**5. The example of calculation of temperature field and phase transformations in welded flats**

In the considered example, it is assumed that the welded material is steel S235 with the following material constants: specific heat *Cp* = 670 J/(kg·K), density *ρ* = 7800 kg/m3 , thermal diffusivity *a* = 1.2 × 10−5 m2 /s, the temperatures at the beginning and end of austenite transfor‐ mation, respectively, *A1* = 996 K (723°C) and *A3* = 1108K (835°C), solidus temperature *TS* = 1763K (1490°C) and the liquidus temperature *TL* = 1793K (1520°C). The model of joining two buttwelded flats with a thickness of 0.012 m and a width of 0.1 m is shown in **Figure 5**.

**Figure 5.** Scheme of the welded joint with technology chamfering.

The speed and power of the movement source are assumed to be *v* = 0.005 m/s and *P* = 18000 W, respectively. Time-varying temperature field was determined according to the formulas (Eqs. (11)–(15)). The temperature field on the upper surface of the welded flats and in the longitudinal section determined by the trace of the source transition is shown in **Figures 6** and **7**.

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

e

e

**transformations in welded flats**

**Figure 5.** Scheme of the welded joint with technology chamfering.

diffusivity *a* = 1.2 × 10−5 m2

solid state material:

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*Trc*

,,,

( ,,, 0 ) *SOL*

In the considered example, it is assumed that the welded material is steel S235 with the

mation, respectively, *A1* = 996 K (723°C) and *A3* = 1108K (835°C), solidus temperature *TS* = 1763K (1490°C) and the liquidus temperature *TL* = 1793K (1520°C). The model of joining two butt-

The speed and power of the movement source are assumed to be *v* = 0.005 m/s and *P* = 18000 W, respectively. Time-varying temperature field was determined according to the formulas (Eqs. (11)–(15)). The temperature field on the upper surface of the welded flats and in the

following material constants: specific heat *Cp* = 670 J/(kg·K), density *ρ* = 7800 kg/m3

welded flats with a thickness of 0.012 m and a width of 0.1 m is shown in **Figure 5**.

**5. The example of calculation of temperature field and phase**

*i PFBM*

=

*i Ai*

<sup>=</sup> å (31)

*x y z t for T T* = > (32)

/s, the temperatures at the beginning and end of austenite transfor‐

, thermal

 jg

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

**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 the source and the direction of its movement. Thus, both this and the further calculation illustrations are shown on the right symmetrical cross-section which is perpendicular to a moving heat source. The image of maximum temperature isotherms in a cross-section of the joint (on the front contact of flats) is shown in **Figure 8**.

The analysis of the cooling speed *v8/5* showed that after complete cooling in both weld and heat-affected zone, wherein there has been complete and partial transformation of ferrite and pearlite into austenite, supercooled austenite resulted in a ferrite-pearlite structure. Changes in the temperature and structure in the considered cross-section for the selected time of the welding cycle are shown in **Figures 11**–**19**. The analysis of these changes was investigated on a symmetrical half of the cross-section of the flats' connection, which is perpendicular to the direction of source movement and *x0* = 0.5 m from the beginning of the flats (in half their length), that is, from the beginning of the coordinate system associated with flats. For *t* = *0*, the beginning of the coordinate systems associated with the source and the welded object overlaps. **Figures 11**–**13** illustrate the condition where the joint area was completely filled with the welded material, there were no cooling phase transitions and the largest participation of austenite is observed. By contrast, **Figures 14**–**19** illustrate the gradual decrease in the share of

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**Figure 10.** TTT-welding diagram for S235 steel.

austenite which forms a ferritic-pearlitic structure following the temperature drop.

