**3. Kinetics of phase transformations in a solid state**

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108 Joining Technologies

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*at t R at t* <sup>=</sup> - +- (12)

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Heating processes of steel lead to the transformation of a primary structure into austenite, while cooling leads to the transformation of austenite into ferrite, pearlite, bainite and martensite. Structural changes of a welded joint, connected with its cooling (also with hard‐ ening), develop heterogeneous image of material structure, which influences the state of stress after welding. The zone with a yield point lesser or greater than that of an indigenous material can occur in the welded joint.

Mechanical properties of the joint mostly depend on the type of welding material (its primary structure and chemical constitution of steel) and the characteristics of heat cycles accompany‐ ing welding. Temperature levels attained during heating, the hold time at a particular temperature and velocity of cooling in the 800–500°C range determine the type of structure present in the joint during and after welding.

**Figure 3.** Characteristic structural areas of a welded joint, depending on the temperature and the share of the carbon in the steel.

**Figure 3** shows the distribution characteristics of the weld joint zone of structural carbon steel [37] with a schematic of the fragment of an iron-carbon system and fragment of the TTTwelding diagram. Together, it is categorised into the following zones:

**–** fusion zone, which undergoes a thorough penetration and is characterised by the dendritic structure of solidification,


Several studies have focused on the description and numerical modelling of steel phase transformations. These studies have been reviewed by Rhode and Jeppson [38].

The type of a newly created phase depends heavily on the kinetics of heating and cooling processes. Kinetics of those processes is described by Johnson-Mehl-Avrami's and Kolomo‐ gorov's (JMAK) rules [39]. The amount of austenite *ϕA* created while heating the ferritepearlitic steel is therefore defined according to the following formula:

$$\log\_A(T) = \sum\_{j} \phi\_j^o \left( 1 - \exp\left( -b\_j(T)t^{u\_j(T)} \right) \right) \tag{16}$$

1

=

transformation (**Figure 4**). Those diagrams are called TTT–welding diagrams.

and *k* denotes the number of structural participations.

*k j j* j

1

The Analysis of Temporary Temperature Field and Phase Transformations in One-Side Butt-Welded Steel Flats

The quantitative description of dependence of the material structure and quality on temper‐ ature and transformation time of overcooled austenite during surfacing is made using the timetemperature-transformation diagram during continuous cooling, which combines the time of cooling *t8/5* (time when material stays within the range of temperature between 500 and 800°C, or the velocity of cooling (*v8/5* – (800 – 500)/t*8/5*) and the temperature with the progress of phase

**Figure 4.** Scheme of phase changes of overcooled austenite depending on cooling velocity within temperature range

Quantitatively, the progress of phase transformation is estimated by volumetric fraction *ϕ<sup>j</sup>*

(27), wherein time *t* is replaced with a new independent variable, temperature *T* [41]:

the created phase, where *i* denotes ferrite (*j*≡*F*), pearlite (*j*≡*P*), bainite (*j*≡*B*) or martensite (*j*≡*M*). The volumetric fraction *ϕ<sup>j</sup>* of the created phase can be expressed using a formula given in Eq.

of

800–500°C.

å <sup>=</sup> (19)

http://dx.doi.org/10.5772/63994

111

where *ϕ<sup>j</sup> 0* constitutes an initial share of ferrite (*j*≡*F*), pearlite (*j*≡*P*) and bainite (*j*≡*B*), while constants *bj* and *nj* are determined using conditions of the beginning and the end of transfor‐ mation:

$$m\_j = \frac{\ln\left(\ln\left(0.99\right)\right)}{\ln\left(A\_1 \, /\, A\_j\right)}, b\_j = \frac{0.01n\_i}{A\_1} \tag{17}$$

In welding processes, the volume fractions of particular phases during cooling depend on the temperature, cooling rate, and the share of austenite (in the zone of incomplete conversion 0≤*ϕ*A≤1). In a quantitative perspective, the progress of phase transformation during cooling is estimated using additivity rule by voluminal fraction *ϕ<sup>j</sup>* of the created phase, which can be expressed analogically in Avrami's formula [40] by equation:

$$\varphi\_j(T, t) = \varphi\_A \varphi\_j^{\max} \left( 1 - \exp \left[ b\_j \left( T \left( \mathbf{v}\_{k \le t} \right) \right) t \left( T \right)^{s \left( T \left( \mathbf{v}\_{k \ge t} \right) \right)} \right] \right) \tag{18}$$

where *φ<sup>j</sup> max* is the maximum volumetric fraction of phase *j* for the determined cooling rate estimated on the basis of the continuous cooling diagram (**Figure 4**), while the integral volumetric fraction equals to:

The Analysis of Temporary Temperature Field and Phase Transformations in One-Side Butt-Welded Steel Flats http://dx.doi.org/10.5772/63994 111

$$\sum\_{j=1}^{k} \varphi\_j = 1 \tag{19}$$

and *k* denotes the number of structural participations.

