Structural (* γ*) and thermal (

*) expansion coefficients of phases.*α

Open access peer-reviewed chapter

Submitted: 05 November 2015 Reviewed: 28 April 2016 Published: 21 September 2016

DOI: 10.5772/63994

In this chapter, the welding method applied for modelling the temperature field and phase transformations is presented. Three-dimensional and temporary temperature field for butt welding with thorough penetration was determined on the basis of analytical methods of an integral transformation and Green’s function. Structural changes of heating and cooling, proceeding in a weld (in the heat-affected zone), were described using the existing formulations of phase transformations. Considerations were illustrated by an example, for which analysis of temperature fields, developed by a moving heat source, and calculations of the distribution of particular phases (structures) were carried out. Metallographic studies of the butt joints, which were arc welded under a flux, were carried out in the empirical part of this work. Their results enabled the verification of the numerical simulation results of the phase transformations.

- butt-welded joint
- temperature field
- HAZ
- phase transformations
- numerical modelling
- metallographic examination

Welding is characterised by many specific features associated with variable temperatures and variable physical and mechanical properties of the welding material. The moving heat source, characteristic of welding, partial melts the joint surface and fuses an electrode. The electrode fills a joint space with liquid metal. Hence, welding elements are subjected to varying temperature ranges, that is, from ambient to that of a liquid metal.

Crystallisation and solidification, segregation of alloy elements and solutes and structural changes caused by intensive cooling occur extensively. Thermal and mechanical states and microstructure directly state about the quality of the welding joint.

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 three-dimensional 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 changes in the welded steel.

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

T = T ( r , t ) = T ( x , y , z , t ) ![]()

E1
∇ 2 T ( r , t ) = 1 a ∂ ∂ t T ( r , t ) − f ( r , t ) λ ![]()

E2
where T(r ,t ) is temperature at r position at t time, a is the coefficient of temperature compensation, λ thermal conductivity and f(r,t) supplied energy per volume and time unit.
T ( r , t ) = a λ ∫ 0 t ∫ ∫ ∫ f ( r ' , t ' ) G ( | r − r ' | , t − t ' ) d x ' d y ' d z ' d t ' ![]()

E3
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.
f ( x , y ) = P 2 π R 2 exp ( − x 2 + y 2 2 R 2 ) , − ∞ < x < ∞ , − ∞ < y < ∞ , ![]()

E4
− ∞ < x < ∞ , − B 1 ≤ y ≤ B 2 , 0 ≤ z ≤ D ![]()

E5
G ( | r − r ' | , t − t ' ) = G ( x ) ( x − x ' , t − t ' ) G ( y ) ( y − y ' , t − t ' ) G ( z ) ( z , t − t ' ) ![]()

E6
G ( x ) = [ 4 π a ( t − t ' ) ] − 1 / 2 exp ( − ( x + v ( t 0 − t ' ) − x ' ) 2 4 a ( t − t ' ) ) ![]()

E7
G ( y ) = [ 4 π a ( t − t ' ) ] − 1 / 2 × [ ∑ n = − ∞ ∞ exp ( − [ y − y ' − 2 n ( B 2 + B 1 ) ] 2 4 a ( t − t ' ) ) + ∑ n = 1 ∞ exp ( − [ y − y ' − 2 n B 2 − 2 ( n − 1 ) B 1 ] 2 4 a ( t − t ' ) ) + ∑ n = 1 ∞ exp ( − [ y − y ' + 2 n B 1 + 2 ( n − 1 ) B 2 ] 2 4 a ( t − t ' ) ) ] ![]()

E8
G ( z ) = [ 4 π a ( t − t ' ) ] − 1 / 2 ∑ n = − ∞ ∞ exp ( − ( z − 2 n D ) 2 4 a ( t − t ' ) ) ![]()

E9
where n is the transformation number of the source.
T ( r , t ) = a λ ∫ 0 t ∫ ∫ f ( x ' , y ' ) G ( | r − r ' | , t − t ' ) d x ' d y ' d t ' ![]()

E10
z′ integral is removed in comparison to Eq. (3).
T ( x , t ) = η P 2 π ρ C p ∫ 0 t u ( t , t ' ) ∑ n = − ∞ ∞ exp ( − ( z − 2 n D ) 2 4 a ( t − t ' ) ) × { ∑ n = − ∞ ∞ exp ( − ( x + v ( t 0 − t ' ) ) ) 2 + ( y − 4 n B ) 2 2 R 2 + 4 a ( t − t ' ) ) F 1 ( y ) + ∑ n = − ∞ ∞ exp ( − ( x + v ( t 0 − t ' ) ) ) 2 + ( y − 4 n B ) 2 2 R 2 + 4 a ( t − t ' ) ) F 2 ( y ) + ∑ n = − ∞ ∞ exp ( − ( x + v ( t 0 − t ' ) ) ) 2 + ( y − 4 n B ) 2 2 R 2 + 4 a ( t − t ' ) ) F 3 ( y ) } d t ' + T 0 ![]()

