Document [original]

metals

Article

Numerical Simulation on the Origin of Solidification

Cracking in Laser Welded Thick-Walled Structures

Nasim Bakir 1,2,*ID , Antoni Artinov 1, Andrey Gumenyuk 1, Marcel Bachmann 1ID and

Michael Rethmeier 1,2

1BAM Federal Institute for Material Research and Testing, 12205 Berlin, Germany;

[email protected] (A.A.); andrey[email protected] (A.G.); [email protected] (M.B.);

[email protected] (M.R.)

2Department of Joining Technology, Technische Universität Berlin, 10623 Berlin, Germany

*Correspondence: [email protected]; Tel.: +49-30-8104-4622

Received: 17 May 2018; Accepted: 29 May 2018; Published: 1 June 2018

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Abstract:

One of the main factors affecting the use of lasers in the industry for welding thick structures

is the process accompanying solidification cracks. These cracks mostly occurring along the welding

direction in the welding center, and strongly affect the safety of the welded components. In the

present study, to obtain a better understanding of the relation between the weld pool geometry, the

stress distribution and the solidification cracking, a three-dimensional computational fluid dynamic

(CFD) model was combined with a thermo-mechanical model. The CFD model was employed to

analyze the flow of the molten metal in the weld pool during the laser beam welding process. The

weld pool geometry estimated from the CFD model was used as a heat source in the thermal model

to calculate the temperature field and the stress development and distributions. The CFD results

showed a bulging region in the middle depth of the weld and two narrowing areas separating the

bulging region from the top and bottom surface. The thermo-mechanical simulations showed a

concentration of tension stresses, transversally and vertically, directly after the solidification during

cooling in the region of the solidification cracking.

Keywords:

laser beam welding; solidification cracking; numerical simulation; CFD model; finite

element method (FEM); weld pool; full penetration

1. Introduction

For several years, solid-state lasers have been widely applied in metal processing. The high-power

of the laser sources and the excellent beam quality allow structures with wall thicknesses of more than

10 mm to be welded in one pass. The high welding speed, low heat input, and low-distortion of the

laser beam welding are all advantages that significantly contribute to increasing productivity during

the fabrication of thick-walled constructions, and reducing rework [

]. However, when using the laser

to weld steels with thickness above 10 mm, the risk of solidification cracking of materials increases

due to carelessly coordinated process parameters and mechanical conditions.

Due to the complex nature of the hot cracking phenomena, many hypotheses and theories have

been presented in the past decades. Apblett and Pellini [

] assumed that solidification cracking first

occurs due to a critical strain. They believed that hot cracks occur at a temperature slightly above

the solidus temperature when a film of liquid is still present between the dendrite structures. The

hot cracks appear when the highly localized strains in the liquid films exceed their critical limit.

Prokhorov [

–

] suggested that the materials show a reduced deformation capacity in a specific

temperature range, known as the Brittle Temperature Range (BTR). If the strain during solidification

exceeds the deformation capacity, hot cracks will occur.

Metals 2018,8, 406; doi:10.3390/met8060406 www.mdpi.com/journal/metals

Metals 2018,8, 406 2 of 15

According to in situ observations of crack formation and numerical simulations, an approach

describing the functional relationship between the resulting stresses in the weld and the hot cracking

formation was presented by Zacharia et al. [

]. The researchers indicated that the transverse

compressive stress immediately behind the pool prevents the formation of hot cracks. The hot cracks

will form if a liquid film is present behind the weld pool, such as when the transverse stress changes

from compressive to tensile within the mushy zone (see Figure 1).

Metals 2018, 8, x FOR PEER REVIEW 2 of 15

According to in situ observations of crack formation and numerical simulations, an approach

describing the functional relationship between the resulting stresses in the weld and the hot cracking

formation was presented by Zacharia et al. [6,7]. The researchers indicated that the transverse

compressive stress immediately behind the pool prevents the formation of hot cracks. The hot cracks

will form if a liquid film is present behind the weld pool, such as when the transverse stress changes

from compressive to tensile within the mushy zone (see Figure 1).

Figure 1. Representation of the influence of the transverse stress development on the hot cracking,

according to [6].

Feurer [8] introduced a metallurgical model in which the ratio of the cooling-related volume

deformation, the so-called rate of shrinkage (ROS), and the rate of feeding (ROF) for closing the cavity

are considered. According to this theory, the critical point will be reached when ROF and ROS are

equal. In this condition, hot cracks can be avoided. In the case of a higher shrinkage than feeding rate,

hot cracks will occur.

In welding of thick-walled structures, the problematic solidification cracking becomes even

more complicated. Because of the complex weld pool shape, which is affected by the penetration

depth, its solidification behavior plays an additional role in the hot cracking phenomena.

Shida et al. [9] observed the change in the weld pool geometry, depending on penetration depth.

In their observations, they found a correlation between the weld pool shape and the solidification

cracking by studying the effect of the welding parameter of electron beam welding (EBW) on the

weld pool shape. However, it was reported that a small cavity in the rear part of the weld pool at

approximately one half of the penetration depth resulted from a certain focus position of the electron

beam. The observed cavity has been shown to have a significant influence on solidification cracking.