**Figure 11.** Volume fraction of austenite in cross-section at time *t* = 109 s from process beginning.

**Figure 8.** Maximum temperatures (°C) in half of cross-section of welded flats.

On the basis of the maximum temperature field achieved in particular areas of the weld joint, specific heat-affected zones were determined (**Figure 9**) specified by limit temperatures *TL*, *TS*, *A3* and *A1*. The weld area is limited by a liquidus isotherm, which implies covering the part of the section wherein complete melting of the material occurred. Incomplete melting is limited by both liquidus and solidus temperatures. The heat-affected zone is determined by solidus isotherms and *A1*. In addition, in the region between the solidus and *A3* temperatures, ferrite and pearlite, the starting components of steel, have been completely transformed into austen‐ ite, whereas in the temperature range *A1*-*A3*, there is an incomplete conversion. In the area where the temperature did not exceed *A1*, certainly, phase transitions did not occur and the structure of the parent material (ferritic-pearlitic) was thus preserved. The participation of the heating structures was determined on the basis of the Fe-C diagram. The participation of the cooling structure was determined on the basis of knowledge of the TTT-welding diagram for welding steel S235 and the variations in temperature during cooling (**Figure 10**) [45].

**Figure 9.** Heat-affected zones (HAZ) with selected points.

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**Figure 10.** TTT-welding diagram for S235 steel.

the source and the direction of its movement. Thus, both this and the further calculation illustrations are shown on the right symmetrical cross-section which is perpendicular to a moving heat source. The image of maximum temperature isotherms in a cross-section of the

On the basis of the maximum temperature field achieved in particular areas of the weld joint, specific heat-affected zones were determined (**Figure 9**) specified by limit temperatures *TL*, *TS*, *A3* and *A1*. The weld area is limited by a liquidus isotherm, which implies covering the part of the section wherein complete melting of the material occurred. Incomplete melting is limited by both liquidus and solidus temperatures. The heat-affected zone is determined by solidus isotherms and *A1*. In addition, in the region between the solidus and *A3* temperatures, ferrite and pearlite, the starting components of steel, have been completely transformed into austen‐ ite, whereas in the temperature range *A1*-*A3*, there is an incomplete conversion. In the area where the temperature did not exceed *A1*, certainly, phase transitions did not occur and the structure of the parent material (ferritic-pearlitic) was thus preserved. The participation of the heating structures was determined on the basis of the Fe-C diagram. The participation of the cooling structure was determined on the basis of knowledge of the TTT-welding diagram for

welding steel S235 and the variations in temperature during cooling (**Figure 10**) [45].

joint (on the front contact of flats) is shown in **Figure 8**.

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**Figure 8.** Maximum temperatures (°C) in half of cross-section of welded flats.

**Figure 9.** Heat-affected zones (HAZ) with selected points.

The analysis of the cooling speed *v8/5* showed that after complete cooling in both weld and heat-affected zone, wherein there has been complete and partial transformation of ferrite and pearlite into austenite, supercooled austenite resulted in a ferrite-pearlite structure. Changes in the temperature and structure in the considered cross-section for the selected time of the welding cycle are shown in **Figures 11**–**19**. The analysis of these changes was investigated on a symmetrical half of the cross-section of the flats' connection, which is perpendicular to the direction of source movement and *x0* = 0.5 m from the beginning of the flats (in half their length), that is, from the beginning of the coordinate system associated with flats. For *t* = *0*, the beginning of the coordinate systems associated with the source and the welded object overlaps. **Figures 11**–**13** illustrate the condition where the joint area was completely filled with the welded material, there were no cooling phase transitions and the largest participation of austenite is observed. By contrast, **Figures 14**–**19** illustrate the gradual decrease in the share of austenite which forms a ferritic-pearlitic structure following the temperature drop.

**Figure 11.** Volume fraction of austenite in cross-section at time *t* = 109 s from process beginning.

**Figure 15.** Volume fraction of ferrite in cross-section at time *t* = 134 s from process beginning.

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**Figure 16.** Volume fraction of pearlite in cross-section at time *t* = 134 s from process beginning.

**Figure 17.** Volume fraction of austenite in cross-section at time *t* = 146 s from process beginning.

**Figure 12.** Volume fraction of ferrite in cross-section at time *t* = 109 s from process beginning.

**Figure 13.** Volume fraction of pearlite in cross-section at time *t* = 109 s from process beginning.

**Figure 14.** Volume fraction of austenite in cross-section at time *t* = 134 s from process beginning.

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**Figure 15.** Volume fraction of ferrite in cross-section at time *t* = 134 s from process beginning.

