**2. Temperature field in the butt-welded joint with thorough penetration**

Welding is characterised by an application of the movable, concentrated heat source, which in turn makes the temperature field movable in time and space:

$$T = T\left(r, t\right) = T\left(\mathbf{x}, \mathbf{y}, z, t\right) \tag{1}$$

**–** heat waste by convection and radiation is negligible,

**Figure 1.** Schematic of heating of a steel sample using welding heat source.

0

l

general expression of temperature

**–** heat of fusion is not taken into consideration.

consideration.

**–** reciprocal interaction of temperature field and phase changes is not taken into account,

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

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105

A sample with thickness of *D* and width of *B1* + *B2*, heated by the movable welding heat source, which is displaced at velocity *v* along the *x*-axis (**Figure 1**), is an illustration of the below

According to Geissler and Bergmann [34, 35], the solution of Eq. (2) can be written as a superposition of Green's function. This leads to the following convolution of integrals as a

> , ', ' ' , ' ' ' ' ' *<sup>t</sup> <sup>a</sup> T r t f r t G r r t t dx dy dz dt*

where *r*(*x*,*y*,*z*) is a vector pointing the place on the sample, while *r′* = (*x′*,*y′*,*z′*) determines the source position. Green's function *G* describes the temperature field in the point of material defined by *r* in time *t*. It is caused by a point heat source acting in the *r′* position and at *t′* < *t* time. *G* depends on the geometry of the sample and can be determined by transformation

<sup>=</sup> - - òòòò (3)

( ) ( ) ( )

method, while *f* defines the cross-section of the welding heat source.

Studies are being conducted to develop models of temperature field. Such models should have a real-time shape and temperature gradients based on the geometrical dimension of the welding element and also time. Referring to the formulated problem, the solution of heat equation for isotropic medium is essential to determine a temporary temperature field:

$$\nabla^2 T(r, t) = \frac{1}{a} \frac{\partial}{\partial t} T(r, t) - \frac{f(r, t)}{\lambda} \tag{2}$$

where T(*r*,*t*) is temperature at *r* position at *t* time, a is the coefficient of temperature compen‐ sation, λ thermal conductivity and f(r,t) supplied energy per volume and time unit.

Analytical method, proposed by Geissler and Bergmann [34, 35], was chosen to solve this differential equation. A short description of the method, described in detail in the abovementioned studies, is presented below.

The following assumptions were accepted in the calculations:

**–** quantities characterising the material properties, such as thermal conductivity, tempera‐ ture compensation and thermal capacity, are constant (independent from temperature),


Crystallisation and solidification, segregation of alloy elements and solutes and structural changes caused by intensive cooling occur extensively. Thermal and mechanical states and

Modelling the temperature field during welding was first initiated by Rosenthal [1] and Rykalin [2], who supposed the point and linear models of heat source, respectively. The adoption of a point heat source, as in the above-mentioned studies, yields results with respect to the points located near the centre of the weld, which are significantly different from the actual temperature values. Therefore, Eagar and Tsai [3] proposed a two-dimensional (2D) Gaussian-distributed heat source model and developed a solution of temperature field in a semi-infinite steel plate. Subsequently, Goldak et al. [4] introduced a double ellipsoidal threedimensional heat source model. There are two ways of modelling the temperature field during welding: analytical [5–14] and numerical (the finite difference methods, infinitesimal heat balances and finite element method) [15–30]. The welding methods and types of joints can be studied through these approaches [6, 20, 21, 31–33]. The construction of numerical models with heightening complexity allows more essential factors for the exact description of the structural

**2. Temperature field in the butt-welded joint with thorough penetration**

Welding is characterised by an application of the movable, concentrated heat source, which in

Studies are being conducted to develop models of temperature field. Such models should have a real-time shape and temperature gradients based on the geometrical dimension of the welding element and also time. Referring to the formulated problem, the solution of heat equation for isotropic medium is essential to determine a temporary temperature field:

> ( ) ( ) ( ) <sup>2</sup> <sup>1</sup> , , , *f rt T rt T rt a t* ¶

where T(*r*,*t*) is temperature at *r* position at *t* time, a is the coefficient of temperature compen‐

Analytical method, proposed by Geissler and Bergmann [34, 35], was chosen to solve this differential equation. A short description of the method, described in detail in the above-

**–** quantities characterising the material properties, such as thermal conductivity, tempera‐ ture compensation and thermal capacity, are constant (independent from temperature),

 l

Ñ= - (2)

¶

sation, λ thermal conductivity and f(r,t) supplied energy per volume and time unit.