Locally delayed solidification in the electron beam welding was studied by Tsukamoto et al. in

[10,11]. A connection between the delay in the solidification, the formation of the solidification

cracking, and the porosity was identified.

Gebhardt et al. [12,13] observed the same phenomena in laser beam welding for thick-walled

structures, with the outcome being called bulging in the weld. Their numerical simulations showed

that the bulging has a significant influence on the temperature field and mechanical stress

distribution, which contributes to the formation of hot cracking.

Barbetta et al. [14] confirmed a high correlation between the bulging in the weld and the

occurrence of solidification cracks, as the solidification crack is always associated with a bulge. These

phenomena, bulging and hot cracking, are related by the delayed solidification at the rear part of the

weld pool.

Schaefer et al. [15] reduced the hot cracks formation in steel by modulation with laser power.

However, additional investigation is still needed to clarify the interaction between the weld pool

geometry, the thermally induced tensile stresses, and the strain in the weld zone.

Figure 1.

Representation of the influence of the transverse stress development on the hot cracking,

according to [6].

Feurer [

] introduced a metallurgical model in which the ratio of the cooling-related volume

deformation, the so-called rate of shrinkage (ROS), and the rate of feeding (ROF) for closing the cavity

are considered. According to this theory, the critical point will be reached when ROF and ROS are

equal. In this condition, hot cracks can be avoided. In the case of a higher shrinkage than feeding rate,

hot cracks will occur.

In welding of thick-walled structures, the problematic solidification cracking becomes even more

complicated. Because of the complex weld pool shape, which is affected by the penetration depth, its

solidification behavior plays an additional role in the hot cracking phenomena.

Shida et al. [

] observed the change in the weld pool geometry, depending on penetration depth.

In their observations, they found a correlation between the weld pool shape and the solidification

cracking by studying the effect of the welding parameter of electron beam welding (EBW) on the

weld pool shape. However, it was reported that a small cavity in the rear part of the weld pool at

approximately one half of the penetration depth resulted from a certain focus position of the electron

beam. The observed cavity has been shown to have a significant influence on solidification cracking.

Locally delayed solidification in the electron beam welding was studied by Tsukamoto et al.

in [

]. A connection between the delay in the solidification, the formation of the solidification

cracking, and the porosity was identified.

Gebhardt et al. [

] observed the same phenomena in laser beam welding for thick-walled

structures, with the outcome being called bulging in the weld. Their numerical simulations showed

that the bulging has a significant influence on the temperature field and mechanical stress distribution,

which contributes to the formation of hot cracking.

Barbetta et al. [

] confirmed a high correlation between the bulging in the weld and the occurrence

of solidification cracks, as the solidification crack is always associated with a bulge. These phenomena,

bulging and hot cracking, are related by the delayed solidification at the rear part of the weld pool.

Schaefer et al. [

] reduced the hot cracks formation in steel by modulation with laser power.

However, additional investigation is still needed to clarify the interaction between the weld pool

geometry, the thermally induced tensile stresses, and the strain in the weld zone.

Metals 2018,8, 406 3 of 15

The aim of this study is to use a computational fluid dynamic (CFD) simulation for investigation

and analysis of the weld pool shape for low-alloyed structural steel with a thickness of 12 mm to

observe the thermally deduced stresses by using a thermo-mechanical model in the weld zone and

analyze its influence on the solidification cracking.

2. Materials and Methods

Low-alloyed steel plates (S355-EN 10025) with 12 mm thickness were used in the welding

experiments. The chemical composition of the steel and the process parameters are listed in Tables 1

and 2, respectively. The joint partners were clamped to a device with known restraint intensity (see

Figure 2).

Table 1. Chemical composition (in wt. %).

C Si Mn P S Cr Cu Mo Fe

0.088 0.34 1.38 0.011 0.002 0.048 0.020 0.007 Bal.

Metals 2018, 8, x FOR PEER REVIEW 3 of 15

The aim of this study is to use a computational fluid dynamic (CFD) simulation for investigation

and analysis of the weld pool shape for low-alloyed structural steel with a thickness of 12 mm to

observe the thermally deduced stresses by using a thermo-mechanical model in the weld zone and

analyze its influence on the solidification cracking.

2. Materials and Methods

Low-alloyed steel plates (S355-EN 10025) with 12 mm thickness were used in the welding

experiments. The chemical composition of the steel and the process parameters are listed in Table 1

and Table 2, respectively. The joint partners were clamped to a device with known restraint intensity

(see Figure 2).

Table 1. Chemical composition (in wt. %).

0.088 0.34 1.38 0.011 0.002 0.048 0.020 0.007 Bal.

Figure 2. Schematic illustration of the experiments.

The concept of the intensity of restraint has been introduced by [16,17]. This represents the

stiffness of the surrounding structure around the weld and its influence on the stress development

during solidification. The total restraint intensity of the presented experimental construction before

welding was 12.6 kN/(mm·mm), estimated according to [18,19].