**Figure 12.** Volume fraction of ferrite in cross-section at time *t* = 109 s from process beginning.

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**Figure 13.** Volume fraction of pearlite in cross-section at time *t* = 109 s from process beginning.

**Figure 14.** Volume fraction of austenite in cross-section at time *t* = 134 s from process beginning.

**Figure 16.** Volume fraction of pearlite in cross-section at time *t* = 134 s from process beginning.

**Figure 17.** Volume fraction of austenite in cross-section at time *t* = 146 s from process beginning.

**Figure 18.** Volume fraction of ferrite in cross-section at time *t* = 146 s from process beginning.

**Figure 19.** Volume fraction of pearlite in cross-section at time *t* = 146 s from process beginning.

Changes in temperature and volume fraction of the individual structural components at the selected points in the cross-section (comp. **Figure 9**) are shown in **Figure 20**.

*α* **[1/°C]** *γ*

**Figure 20.** Welding thermal cycles and volume fractions of phases at selected points: T—temperature, A—austenite, P

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Dilatometric graphs (thermal and structural strains as a function of temperature) for selected points of the section (comp. **Figure 9**) are shown in **Figure 21**. In **Figure 21a**, a dilatometric graph for point *1* of the weld area is presented, where the material is deposited in the liquid phase as the molten material of the electrode, flux and partially melted edges of the flats; therefore, we observe only shrinkage "negative" strains of the cooling metal with a clear fault

Austenite 2.178 × 10−5 *γ*F,P,S→A 1.986 × 10−3 Ferrite 1.534 × 10−5 γB→A 1.440 × 10−3 Pearlite 1.534 × 10−5 *γ*A→F,P 3.055 × 10−3 Bainite 1.171 × 10−5 *γ*A→B 4.0 × 10−3

**Table 1.** Structural (*γ*) and thermal (*α*) expansion coefficients of phases.

Martensite 1.36 × 10−5

—pearlite, F—ferrite.

Point *1* is located in the area of chamfering of flats; hence, the graph of temperature and austenite volume fraction starts at the moment of the joint execution. At points *2* (fusion zone) and *3* (full-transformation zone), a complete austenite transformation occurs. But at point *2*, apart from phase transformations in the solid state, melting and solidification phase transfor‐ mations occur, resulting in a diverse dendritic structure shown during metallographic testing. At point 4, a partial transformation of the primary structure into austenite occurs, which is visible on the graph.

In strain calculations, linear expansion coefficients of the particular structural elements were assumed and structural stresses (**Table 1**) were determined on the basis of the author's own dilatometric research [46].

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**Figure 20.** Welding thermal cycles and volume fractions of phases at selected points: T—temperature, A—austenite, P —pearlite, F—ferrite.


**Table 1.** Structural (*γ*) and thermal (*α*) expansion coefficients of phases.

**Figure 18.** Volume fraction of ferrite in cross-section at time *t* = 146 s from process beginning.

**Figure 19.** Volume fraction of pearlite in cross-section at time *t* = 146 s from process beginning.

selected points in the cross-section (comp. **Figure 9**) are shown in **Figure 20**.

visible on the graph.

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dilatometric research [46].

Changes in temperature and volume fraction of the individual structural components at the

Point *1* is located in the area of chamfering of flats; hence, the graph of temperature and austenite volume fraction starts at the moment of the joint execution. At points *2* (fusion zone) and *3* (full-transformation zone), a complete austenite transformation occurs. But at point *2*, apart from phase transformations in the solid state, melting and solidification phase transfor‐ mations occur, resulting in a diverse dendritic structure shown during metallographic testing. At point 4, a partial transformation of the primary structure into austenite occurs, which is