*T T rt T xyzt* = = ( , ,,, ) ( ) (1)

microstructure directly state about the quality of the welding joint.

turn makes the temperature field movable in time and space:

changes in the welded steel.

104 Joining Technologies

mentioned studies, is presented below.

The following assumptions were accepted in the calculations:

A sample with thickness of *D* and width of *B1* + *B2*, heated by the movable welding heat source, which is displaced at velocity *v* along the *x*-axis (**Figure 1**), is an illustration of the below consideration.

**Figure 1.** Schematic of heating of a steel sample using welding heat source.

According to Geissler and Bergmann [34, 35], the solution of Eq. (2) can be written as a superposition of Green's function. This leads to the following convolution of integrals as a general expression of temperature

$$T\left(r,t\right) = \frac{a}{\lambda} \Big| \int\limits\_{0} \int\limits\_{0} \int\int f\left(r',t'\right)G\left(\left|r-r\right|,t-t'\right)d\mathbf{x}'d\mathbf{y}'d\mathbf{z}'dt'\tag{3}$$

where *r*(*x*,*y*,*z*) is a vector pointing the place on the sample, while *r′* = (*x′*,*y′*,*z′*) determines the source position. Green's function *G* describes the temperature field in the point of material defined by *r* in time *t*. It is caused by a point heat source acting in the *r′* position and at *t′* < *t* time. *G* depends on the geometry of the sample and can be determined by transformation method, while *f* defines the cross-section of the welding heat source.

A three-dimensional temperature field with the possibility of acceptance of different geome‐ tries of samples as well as the shape of the heat source can be determined from Eq. (3).

In the case of the Gauss model of heat source, we have

$$f\left(\mathbf{x},\mathbf{y}\right) = \frac{P}{2\pi R^2} \exp\left(-\frac{\mathbf{x}^2 + \mathbf{y}^2}{2R^2}\right), -\infty < \mathbf{x} < \infty, -\infty < \mathbf{y} < \infty,\tag{4}$$

Boundary conditions defining the surface and Green's function were taken from the study by

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

( ) ( ) ( ( ) )


1/2 <sup>0</sup> ' ' 4 ' exp 4 ' *<sup>x</sup>*

( ) ( ) ( )

*y y nB B G at t*

å

=-¥

4 ' exp 4 '

*n*

æ ö é- - - - ù ë û + -ç ÷ - è ø

æ öù é- + + - ù ë û + -ç ÷ú

exp 4 '

¥ -

exp 4 '

( ) ( ) ( ) ( ) ( ) ( ) ', ' ', ' ', ' , ' ( ) *G r r t t G x x t t G y y t t G zt t xy z* - -= - - - - - (6)

( )

2

' 2

<sup>ë</sup> - è ø

2

( )

<sup>=</sup> - - òòò (10)

*z nD*

*at t*

2

å (9)

( )

*at t*

*x vt t x*

*at t*

( )

2 1

1/2 2 1

( )

1 2

1/2 2

( )

'2 2 1

*y y nB n B at t*


( ) ( ) ( )

¥ -

4 ' exp 4 ' *<sup>z</sup> n*

( ) ( ) ( )

, ', ' ' , ' ' ' ' *<sup>t</sup> <sup>a</sup> T r t f x y G r r t t dx dy dt*

Temperature distribution can be calculated after substituting Green's function and Eq. (4) into Eq. (10). The integral over a range of variables can be evaluated, which yields the following

0

l

*z′* integral is removed in comparison to Eq. (3).

=-¥ æ ö - =é - ù -ç ÷ ë û - è ø

The relationship of movement of the welding heat source to the welding element is included in *G*(*x*) function. Because a flat model of heat source was assumed, the *z′* coordinate is not present in *G*(*z*) function. The coordinate systems connected with the heat source and welding material coincide with *t0* time. Considering that the shape of the heat source is independent of time, *z′* is dependent on *f* and *G* is eliminated; a modified integral to count temperature profile

'2 2 1

æ ö <sup>é</sup> é- - + ù ë û =é - ù ´ -ç ÷ ë û <sup>ê</sup>

*y y nB n B at t*

( )

2

2

(7)

107

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

(8)

Carslaw and Jaeger [36]. Green's function takes the following form:

*G at t*

1

¥ =

å

p

*n*

¥ =

å

*y*

is thus obtained:

result:

1

*G at t*

where *n* is the transformation number of the source.

p

*n*

p

where the power of source is denoted by *P* and determined for *R* radius and corresponds to 1/e of its peak value (**Figure 2**).