Table 2. Welding parameters used in the experiments.

Parameters

Value

Laser power

14 kW

Welding speed

2 m/min

Focal position

−2 mm

Weld seam length

100 mm

Plate thickness 12 mm

Shielding gas

Restraint intensity

12.6 kN/(mm·mm)

To study more deeply the effect of the weld pool shape on the temporal and spatial distribution

of the stresses, the model was divided into three simulation phases:

 CFD model: Calculation of the heat and mass transport in the weld pool to analyze the weld

pool shape.

Figure 2. Schematic illustration of the experiments.

The concept of the intensity of restraint has been introduced by [

]. This represents the

stiffness of the surrounding structure around the weld and its influence on the stress development

during solidification. The total restraint intensity of the presented experimental construction before

welding was 12.6 kN/(mm·mm), estimated according to [18,19].

Table 2. Welding parameters used in the experiments.

Parameters Value

Laser power 14 kW

Welding speed 2 m/min

Focal position −2 mm

Weld seam length 100 mm

Plate thickness 12 mm

Shielding gas N2

Restraint intensity 12.6 kN/(mm·mm)

To study more deeply the effect of the weld pool shape on the temporal and spatial distribution

of the stresses, the model was divided into three simulation phases:

Metals 2018,8, 406 4 of 15

•

CFD model: Calculation of the heat and mass transport in the weld pool to analyze the weld

pool shape.

•

Thermal FEM model: The CFD-obtained weld pool geometry is used as a heat source in the

calculation of the temperature field.

•

Mechanical FEM model: Using the calculated transient temperature field as a thermal load in the

mechanical model to evaluate the stress field.

2.1. CFD Model

The proposed mathematical model in this paper was implemented and solved with the commercial

Software ANSYS Fluent (ANSYS Inc., Canonsburg, PA, USA). In addition, the CFD simulation

of full-penetration keyhole laser beam welding was used to obtain the weld pool geometry by

considering the most relevant physical effects, such as Marangoni and natural convection, fusion

heat and temperature-dependent material properties up to evaporation temperature. The geometrical

dimensions of the computational domain were 40 mm

25 mm

12 mm (see Figure 3b). A symmetry

plane was applied to reduce the numerical effort and computational time while the computational

domain was discretized by a polygonal mesh of tetrahedral elements. The total number of mesh

elements was about 9

(see Figure 3b), allowing for a minimum element size of 0.08 mm at

the free surfaces and the keyhole wall to be used. Because the physical phenomena behind the laser

beam welding process are strongly coupled and temperature dependent, a highly nonlinear system of

equations must be solved to obtain a solution.

Metals 2018, 8, x FOR PEER REVIEW 4 of 15

 Thermal FEM model: The CFD-obtained weld pool geometry is used as a heat source in the

calculation of the temperature field.

 Mechanical FEM model: Using the calculated transient temperature field as a thermal load in

the mechanical model to evaluate the stress field.

2.1. CFD Model

The proposed mathematical model in this paper was implemented and solved with the

commercial Software ANSYS Fluent (ANSYS Inc., Canonsburg, PA, USA). In addition, the CFD

simulation of full-penetration keyhole laser beam welding was used to obtain the weld pool geometry

by considering the most relevant physical effects, such as Marangoni and natural convection, fusion

heat and temperature-dependent material properties up to evaporation temperature. The geometrical

dimensions of the computational domain were 40 mm × 25 mm × 12 mm (see Figure 3b). A symmetry

plane was applied to reduce the numerical effort and computational time while the computational

domain was discretized by a polygonal mesh of tetrahedral elements. The total number of mesh

elements was about 9 × 105 (see Figure 3b), allowing for a minimum element size of 0.08 mm at the

free surfaces and the keyhole wall to be used. Because the physical phenomena behind the laser beam

welding process are strongly coupled and temperature dependent, a highly nonlinear system of

equations must be solved to obtain a solution.

(a)

(b)

Figure 3. (a) Boundary condition of the computational fluid dynamic (CFD) model according to

[20,21]. (b) The model after meshing.

Here, a simplified form of the numerical model was used to guarantee numerical stability and

reasonable computing time. The main assumptions in the model were similar to those used in [22],

and are given as follows:

 Transient approach until reaching a quasi-steady-state.

 Adapted size of the computational domain.

 Fixed free surface geometry.

 Approximated simplified and fixed keyhole geometry from the weld cross-section (see Figure

4), used as a model parameter to adapt the numerical to the experimental results. Thus, effects

caused by keyhole oscillations were not considered.

 Shear stress due to the interaction of metal vapor and liquid metal was not considered.

 Heat losses by radiation were neglected due to the high relation of volume versus surface of the

plate.

Figure 3.

(

) Boundary condition of the computational fluid dynamic (CFD) model according to [

(b) The model after meshing.