**–** partial joint penetration, where material is in a semi-fluid state and creates the border between the melted material and the material being converted into austenite,

**–** proper transformation, where perfect conversion of primary structure into austenite

**–** partial transformation between temperature A1 at the beginning of austenisation and A3 at the end of austenisation, where only a part of the structure changes into austenite,

Several studies have focused on the description and numerical modelling of steel phase

The type of a newly created phase depends heavily on the kinetics of heating and cooling processes. Kinetics of those processes is described by Johnson-Mehl-Avrami's and Kolomo‐ gorov's (JMAK) rules [39]. The amount of austenite *ϕA* created while heating the ferrite-

( ) ( ) ( ) ( ( )) <sup>0</sup> 1 exp *<sup>j</sup> n T*

constitutes an initial share of ferrite (*j*≡*F*), pearlite (*j*≡*P*) and bainite (*j*≡*B*), while

and *nj* are determined using conditions of the beginning and the end of transfor‐

*i*

*<sup>T</sup>* = -- å *b Tt* (16)

*AA A* = = (17)

*b T v tT* é ù = - ê ú ë û (18)

of the created phase, which can be

transformations. These studies have been reviewed by Rhode and Jeppson [38].

pearlitic steel is therefore defined according to the following formula:

j

estimated using additivity rule by voluminal fraction *ϕ<sup>j</sup>*

 jj

j

volumetric fraction equals to:

expressed analogically in Avrami's formula [40] by equation:

*<sup>j</sup> T t A j <sup>j</sup>*

*Aj j j*

( ( ))

*j j <sup>n</sup> n b*

( 1 3 ) <sup>1</sup> ln ln 0.99 0.01 , ln /

In welding processes, the volume fractions of particular phases during cooling depend on the temperature, cooling rate, and the share of austenite (in the zone of incomplete conversion 0≤*ϕ*A≤1). In a quantitative perspective, the progress of phase transformation during cooling is

> ( ) ( ( )) ( ) ( ( )) { } max 8/5 8/5 , 1 exp *nT v*

estimated on the basis of the continuous cooling diagram (**Figure 4**), while the integral

*max* is the maximum volumetric fraction of phase *j* for the determined cooling rate

 j

**–** the course-grained structure, the so-called overheating zone,

occurs,

110 Joining Technologies

**–** recrystallisation.

where *ϕ<sup>j</sup>*

mation:

where *φ<sup>j</sup>*

constants *bj*

*0*

The quantitative description of dependence of the material structure and quality on temper‐ ature and transformation time of overcooled austenite during surfacing is made using the timetemperature-transformation diagram during continuous cooling, which combines the time of cooling *t8/5* (time when material stays within the range of temperature between 500 and 800°C, or the velocity of cooling (*v8/5* – (800 – 500)/t*8/5*) and the temperature with the progress of phase transformation (**Figure 4**). Those diagrams are called TTT–welding diagrams.

**Figure 4.** Scheme of phase changes of overcooled austenite depending on cooling velocity within temperature range 800–500°C.

Quantitatively, the progress of phase transformation is estimated by volumetric fraction *ϕ<sup>j</sup>* of the created phase, where *i* denotes ferrite (*j*≡*F*), pearlite (*j*≡*P*), bainite (*j*≡*B*) or martensite (*j*≡*M*). The volumetric fraction *ϕ<sup>j</sup>* of the created phase can be expressed using a formula given in Eq. (27), wherein time *t* is replaced with a new independent variable, temperature *T* [41]:

$$
\rho \sigma\_j = \rho\_A \rho\_j^{\text{max}} \left( 1 - \exp\left( -b\_j T^{n\_j} \right) \right) + \rho\_j^0 \tag{20}
$$

different densities of the given structures. Then, the strain caused during heating is calculated

The Analysis of Temporary Temperature Field and Phase Transformations in One-Side Butt-Welded Steel Flats

( ) ( ) ( ) ( ) ( )

*T T HT T T T HT THT T*

*i i A ii A A A*

*i iA*

 jg

where *γiA* is the structural strain of the *i*-th structure in austenite, *T0* is the initial temperature,

is the linear thermal expansion coefficient of the *i*-th structure and *H*(*x*) is the function defined

1 0 0,5 0 0 0

*for x*

During cooling, the total strain (similarly as during heating) is the sum of strains associated with thermal expansion (in this case, the shrinkage of the material) as well as structural strains. Volumetric increase can be attributed to the high density of austenite (highest among the hardening structures such as martensite, bainite, ferrite and pearlite). The strain

> *C Tc Trc* eee

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a

+ -- å (30)