E11
where
u ( t , t ' ) = 1 2 π a ( t − t ' ) ( R 2 + 2 a ( t − t ' ) ) ![]()

E12
F 1 ( y ) = e r f ( R 2 + 2 a ( t − t ' ) 4 a R 2 ( t − t ' ) ( B 2 − R 2 ( y − 4 n B ) R 2 + 2 a ( t − t ' ) ) ) + − e r f ( R 2 + 2 a ( t − t ' ) 4 a R 2 ( t − t ' ) ( − B 1 − R 2 ( y − 4 n B ) R 2 + 2 a ( t − t ' ) ) ) ![]()

E13
F 2 ( y ) = e r f ( R 2 + 2 a ( t − t ' ) 4 a R 2 ( t − t ' ) ( B 2 − R 2 ( y − 2 ( 2 n − 1 ) B ) R 2 + 2 a ( t − t ' ) ) ) + − e r f ( R 2 + 2 a ( t − t ' ) 4 a R 2 ( t − t ' ) ( − B 1 − R 2 ( y − 2 ( 2 n − 1 ) B ) R 2 + 2 a ( t − t ' ) ) ) ![]()

E14
F 3 ( y ) = e r f ( R 2 + 2 a ( t − t ' ) 4 a R 2 ( t − t ' ) ( B 2 − R 2 ( y + 2 ( 2 n − 1 ) B ) R 2 + 2 a ( t − t ' ) ) ) + − e r f ( R 2 + 2 a ( t − t ' ) 4 a R 2 ( t − t ' ) ( − B 1 − R 2 ( y + 2 ( 2 n − 1 ) B ) R 2 + 2 a ( t − t ' ) ) ) ![]()

E15

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

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:

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-mentioned studies, is presented below.

The following assumptions were accepted in the calculations:

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

heat waste by convection and radiation is negligible,

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

heat of fusion is not taken into consideration.

A sample with thickness of * D* and width of

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

A three-dimensional temperature field with the possibility of acceptance of different geometries 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

where the power of source is denoted by * P* and determined for

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:

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:

The relationship of movement of the welding heat source to the welding element is included in * G*(

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:

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

ϕ A ( T ) = ∑ j ϕ j 0 ( 1 − exp ( − b j ( T ) t n j ( T ) ) ) ![]()

E16
where ϕ_{j}^{0} constitutes an initial share of ferrite (j ≡F ), pearlite (j ≡P ) and bainite (j ≡B ), while constants b_{j} and n_{j} are determined using conditions of the beginning and the end of transformation:
n j = ln ( ln ( 0.99 ) ) ln ( A 1 / A 3 ) , b j = 0.01 n i A 1 ![]()

E17
ϕ j ( T , t ) = ϕ A ϕ j max { 1 − exp [ b j ( T ( v 8 / 5 ) ) t ( T ) n ( T ( v 8 / 5 ) ) ] } ![]()

E18
where φ_{j}^{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:
∑ j = 1 k ϕ j = 1 ![]()

E19
and k denotes the number of structural participations.
ϕ j = ϕ A ϕ j max ( 1 − exp ( − b j T n j ) ) + ϕ j 0 ![]()

E20
where
n j = ln ( ln ( 1 − ϕ j s ) / ln ( 1 − ϕ j f ) ) ln ( T j s / T j f ) , b j = n j ( 1 − ϕ j f ) T j s ![]()

E21
ϕ j s ϕ j max = 0 , 01 , ϕ j f ϕ j max = 0 , 99 ![]()

E22
ϕ_{j}^{0} is the volumetric participation of j -th structural component, which has not been converted during austenitisation; T_{j}^{s} = T_{j}^{s} (v_{8/5} ) and T_{j}f^{} = T_{j}^{f} (v_{8/5} ) are, respectively, the initial and final temperature of phase transformation of this component.
ϕ M ( T ) = ϕ A / ϕ M max { 1 − exp [ − μ ( M s − T ) ] } , μ = − ln ( ϕ M min = 0.1 ) ) M s − M f ![]()

E23
where ϕ_{m} denotes the volumetric fraction of martensite; M_{s} and M_{f} denote the initial and final temperature of martensite transformation, respectively; T is the current temperature of the process.

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 hardening), 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 accompanying 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 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-welding 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,

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,

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

proper transformation, where perfect conversion of primary structure into austenite occurs,

partial transformation between temperature A

_{1}at the beginning of austenisation and A_{3}at the end of austenisation, where only a part of the structure changes into austenite,recrystallisation.