**Figure 2.** Gauss distribution of power of the heat source.

An infinitely long bar with the above-mentioned dimensions of cross-section was accepted in the considered example (**Figure 1**).

This can be written in the Cartesian coordinate system as follows:

$$-\infty < \infty < \infty, -B\_1 \le y \le B\_2, \, 0 \le z \le D \tag{5}$$

Boundary conditions defining the surface and Green's function were taken from the study by Carslaw and Jaeger [36]. Green's function takes the following form:

$$G\left(\left|r-r\right|,t-t'\right) = G\_{(\iota)}\left(\mathbf{x}-\mathbf{x}',t-t'\right)G\_{(\jmath)}\left(\mathbf{y}-\mathbf{y}',t-t'\right)G\_{(\varepsilon)}\left(z,t-t'\right)\tag{6}$$

$$G\_{(\mathbf{x})} = \left[4\pi a \left(t - t^\*\right)\right]^{-1/2} \exp\left(-\frac{\left(\mathbf{x} + \nu \left(t\_0 - t^\*\right) - \mathbf{x}^\*\right)^2}{4a \left(t - t^\*\right)}\right) \tag{7}$$

$$\begin{split} G\_{(\nu)} &= \left[ 4\pi a \left( t - t' \right) \right]^{-1/2} \times \left[ \sum\_{s=-\alpha}^{\alpha} \exp \left( -\frac{\left[ \left. y - \left. y' - 2n \left( B\_{2} + B\_{1} \right) \right] \right]^{2}}{4a \left( t - t' \right)} \right) \right. \\ &+ \sum\_{s=1}^{\alpha} \exp \left( -\frac{\left[ \left. y - \left. y' - 2nB\_{2} - 2 \left( n - 1 \right) B\_{1} \right] \right\}^{2}}{4a \left( t - t' \right)} \right) \\ &+ \sum\_{s=1}^{\alpha} \exp \left( -\frac{\left[ \left. y - \left. y' + 2nB\_{1} + 2 \left( n - 1 \right) B\_{2} \right] \right\}^{2}}{4a \left( t - t' \right)} \right) \right] \end{split} \tag{8}$$

$$G\_{\left(\varepsilon\right)} = \left[4\pi a \left(t - t'\right)\right]^{-1/2} \sum\_{n=-\varepsilon}^{\alpha} \exp\left(-\frac{\left(\varepsilon - 2nD\right)^2}{4a\left(t - t'\right)}\right) \tag{9}$$

where *n* is the transformation number of the source.

A three-dimensional temperature field with the possibility of acceptance of different geome‐ tries of samples as well as the shape of the heat source can be determined from Eq. (3).

> 2 2 , exp , , , 2 2 *P xy f xy x y*

= - -¥< <¥ -¥< <¥ ç ÷

where the power of source is denoted by *P* and determined for *R* radius and corresponds to

An infinitely long bar with the above-mentioned dimensions of cross-section was accepted in

1 2 -¥ < < ¥ - £ £ £ £ *x B yB zD* , , 0 (5)

è ø (4)

æ ö +

In the case of the Gauss model of heat source, we have

p

1/e of its peak value (**Figure 2**).

106 Joining Technologies

**Figure 2.** Gauss distribution of power of the heat source.

This can be written in the Cartesian coordinate system as follows:

the considered example (**Figure 1**).

( ) 2 2

*R R*

The relationship of movement of the welding heat source to the welding element is included in *G*(*x*) function. Because a flat model of heat source was assumed, the *z′* coordinate is not present in *G*(*z*) function. The coordinate systems connected with the heat source and welding material coincide with *t0* time. Considering that the shape of the heat source is independent of time, *z′* is dependent on *f* and *G* is eliminated; a modified integral to count temperature profile is thus obtained:

$$T\left(r,t\right) = \frac{a}{\lambda} \Big| \int \int f\left(\mathbf{x}',\mathbf{y}'\right) G\left(\left|r-r\right|,t-t'\right) d\mathbf{x}' d\mathbf{y}' dt' \tag{10}$$