*T T HT T T T HT T T T HT T*

= - - + - -+

*A SOL s A s SOL s i i si si*

î <

*for x*

<sup>ì</sup> <sup>&</sup>gt; <sup>ï</sup> = = <sup>í</sup>

a j

= - -+ - - - +

,,,

*i PFBM*

=

1 1 3 1

= - (25)

http://dx.doi.org/10.5772/63994

<sup>=</sup> å (27)

= + (29)

(26)

113

(28)

*H Th Trh* eee

where *εTh* is the strain caused by thermal expansion of the material:

)

*Trh*

( )

caused during cooling can be described by the following relation [44]:

where *εTc* is the strain caused by thermal shrinkage of material:

a j

,,,,

*i APF BM*

=

*Tc*

e a *H x for x*

ï

e

( ) ( )

*AA A*

*T T HT T*

,,,,

*i APF BM*

å

=

a

(

a j

+- -

3 3

while *εTrh* is the phase transformation strain during heating:

0 0

as follows:

*Th*

e

*αi*

as follows:

where

$$m\_j = \frac{\ln\left(\ln\left(1-\phi\_j^s\right) / \ln\left(1-\phi\_j^{\prime}\right)\right)}{\ln\left(T\_j^s / T\_j^{\prime}\right)},\\b\_j = \frac{n\_j\left(1-\phi\_j^{\prime}\right)}{T\_j^s} \tag{21}$$

$$\frac{\wp\_{j}^{\circ}}{\wp\_{j}^{\max}} = 0,01,\frac{\wp\_{j}^{\circ}}{\wp\_{j}^{\max}} = 0,99\tag{22}$$

*ϕj <sup>0</sup>* is the volumetric participation of *j*-th structural component, which has not been converted during austenitisation; *Tj s* = *Tj s* (*v8/5*) and *Tj f* = *Tj f* (*v8/5*) are, respectively, the initial and final temperature of phase transformation of this component.

The fraction of martensite formed below the temperature *Ms* is calculated using the Koistinen-Marburger formula [42, 43]:

$$\log\_M\left(T\right) = \left.\varphi\_A \mid \varphi\_M^{\text{max}}\left\{1 - \exp\left[-\mu\left(M\_s - T\right)\right]\right\}, \mu = -\frac{\ln\left(\varphi\_M^{\text{min}} = 0.1\right)}{M\_s - M\_f}\tag{23}$$

where *ϕ<sup>m</sup>* denotes the volumetric fraction of martensite; *Ms* and *Mf* denote the initial and final temperature of martensite transformation, respectively; *T* is the current temperature of the process.

#### **4. Thermal and phase transformation strains**

Changes in temperature during welding cause deformations associated with the thermal expansion and deformation of the material resulting from the structural phase transformation. Deformation during the whole thermal cycle is the total deformation created during heating and cooling [44]:

$$
\varepsilon\left(\mathbf{x}, \mathbf{y}, \mathbf{z}, t\right) = \varepsilon^H + \varepsilon^C \tag{24}
$$

where *ε*H and *ε<sup>C</sup>* denote the thermal and phase transformation strains during heating and cooling, respectively.

Heating leads to an increase in the material volume, while transformation of the initial structure (ferritic, pearlitic or bainitic) in austenite causes shrinkage which is associated with The Analysis of Temporary Temperature Field and Phase Transformations in One-Side Butt-Welded Steel Flats http://dx.doi.org/10.5772/63994 113

different densities of the given structures. Then, the strain caused during heating is calculated as follows:

$$\boldsymbol{\varepsilon}^{H} = \boldsymbol{\varepsilon}^{Th} - \boldsymbol{\varepsilon}^{Th} \tag{25}$$

where *εTh* is the strain caused by thermal expansion of the material:

( ( )) max <sup>0</sup> 1 exp *<sup>j</sup> <sup>n</sup> j Aj j j*

ln ln 1 / ln 1 (<sup>1</sup> ) , ln /

max max 0,01, 0,99 *s f j j j j*

 j

 j

 j

*j j s f s*

*s f f j j j j*


*j j j*

*T T T*

*<sup>0</sup>* is the volumetric participation of *j*-th structural component, which has not been converted

*f* = *Tj f*

The fraction of martensite formed below the temperature *Ms* is calculated using the Koistinen-

( ) { ( ) } ( ) min max ln 0.1) / 1 exp , *<sup>M</sup>*

where *ϕ<sup>m</sup>* denotes the volumetric fraction of martensite; *Ms* and *Mf* denote the initial and final temperature of martensite transformation, respectively; *T* is the current temperature of the

Changes in temperature during welding cause deformations associated with the thermal expansion and deformation of the material resulting from the structural phase transformation. Deformation during the whole thermal cycle is the total deformation created during heating

( ,,, ) *H C*

 ee

where *ε*H and *ε<sup>C</sup>* denote the thermal and phase transformation strains during heating and

Heating leads to an increase in the material volume, while transformation of the initial structure (ferritic, pearlitic or bainitic) in austenite causes shrinkage which is associated with

e

 m<sup>=</sup> = - é- - ù = - ë û - (23)

 m j

j = = (21)

= = (22)

*s f*

*M M* j

*xyzt* = + (24)

(*v8/5*) are, respectively, the initial and final

(20)

= -- + *b T*

( ( ) ( )) ( )

(*v8/5*) and *Tj*

j

j

*s* = *Tj s*

temperature of phase transformation of this component.