In strain calculations, linear expansion coefficients of the particular structural elements were assumed and structural stresses (**Table 1**) were determined on the basis of the author's own Dilatometric graphs (thermal and structural strains as a function of temperature) for selected points of the section (comp. **Figure 9**) are shown in **Figure 21**. In **Figure 21a**, a dilatometric graph for point *1* of the weld area is presented, where the material is deposited in the liquid phase as the molten material of the electrode, flux and partially melted edges of the flats; therefore, we observe only shrinkage "negative" strains of the cooling metal with a clear fault reflecting the strain of austenite transformation into ferrite and pearlite. Point *2* (**Figure 21b**) is located in the area of melting of the starting material, and the dashed line in the figure reflects the change of state from solid to liquid. The solid line (bottom) is initiated by weld solidifica‐ tion.

of ferrite and pearlite in austenite was observed, that is, at that point, the temperature during heating has exceeded the temperature of the beginning of austenitizing *A1*, but has not reached the final temperature of austenitizing *A3*. The stage of the austenite transformation is deter‐ mined linearly in a numerical model by making the austenite transformation progress conditionally on the ratio (*Tmax* – *A1*)/(*A3* – *A1*), where *Tmax* is the maximum temperature reached

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**6. Verification of the results of numerical simulation of phase transitions**

In order to verify the results of the numerical calculations, metallographic tests of the buttwelded joint were carried out. For this purpose, two flats with identical geometry as calculated, that is, two flats with a thickness of 0.012 m and width of 0.1 m, were welded. The material of the flats was steel S235. Before making the joint, sheet chamfering was conducted. Then, welding was carried out with the submerged arc welding method, under welding flux Taste-3 and with SPG*Φ*15 wire. Welding parameters were voltage U = 30 V, current I = 600 A and welding speed 20 m/h. The diagram of the elements prepared for joining is presented in **Figure 5** (identical to the welded joint adopted for numerical analysis). Cross-section of the weld joint (the image of the sample taken for metallographic examinations) is presented in **Figure 22**. Metallographic analysis was performed for specific zones of a welded joint, that is, for the area of weld, heat-affected zone and parent material. **Figure 23** shows an image of the middle part of the welded joint (at the junction of welded flats) with a clearly visible dendritic structure which is characteristic of solidification. **Figure 24** shows an image of the structure in the right symmetrical part of the welded joint in the area from the weld to the ferrite-pearlite structure of the parent material. On the border of the weld, dendrites are visible, which change in the heat-affected zone into a structure with the Widmannstatten structure elements.

**Figure 22.** Cross-section of a welded joint: sample taken for metallographic analysis, magnification 2×.

at a particular point of the weld joint.

**with the results of metallographic research**

**Figure 21.** Dilatometric curves for selected points of cross-section.

Dilatometric graph in point *3* (**Figure 21c**) illustrates the history of strain changes for a full thermal cycle. In the considered point, the material with a ferritic-pearlitic structure is completely transformed into austenite during heating (lower line), which is shown with a fault in the diagram in the temperature range 723–835°C. At point 4 (**Figure 21d**) partial conversion of ferrite and pearlite in austenite was observed, that is, at that point, the temperature during heating has exceeded the temperature of the beginning of austenitizing *A1*, but has not reached the final temperature of austenitizing *A3*. The stage of the austenite transformation is deter‐ mined linearly in a numerical model by making the austenite transformation progress conditionally on the ratio (*Tmax* – *A1*)/(*A3* – *A1*), where *Tmax* is the maximum temperature reached at a particular point of the weld joint.

reflecting the strain of austenite transformation into ferrite and pearlite. Point *2* (**Figure 21b**) is located in the area of melting of the starting material, and the dashed line in the figure reflects the change of state from solid to liquid. The solid line (bottom) is initiated by weld solidifica‐

**Figure 21.** Dilatometric curves for selected points of cross-section.

Dilatometric graph in point *3* (**Figure 21c**) illustrates the history of strain changes for a full thermal cycle. In the considered point, the material with a ferritic-pearlitic structure is completely transformed into austenite during heating (lower line), which is shown with a fault in the diagram in the temperature range 723–835°C. At point 4 (**Figure 21d**) partial conversion

tion.

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