#### *z′* integral is removed in comparison to Eq. (3).

Temperature distribution can be calculated after substituting Green's function and Eq. (4) into Eq. (10). The integral over a range of variables can be evaluated, which yields the following result:

$$\begin{split} T\left(x,t\right) &= \frac{\eta P}{2\pi\rho Cp} \Big[ u\left(t,t'\right) \Big|\_{s=-\alpha}^{s} \exp\left(-\frac{\left(z-2nD\right)^{2}}{4a\left(t-t'\right)}\right) \\ &+ \left(\sum\_{s=-\alpha}^{\alpha} \exp\left(-\frac{\left(x+\nu\left(t\_{0}-t'\right)\right)^{2}+\left(y-4nB\right)^{2}}{2R^{2}+4a\left(t-t'\right)}\right) F\_{1}\left(y\right) \\ &+ \sum\_{s=-\alpha}^{\alpha} \exp\left(-\frac{\left(x+\nu\left(t\_{0}-t'\right)\right)^{2}+\left(y-4nB\right)^{2}}{2R^{2}+4a\left(t-t'\right)}\right) F\_{2}\left(y\right) \\ &+ \sum\_{s=-\alpha}^{\alpha} \exp\left(-\frac{\left(x+\nu\left(t\_{0}-t'\right)\right)^{2}+\left(y-4nB\right)^{2}}{2R^{2}+4a\left(t-t'\right)}\right) F\_{3}\left(y\right) \Bigg| dt' + T\_{0} \end{split} \tag{11}$$

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

can occur in the welded joint.

the steel.

structure of solidification,

present in the joint during and after welding.

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

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

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109

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

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

**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 TTT-

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

welding diagram. Together, it is categorised into the following zones:

where

$$u\left(t, t'\right) = \frac{1}{2\sqrt{\pi a \left(t - t'\right)} \left(R^2 + 2a\left(t - t'\right)\right)}\tag{12}$$

$$\begin{split} F\_{i(\boldsymbol{y})} &= \operatorname{erf}\left(\sqrt{\frac{R^2 + 2a(t-t')}{4aR^2\left(t-t'\right)}} \left(B\_2 - \frac{R^2\left(\boldsymbol{y} - 4nB\right)}{R^2 + 2a\left(t-t'\right)}\right)\right) + \\ &- \operatorname{erf}\left(\sqrt{\frac{R^2 + 2a(t-t')}{4aR^2\left(t-t'\right)}} \left(-B\_1 - \frac{R^2\left(\boldsymbol{y} - 4nB\right)}{R^2 + 2a\left(t-t'\right)}\right)\right) \end{split} \tag{13}$$

$$\begin{split} F\_{z(\boldsymbol{\nu})} &= \operatorname{erf}\left(\sqrt{\frac{R^2 + 2a\left(t - t'\right)}{4aR^2\left(t - t'\right)}} \bigg(B\_2 - \frac{R^2\left(\boldsymbol{\nu} - 2\left(2n - 1\right)B\right)}{R^2 + 2a\left(t - t'\right)}\right) \right) + \\ &- \operatorname{erf}\left(\sqrt{\frac{R^2 + 2a\left(t - t'\right)}{4aR^2\left(t - t'\right)}} \bigg(-B\_1 - \frac{R^2\left(\boldsymbol{\nu} - 2\left(2n - 1\right)B\right)}{R^2 + 2a\left(t - t'\right)}\right) \right) \end{split} \tag{14}$$

$$\begin{split} F\_{\mathfrak{z}(\boldsymbol{\nu})} &= \operatorname{erf}\left(\sqrt{\frac{R^{2} + 2a\left(t - t'\right)}{4aR^{2}\left(t - t'\right)}} \bigg(B\_{2} - \frac{R^{2}\left(\boldsymbol{\nu} + 2\left(2n - 1\right)B\right)}{R^{2} + 2a\left(t - t'\right)}\right) \right) + \\ &- \operatorname{erf}\left(\sqrt{\frac{R^{2} + 2a\left(t - t'\right)}{4aR^{2}\left(t - t'\right)}} \bigg(-B\_{1} - \frac{R^{2}\left(\boldsymbol{\nu} + 2\left(2n - 1\right)B\right)}{R^{2} + 2a\left(t - t'\right)}\right) \right) \end{split} \tag{15}$$