Here, a simplified form of the numerical model was used to guarantee numerical stability and

reasonable computing time. The main assumptions in the model were similar to those used in [

and are given as follows:

•Transient approach until reaching a quasi-steady-state.

•Adapted size of the computational domain.

•Fixed free surface geometry.

•

Approximated simplified and fixed keyhole geometry from the weld cross-section (see Figure 4),

used as a model parameter to adapt the numerical to the experimental results. Thus, effects

caused by keyhole oscillations were not considered.

•Shear stress due to the interaction of metal vapor and liquid metal was not considered.

•

Heat losses by radiation were neglected due to the high relation of volume versus surface of

the plate.

Metals 2018,8, 406 5 of 15

Metals 2018, 8, x FOR PEER REVIEW 5 of 15

Figure 4. Cross-section of the weld and the used keyhole geometry in the CFD model.

The thermophysical properties of the low-alloyed steel used for the calculation can be taken from

[23]. The velocity, pressure and temperature fields of the incompressible flow were approximated by

the numerical solution of the mass, the momentum and the energy conservation equations by making

use of the simulation framework of ANSYS Fluent. The numerical setup including the geometry of

the workpiece, the initial state, and the boundary conditions can be seen in Figure 3a. Note, the heat

input was considered as a boundary condition at the keyhole surface by setting the temperature there

equal to evaporation temperature. A turbulent flow pattern, based on the high velocities on both

upper and lower sides—caused by the Marangoni-driven flow—and the influence of the keyhole

geometry on the flow, was considered by combining the Reynolds-Averaged-Navier–Stokes (RANS)

equations with the κ-ε turbulence model. Natural convection and buoyancy-driven flows due to

gravity were considered by the Boussinesq approximation and the enthalpy-porosity technique was

applied to simulate the solid-liquid phase transformation. The heat conductivity was modified by the

Kays and Crawford heat transport turbulence model and accounts for the amount of produced

turbulent heat conductivity. To consider the latent heat of fusion by the solid-liquid and liquid-solid

phase transformation, the method of apparent heat capacity was included.

2.2. Thermo-Mechanical Model

A plane strain two-dimensional model was employed to perform the thermo-mechanical

simulation. All out-of-plane strain components were neglected. The thermo-physical material

properties were taken from [22].

The stress-strain diagram was taken from the Sysweld material database (ESI Group, Paris,

France) [23] and the data was supplied for S355J2G3. The material was assumed to follow an elasto-

plastic law with isotropic hardening behaviour (von Mises plasticity model).

The phase transformation is also considered in the model. The material properties of all elements

reaching the austenite end temperature (AC3) changed during cooling to austenite. When austenite is

cooled to the martensite finish temperature (MF), the material properties of the elements changed to

martensite. Elements that do not reach the austenite start temperature (AC1) maintain the properties

of the base material. Another material model was used for the element, which reaches a temperature

above AC1 and below AC3. This model was composed of 50% of base material and martensite

properties below MF and 50% of base material and austenite properties above MF, until the martensite

start temperature (MS).

Figure 4. Cross-section of the weld and the used keyhole geometry in the CFD model.

The thermophysical properties of the low-alloyed steel used for the calculation can be taken

from [

]. The velocity, pressure and temperature fields of the incompressible flow were approximated

by the numerical solution of the mass, the momentum and the energy conservation equations by

making use of the simulation framework of ANSYS Fluent. The numerical setup including the

geometry of the workpiece, the initial state, and the boundary conditions can be seen in Figure 3a.

Note, the heat input was considered as a boundary condition at the keyhole surface by setting the

temperature there equal to evaporation temperature. A turbulent flow pattern, based on the high

velocities on both upper and lower sides—caused by the Marangoni-driven flow—and the influence of

the keyhole geometry on the flow, was considered by combining the Reynolds-Averaged-Navier–Stokes

(RANS) equations with the

turbulence model. Natural convection and buoyancy-driven flows due

to gravity were considered by the Boussinesq approximation and the enthalpy-porosity technique was

applied to simulate the solid-liquid phase transformation. The heat conductivity was modified by

the Kays and Crawford heat transport turbulence model and accounts for the amount of produced

turbulent heat conductivity. To consider the latent heat of fusion by the solid-liquid and liquid-solid

phase transformation, the method of apparent heat capacity was included.

2.2. Thermo-Mechanical Model

A plane strain two-dimensional model was employed to perform the thermo-mechanical

simulation. All out-of-plane strain components were neglected. The thermo-physical material

properties were taken from [22].

The stress-strain diagram was taken from the Sysweld material database (ESI Group, Paris,

France) [

] and the data was supplied for S355J2G3. The material was assumed to follow an

elasto-plastic law with isotropic hardening behaviour (von Mises plasticity model).

The phase transformation is also considered in the model. The material properties of all elements

reaching the austenite end temperature (A

) changed during cooling to austenite. When austenite is

cooled to the martensite finish temperature (M

), the material properties of the elements changed to

martensite. Elements that do not reach the austenite start temperature (A

) maintain the properties of

the base material. Another material model was used for the element, which reaches a temperature

above A

and below A

. This model was composed of 50% of base material and martensite

properties below M

and 50% of base material and austenite properties above M

until the martensite

start temperature (MS).