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 Kolomogorov’s (JMAK) rules [39]. The amount of austenite * ϕ* created while heating the ferrite-pearlitic steel is therefore defined according to the following formula:

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≤_{A}≤1). In a quantitative perspective, the progress of phase transformation during cooling is estimated using additivity rule by voluminal fraction * ϕ* of the created phase, which can be expressed analogically in Avrami’s formula [40] by equation:

The quantitative description of dependence of the material structure and quality on temperature and transformation time of overcooled austenite during surfacing is made using the time-temperature-transformation diagram during continuous cooling, which combines the time of cooling * t* (time when material stays within the range of temperature between 500 and 800°C, or the velocity of cooling (

Quantitatively, the progress of phase transformation is estimated by volumetric fraction * ϕ* of the created phase, where

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

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## 4. Thermal and phase transformation strains

ε ( x , y , z , t ) = ε H + ε C ![]()

E24
where ε ^{H} and ε^{C} denote the thermal and phase transformation strains during heating and cooling, respectively.
ε H = ε T h − ε T r h ![]()

E25
where ε^{Th} is the strain caused by thermal expansion of the material:
ε T h = ∑ i = A , P , F , B , M ( α i ϕ i 0 ( T − T 0 ) H ( T A 1 − T ) + α i ϕ i ( T − T A 1 ) H ( T A 3 − T ) H ( T − T A 1 ) + + α A ( T − T A 3 ) H ( T − T A 3 ) ) ![]()

E26
while ε^{Trh} is the phase transformation strain during heating:
ε T r h = ∑ i = P , F , B , M ϕ i γ i A ![]()

E27
where γ_{iA} is the structural strain of the i -th structure in austenite, T_{0} is the initial temperature, α_{i} is the linear thermal expansion coefficient of the i -th structure and H (x ) is the function defined as follows:
H ( x ) = { 1 f o r x > 0 0 , 5 f o r x = 0 0 f o r x < 0 ![]()

E28
ε C = ε T c + ε T r c ![]()

E29
ε T c = α A ( T − T S O L ) H ( T − T s ) + α A ( T s − T S O L ) H ( T s − T ) + + ∑ i = A , P , F , B , M α i ϕ i ( T − T s i ) H ( T s i − T ) ![]()

E30
ε T r c = ∑ i = P , F , B , M ϕ i γ A i ![]()

E31
ε ( x , y , z , t ) = 0 f o r T > T S O L ![]()

E32

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]:

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

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]:

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

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

where * T* denotes solidus temperature,

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## 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 * C* = 670 J/(kg·K), density

The speed and power of the movement source are assumed to be * v* = 0.005 m/s and

The quantities * B* =

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 * T*,

The analysis of the cooling speed * v* 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

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.

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

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].

[1/°C] |
|||
---|---|---|---|

Austenite | 2.178 × 10^{−5} |
_{F,P,S→A} |
1.986 × 10^{−3} |

Ferrite | 1.534 × 10^{−5} |
γ_{B→A} |
1.440 × 10^{−3} |

Pearlite | 1.534 × 10^{−5} |
_{A→F,P} |
3.055 × 10^{−3} |

Bainite | 1.171 × 10^{−5} |
_{A→B} |
4.0 × 10^{−3} |

Martensite | 1.36 × 10^{−5} |

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

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

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## 6. Verification of the results of numerical simulation of phase transitions with the results of metallographic research

In order to verify the results of the numerical calculations, metallographic tests of the butt-welded 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.

Next, the images of structures of individual areas are presented. In the area of the weld (Figure 25), we observe the ferritic-pearlitic structure with a small amount of supercooled pearlite and island bainite in the dendritic system characteristic of solidified structures. In the heat-affected zone (Figure 26), ferrite, pearlite and pearlite balls supercooled with a number of non-metallic inclusions are visible. Primary structure of the parent material (Figure 27) consists of ferrite and pearlite in the band system (hardly visible).

The results of metallographic tests show high conformity with the results of numerical simulation and testify to the correctness of the developed numerical model.

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## 7. Conclusion

The proposed model for determining temperature field allows to obtain characteristic zones of the weld joint with shapes and dimensions similar to the real ones. Metallographic verification of the results of numerical simulation of phase transitions also provided satisfactory results.

Analysis of phase transformations occurring during the welding process allowed determination of the quantitative composition of the cooling structures in the weld area (in the heat-affected zone).

This approach allows for an accurate tracking of changes in phase participations and the course of thermal and phase strains, which enable to determine, within the framework of thermoplasticity theory, welding stresses temporary and residual welding stresses.

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Submitted: 05 November 2015 Reviewed: 28 April 2016 Published: 21 September 2016

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