds. This corresponds to an intensive forging or stamping.

Acceleration of deformation is also affects stress level in material. It leads to some increasing of ferrite transformation temperature start. But competently with deformation degree increasing strain velocity increasing leads to some increasing of globular ferrite amount and reduction of acicularis ferrite. It also decreases amount of pearlite content forming during faster cooling rates. The slope corresponding to a decrease in the temperature of martensitic transformation start with an increase in the cooling rate becomes more acclivous. But in general, the effect of strain velocity is much lesser then the effect of deformation degree.


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Figure 4. CCT-diagram for low-carbon steel alloyed with Mn and Ni based on modeling (predeformation 30 % at 927 ºC during 0.3 seconds)

Conclusion

Using a developed computer model, simulations were conducted to study austenite transformation kinetic in low-carbon steel alloyed by Mn, Ni and some amount of Mo. The simulations were carried out for a wide range of cooling rates. That allows us to build CCT-diagrams for this steel. Such diagram for this type of steel was not found accessible sources, especially for previously deformed martial, lack which is an essential problem.

Because of the model takes into account previous deformation of austenite at high temperature, studied in the article in were the effects of deformation degree and strain velocity. A significant influence of deformation is shown on ferrite, martensite and pearlite transformations, especially at fast cooling rates.


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It is shown that previous deformation of austenite leads to acceleration of ferrite transformation start. It leads to appearance of large amount of acicularis ferrite and reducing of martensite part in the final structure after fast cooling. Effect of deformation on bainite transformation and process at slow cooling rates is not very significant.

A fading character of the effect of extra increasing of deformation degree was found out. Increasing of deformation degree affects in the same direction but less and less more.

The effect of strain velocity is also exists, but it less significant than the influence of deformation degree. These effects are as follows: a more acclivous decrease of the martensitic transformation start with the cooling rate decreasing; a small increase in the amount of globular ferrite in comparison with the acicular one at high cooling rates.

The results obtained and the model itself is very useful for developing of technologies for thermo-mechanical processing of low-carbon steels. In this way it is possible to predict the structure obtained and feet the cooling rate and deformation parameters to receive the desirable one. Also the usage of the computer modeling allows us to do this rather fast and without conduction of difficult and expensive experiments.

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Authors' Information

Vladislav Kaverinsky – Institute for Problems of Material Science NAS of Ukraine; Ph.D. Senior Researcher, Krzhizhanovsky st. 3, Kiev, Ukraine;

e-mail: [email protected]

Major Fields of Scientific Research: Theory and mathematical modeling of phase and structure transformation processes, evolution of the distribution function of dispersed systems, foundry and castings modifications, natural language processing for text analysis and synthesis in inflectional languages

computer modeling of deformation influence on cct-diagram ...austenite decomposition was built for a...

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