Metals 2018,8, 406 6 of 15

The clamping has been replaced in the model with one-dimensional spring elements that have

similar stiffness. Figure 5shows the dimensions and the applied boundary conditions used in the

thermo-mechanical model. This simplification has been used in [

]. The Birth and Death feature has

been used for all elements receiving an average temperature of 1440

◦

C. All these elements, reaching

the melting temperature, were “killed”. The stiffness matrix for these elements was multiplied by a

penalty factor and all strains, including the plastic strains, were eliminated. The computed weld pool

geometry from CFD simulation was used as a heat source to calculate the temperature field in the

thermal model.

Metals 2018, 8, x FOR PEER REVIEW 6 of 15

The clamping has been replaced in the model with one-dimensional spring elements that have

similar stiffness. Figure 5 shows the dimensions and the applied boundary conditions used in the

thermo-mechanical model. This simplification has been used in [24,25]. The Birth and Death feature

has been used for all elements receiving an average temperature of 1440 °C. All these elements,

reaching the melting temperature, were “killed”. The stiffness matrix for these elements was

multiplied by a penalty factor and all strains, including the plastic strains, were eliminated. The

computed weld pool geometry from CFD simulation was used as a heat source to calculate the

temperature field in the thermal model.

Figure 5. Boundary conditions used in the thermo-mechanical analysis.

3. Results and Discussion

Radiographic tests showed that cracks formed in the weld interior (see Figure 6a). In Figure 6b,

a micrograph of the weld cross-section presents a typical vertical solidification crack in the weld

centerline. It should be noted that small cracks that can be observed in the weld and the heat-affected

zone were not taken into consideration in this study. Moreover, the low-alloyed steel was not

susceptible to either liquation cracks or Ductility Dip Cracking (DDC), or cold cracks [26]. These

findings are similar to those reported by [25], where they welded 15 mm plate under similar restraint

conditions.

(

)

(

)

Figure 6. (a) Radiographic film of a welded sample. (b) Weld cross-section at the crack location.

Figure 5. Boundary conditions used in the thermo-mechanical analysis.

3. Results and Discussion

Radiographic tests showed that cracks formed in the weld interior (see Figure 6a). In Figure 6b,

a micrograph of the weld cross-section presents a typical vertical solidification crack in the weld

centerline. It should be noted that small cracks that can be observed in the weld and the heat-affected

zone were not taken into consideration in this study. Moreover, the low-alloyed steel was not

susceptible to either liquation cracks or Ductility Dip Cracking (DDC), or cold cracks [

]. These

findings are similar to those reported by [

], where they welded 15 mm plate under similar

restraint conditions.

Metals 2018, 8, x FOR PEER REVIEW 6 of 15

The clamping has been replaced in the model with one-dimensional spring elements that have

similar stiffness. Figure 5 shows the dimensions and the applied boundary conditions used in the

thermo-mechanical model. This simplification has been used in [24,25]. The Birth and Death feature

has been used for all elements receiving an average temperature of 1440 °C. All these elements,

reaching the melting temperature, were “killed”. The stiffness matrix for these elements was

multiplied by a penalty factor and all strains, including the plastic strains, were eliminated. The

computed weld pool geometry from CFD simulation was used as a heat source to calculate the

temperature field in the thermal model.

Figure 5. Boundary conditions used in the thermo-mechanical analysis.

3. Results and Discussion

Radiographic tests showed that cracks formed in the weld interior (see Figure 6a). In Figure 6b,

a micrograph of the weld cross-section presents a typical vertical solidification crack in the weld

centerline. It should be noted that small cracks that can be observed in the weld and the heat-affected

zone were not taken into consideration in this study. Moreover, the low-alloyed steel was not

susceptible to either liquation cracks or Ductility Dip Cracking (DDC), or cold cracks [26]. These

findings are similar to those reported by [25], where they welded 15 mm plate under similar restraint

conditions.

(

)

(

)

Figure 6. (a) Radiographic film of a welded sample. (b) Weld cross-section at the crack location.

Metals 2018,8, 406 7 of 15

However, such cracks are related to a high weld depth-to-width ratio. This ratio leads to a

pronounced intersection of columnar grains, along with a plane at the weld centerline and the

accumulation of segregations in this zone. It is believed that if the weld metal is unable to accommodate

the contractional strains of solidification and cooling, hot cracks will inevitably originate.

However, the experiments were conducted to create appropriate conditions for solidification

cracking initiation. Those boundary conditions and the results of the crack location, as well as the

cross-section and the temperature measurement were used for the calibration of the numerical models.

The aim of the CFD model was to predict the geometry of the quasi-steady-state weld pool in

the stable zone of the weld. The three-dimensional weld pool geometry is defined by the liquidus

temperature (see Figure 7a), and part of this isotherm can be seen in the symmetry plane of the CFD

model (see Figure 7b). In Figure 7, three regions of the weld pool can be recognized. The weld pool

shape is strongly influenced by the Marangoni convection in the upper and lower sides of the part.