*M AM s*

**4. Thermal and phase transformation strains**

*T M T*


*<sup>n</sup> n b*

j jj

where

112 Joining Technologies

*ϕj*

process.

and cooling [44]:

cooling, respectively.

during austenitisation; *Tj*

Marburger formula [42, 43]:

j

 jj

$$\begin{aligned} \boldsymbol{\varepsilon}^{Th} &= \sum\_{i=A,F,F,B,M} \left( \alpha\_i \boldsymbol{\rho}\_{i0} \left( T - T\_0 \right) H \left( T\_{A\_i} - T \right) + \alpha\_i \boldsymbol{\rho}\_i \left( T - T\_{A\_i} \right) H \left( T\_{A\_i} - T \right) H \left( T - T\_{A\_i} \right) + \alpha\_i \boldsymbol{\rho}\_i \left( T - T\_{A\_i} \right) \right) \\ &+ \alpha\_A \left( T - T\_{A\_i} \right) H \left( T - T\_{A\_i} \right) \end{aligned} \tag{26}$$

while *εTrh* is the phase transformation strain during heating:

$$\varepsilon^{\mathcal{T}\mathbb{A}} = \sum\_{i=P,F,B,M} \varphi\_i \mathcal{Y}\_{i4} \tag{27}$$

where *γiA* is the structural strain of the *i*-th structure in austenite, *T0* is the initial temperature, *αi* is the linear thermal expansion coefficient of the *i*-th structure and *H*(*x*) is the function defined as follows:

$$H(\mathbf{x}) = \begin{cases} 1 & \text{for } \mathbf{x} > 0 \\ 0, \mathbf{S} \text{ for } \mathbf{x} = \mathbf{0} \\ 0 & \text{for } \mathbf{x} < \mathbf{0} \end{cases} \tag{28}$$

During cooling, the total strain (similarly as during heating) is the sum of strains associated with thermal expansion (in this case, the shrinkage of the material) as well as structural strains. Volumetric increase can be attributed to the high density of austenite (highest among the hardening structures such as martensite, bainite, ferrite and pearlite). The strain caused during cooling can be described by the following relation [44]:

$$
\mathfrak{e}^{\mathcal{C}} = \mathfrak{e}^{\mathcal{T}\mathfrak{e}} + \mathfrak{e}^{\mathcal{T}\mathfrak{e}} \tag{29}
$$

where *εTc* is the strain caused by thermal shrinkage of material:

$$\begin{split} \omega^{\text{Tr}} &= \alpha\_{\text{A}} \left( T - T\_{\text{SOL}} \right) H \left( T - T\_{\text{s}} \right) + \alpha\_{\text{A}} \left( T\_{\text{s}} - T\_{\text{SOL}} \right) H \left( T\_{\text{s}} - T \right) + \\ &+ \sum\_{i=A, P, F, R, M} \alpha\_{i} \rho\_{i} \left( T - T\_{\text{s}} \right) H \left( T\_{\text{s}i} - T \right) \end{split} \tag{30}$$

while *εTrc* is the strain caused by phase transformation during cooling:

$$\mathfrak{s}^{\text{Pr}} = \sum\_{l=P,F,B,M} \mathfrak{q}\_l \mathfrak{y}\_{Al} \tag{31}$$

longitudinal section determined by the trace of the source transition is shown in **Figures 6**

The Analysis of Temporary Temperature Field and Phase Transformations in One-Side Butt-Welded Steel Flats

http://dx.doi.org/10.5772/63994

115

**Figure 6.** Temperature distribution (°C) on the surface of flats at time *t* = 102 s from beginning of welding.

**Figure 7.** Temperature distribution (°C) in longitudinal section at time *t* = 102 s from beginning of welding.

The quantities *B1* = *B2* = *B* are considered in calculations, which are equal to the width of one flat, thereby obtaining a temperature field symmetrical to the plane defined by vertical axis of

and **7**.

where *TSOL* denotes solidus temperature, *Ts* the initial temperature of phase transformation, *Tsi* the initial temperature of austenite transformation in the *i*-th structure and *γAi* the structural strain of austenite in the *i*-th structure. In addition, due to the limitation of the existence of solid state material:

$$
\sigma(x, y, z, t) = 0 \quad \text{for} \quad T > T\_{\text{sol}} \tag{32}
$$