Together with the movement of the laser source, three regions are developed: an upper, a bulging

(middle), and a lower region. The observed velocities of the melt in the bulging region were very

small compared to the values in the upper and lower regions. Hence, the edge of the middle region is

nearly a parallel shift of the keyhole edge, which forms together with the two backflows due to the

thermo-capillary-driven flow the observed bulge.

The quasi-steady-state weld pool represents the state of a mass and energy equilibrium defined

by the moving energy source and the solidification speed at the trailing part of the weld pool. Due to

the movement of the energy source along the partly cold material, new material is melted and added

to the weld pool. This is then transferred by the different flows in the molten pool. Simultaneously,

through the heat transfer, the same amount of molten material solidifies at the trailing part of the weld.

In the present model, the flows in the upper and lower regions dominate and transfer cooler material

back to the vicinity of the keyhole, ensuring mass conservation. The two necking areas are formed and,

consequently, also the bulge (see Figure 7).

Metals 2018, 8, x FOR PEER REVIEW 7 of 15

However, such cracks are related to a high weld depth-to-width ratio. This ratio leads to a

pronounced intersection of columnar grains, along with a plane at the weld centerline and the

accumulation of segregations in this zone. It is believed that if the weld metal is unable to

accommodate the contractional strains of solidification and cooling, hot cracks will inevitably

originate.

However, the experiments were conducted to create appropriate conditions for solidification

cracking initiation. Those boundary conditions and the results of the crack location, as well as the

cross-section and the temperature measurement were used for the calibration of the numerical

models.

The aim of the CFD model was to predict the geometry of the quasi-steady-state weld pool in

the stable zone of the weld. The three-dimensional weld pool geometry is defined by the liquidus

temperature (see Figure 7a), and part of this isotherm can be seen in the symmetry plane of the CFD

model (see Figure 7b). In Figure 7, three regions of the weld pool can be recognized. The weld pool

shape is strongly influenced by the Marangoni convection in the upper and lower sides of the part.

Together with the movement of the laser source, three regions are developed: an upper, a bulging

(middle), and a lower region. The observed velocities of the melt in the bulging region were very

small compared to the values in the upper and lower regions. Hence, the edge of the middle region

is nearly a parallel shift of the keyhole edge, which forms together with the two backflows due to the

thermo-capillary-driven flow the observed bulge.

The quasi-steady-state weld pool represents the state of a mass and energy equilibrium defined

by the moving energy source and the solidification speed at the trailing part of the weld pool. Due to

the movement of the energy source along the partly cold material, new material is melted and added

to the weld pool. This is then transferred by the different flows in the molten pool. Simultaneously,

through the heat transfer, the same amount of molten material solidifies at the trailing part of the

weld. In the present model, the flows in the upper and lower regions dominate and transfer cooler

material back to the vicinity of the keyhole, ensuring mass conservation. The two necking areas are

formed and, consequently, also the bulge (see Figure 7).

(

)

(

)

Figure 7. (a) The weld pool geometry defined by the liquidus temperature. (b) Temperature field and

velocity streamlines in the symmetry plane of the model.

The diameter of the keyhole was used as a parameter to achieve a good match between the

experiments and the model. This parameter was adjusted until an error of less than 5% was obtained.

The fused zone, highlighted with dashed lines, agrees well with the experiment (see Figure 8).

Figure 7.

(

) The weld pool geometry defined by the liquidus temperature. (

) Temperature field and

velocity streamlines in the symmetry plane of the model.

The diameter of the keyhole was used as a parameter to achieve a good match between the

experiments and the model. This parameter was adjusted until an error of less than 5% was obtained.

The fused zone, highlighted with dashed lines, agrees well with the experiment (see Figure 8).

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Figure 8. Comparison of the fused zone between experiment and CFD model.

Figure 9 shows the temperature measurements over time using thermocouples on the top and

the bottom of the specimen surface compared to measurements obtained from calculations.

Figure 9. Comparison of the temperature measurements between experiments and the computational

model.

To use the computed weld pool geometry as a heat source in the two-dimensional model, it was

assumed that the temperature within the melt isotherm is homogeneous, constant, and equal to

melting temperature (1440 °C). Then, the weld pool simply consists of multiple layers of two

combined half-ellipses, as shown schematically in Figure 10.

Figure 8. Comparison of the fused zone between experiment and CFD model.

Figure 9shows the temperature measurements over time using thermocouples on the top and the

bottom of the specimen surface compared to measurements obtained from calculations.

Metals 2018, 8, x FOR PEER REVIEW 8 of 15

Figure 8. Comparison of the fused zone between experiment and CFD model.

Figure 9 shows the temperature measurements over time using thermocouples on the top and

the bottom of the specimen surface compared to measurements obtained from calculations.

Figure 9. Comparison of the temperature measurements between experiments and the computational

model.

To use the computed weld pool geometry as a heat source in the two-dimensional model, it was

assumed that the temperature within the melt isotherm is homogeneous, constant, and equal to

melting temperature (1440 °C). Then, the weld pool simply consists of multiple layers of two

combined half-ellipses, as shown schematically in Figure 10.

Figure 9.

Comparison of the temperature measurements between experiments and the

computational model.

To use the computed weld pool geometry as a heat source in the two-dimensional model, it

was assumed that the temperature within the melt isotherm is homogeneous, constant, and equal to

melting temperature (1440

◦

C). Then, the weld pool simply consists of multiple layers of two combined

half-ellipses, as shown schematically in Figure 10.

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Metals 2018, 8, x FOR PEER REVIEW 9 of 15

Figure 10. Schematic representation of the weld pool geometry and the parameters of two combined

half-ellipses.

The equation of the combination of two half-ellipses can be given as flowing:









(





)









(



)

where:



(





)



󰇥







≤







The front part of the combination of the ellipses is defined by parameters b and c1 and the rear

of the ellipse is defined by b and c2. The parameter c is a function of the weld pool length (x-

coordinate) and the weld pool depth (z-coordinate), and the parameter b is a function of the weld

pool depth.

The parameters c1, c2, and b were fitted with fourth-degree polynomial function (see Figure 11).

The obtained quartic functions were employed for the calculation of those parameters in the thermal

model.

The developed heat source will check at each time step, whether the nodes are located inside

one of the ellipses, and if so, the temperature will be changed to the melt temperature. If not, the

temperature will be maintained.

Figure 12 shows a comparison of the temperature field between calculations, using the CFD

model on the right and using the developed heat source, which was previously described, on the

opposite side of the temperature field. A very slight difference can be observed between fusion lines.

This difference is a result of the fitting of the heat source parameters. However, the results of this

technique showed high efficiency in the transfer of computed weld pool geometry, which cannot be

achieved by using common heat sources, such as the conical Gaussian heat source model [27], the

cylindrical heat source model [28], or the combination of both [29,30].

Figure 10.

Schematic representation of the weld pool geometry and the parameters of two

combined half-ellipses.

The equation of the combination of two half-ellipses can be given as flowing:

c2(x,z)+y2

b2(z)=1

where:

c(x,z)=(c=c1 if x≤0

c=c2 if x>0

The front part of the combination of the ellipses is defined by parameters b and c1 and the rear of

the ellipse is defined by band c2. The parameter cis a function of the weld pool length (x-coordinate)

and the weld pool depth (z-coordinate), and the parameter b is a function of the weld pool depth.

The parameters c1, c2, and bwere fitted with fourth-degree polynomial function (see Figure 11).

The obtained quartic functions were employed for the calculation of those parameters in the

thermal model.

The developed heat source will check at each time step, whether the nodes are located inside

one of the ellipses, and if so, the temperature will be changed to the melt temperature. If not, the

temperature will be maintained.

Figure 12 shows a comparison of the temperature field between calculations, using the CFD model

on the right and using the developed heat source, which was previously described, on the opposite

side of the temperature field. A very slight difference can be observed between fusion lines. This

difference is a result of the fitting of the heat source parameters. However, the results of this technique

showed high efficiency in the transfer of computed weld pool geometry, which cannot be achieved by

using common heat sources, such as the conical Gaussian heat source model [

], the cylindrical heat

source model [28], or the combination of both [29,30].

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Metals 2018, 8, x FOR PEER REVIEW 10 of 15

Figure 11. The fitting of the parameters c1, c2 and b as a function of the weld depth.

Figure 12. Comparison of the temperature fields of the CFD and the thermal model.

These heat source models were mainly developed based on the cross-section of the weld,

without considering the weld pool in the longitudinal section. However, this fulfills the purpose of

the calculation of residual stresses and the distortions.

The transversal stress distribution and the temperature distribution of approximately 230 ms

after complete solidification of the melt (i.e., at 1100 °C in the middle of the weld) are shown in Figure

13. A concentration of the tensile stress can be observed in the middle of the weld in the bulging

Figure 11. The fitting of the parameters c1, c2 and bas a function of the weld depth.

Metals 2018, 8, x FOR PEER REVIEW 10 of 15

Figure 11. The fitting of the parameters c1, c2 and b as a function of the weld depth.

Figure 12. Comparison of the temperature fields of the CFD and the thermal model.

These heat source models were mainly developed based on the cross-section of the weld,

without considering the weld pool in the longitudinal section. However, this fulfills the purpose of

the calculation of residual stresses and the distortions.

The transversal stress distribution and the temperature distribution of approximately 230 ms

after complete solidification of the melt (i.e., at 1100 °C in the middle of the weld) are shown in Figure

13. A concentration of the tensile stress can be observed in the middle of the weld in the bulging

Figure 12. Comparison of the temperature fields of the CFD and the thermal model.

These heat source models were mainly developed based on the cross-section of the weld, without

considering the weld pool in the longitudinal section. However, this fulfills the purpose of the

calculation of residual stresses and the distortions.

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The transversal stress distribution and the temperature distribution of approximately 230 ms after

complete solidification of the melt (i.e., at 1100

◦

C in the middle of the weld) are shown in Figure 13.

A concentration of the tensile stress can be observed in the middle of the weld in the bulging region.

Regions subject to high compressive stresses can be observed at the top and bottom of this region. The

same observation can be found for the vertical stress (see Figure 14).

Metals 2018, 8, x FOR PEER REVIEW 11 of 15

region. Regions subject to high compressive stresses can be observed at the top and bottom of this

region. The same observation can be found for the vertical stress (see Figure 14).

Figure 13. Transversal stress distribution and temperature distribution after solidification of the weld

pool.

Figure 14. Vertical stress distribution and corresponding temperature distribution after solidification

of the weld pool.

To closely evaluate the observed stress distribution at this moment, three evaluation points were

chosen, which were positioned below each other. P1 represents the upper narrowing region, P2 the

bulging region, and P3 the lower narrowing region (see Figure 13). Figure 15 and Figure 16 indicate

the transversal and vertical stress evolution versus the time during cooling at the locations P1, P2,

and P3. On the narrowing regions (P1 and P3), the transversal stresses developed in a very similar

manner because immediately after solidification, tensile stresses appeared for a short period and soon

became compressive stresses until reaching the value of −30 MPa. Nevertheless, the vertical stresses

in all regions show tensile development.

Figure 13.

Transversal stress distribution and temperature distribution after solidification of the

weld pool.

Metals 2018, 8, x FOR PEER REVIEW 11 of 15

region. Regions subject to high compressive stresses can be observed at the top and bottom of this

region. The same observation can be found for the vertical stress (see Figure 14).

Figure 13. Transversal stress distribution and temperature distribution after solidification of the weld

pool.

Figure 14. Vertical stress distribution and corresponding temperature distribution after solidification

of the weld pool.

To closely evaluate the observed stress distribution at this moment, three evaluation points were

chosen, which were positioned below each other. P1 represents the upper narrowing region, P2 the

bulging region, and P3 the lower narrowing region (see Figure 13). Figure 15 and Figure 16 indicate

the transversal and vertical stress evolution versus the time during cooling at the locations P1, P2,

and P3. On the narrowing regions (P1 and P3), the transversal stresses developed in a very similar

manner because immediately after solidification, tensile stresses appeared for a short period and soon

became compressive stresses until reaching the value of −30 MPa. Nevertheless, the vertical stresses

in all regions show tensile development.

Figure 14.

Vertical stress distribution and corresponding temperature distribution after solidification

of the weld pool.

To closely evaluate the observed stress distribution at this moment, three evaluation points were

chosen, which were positioned below each other. P1 represents the upper narrowing region, P2 the

bulging region, and P3 the lower narrowing region (see Figure 13). Figures 15 and 16 indicate the

transversal and vertical stress evolution versus the time during cooling at the locations P1, P2, and P3.

On the narrowing regions (P1 and P3), the transversal stresses developed in a very similar manner

Metals 2018,8, 406 12 of 15

because immediately after solidification, tensile stresses appeared for a short period and soon became

compressive stresses until reaching the value of

−

30 MPa. Nevertheless, the vertical stresses in all

regions show tensile development.

Metals 2018, 8, x FOR PEER REVIEW 12 of 15

Figure 15. Transversal stress evaluation on the points P1, P2, and P3.

Figure 16. Vertical stress evaluation on the points P1, P2, and P3.

In the bulging region (P2), transversal and vertical tensile stresses development can be observed

immediately after solidification, continuing to rise until reaching a value of 80 MPa and 110 MPa,

respectively. The vertical tensile stress along the weld depth can be returned to the restraint of the

plate to the vertical shrinkage of the weld. However, since the transversal stresses have an opposite

direction to the dendrites growth, the stresses are held responsible for the formation of these cracks.

As previously observed, the chronological order of the solidification of the weld cannot be

separated from the nature and distribution of the stresses in the weld zone. To explain this mutual

interaction in the weld during cooling, Figure 17 schematically shows the melt and the solid

distribution in a considered plane orthogonal to the welding direction, while the weld pool passes

through it, at three consecutive moments of time. Moment T1 represents the section on the widest

location of the weld pool crossing the plane. Until this moment, the weld region is heated from room

temperature to melt temperature, and the weld volume expands. This expansion is restrained by the

surrounding cold material, in addition to the restraint of the clamping causing compressive stress in

the surrounding material. In this moment, no stresses have taken place in the weld zone because the

melt has nil strength [2]. It must be noted that the stress resulting from the thermal strains during

solidification is relieved as long as the connection to the melt is available.

As the weld pool moves slightly forward there are still melted regions within the bulging top

and bottom parts, whereas the narrowing regions solidify between them (T2). As long as these liquid

regions have access to the melt or to the top and bottom free surfaces, the resulting thermal stresses

can be relieved. This corresponds to the observations at points P1 and P3.