1. Introduction
Electron beam welding (EBW) stands out as a fusion welding technique utilizing a concentrated beam of high-velocity electrons focused on a narrow area of the materials being joined. As the electrons strike the surfaces, their kinetic energy transforms into thermal energy, causing the materials to melt and subsequently solidify to form a weld joint1). This method sets a benchmark for achieving superior quality and efficiency in joining components or materials commonly found in engineering sectors. Notably, EBW excels in producing high- quality joints, particularly in thicker components, owing to its high energy density. The intense energy of the electron beam vaporizes the material, creating a keyhole in the molten pool. Material flow within this pool, guided by the electron beam, effectively closes gaps through a coalescence process, ultimately forming the weld joint2). The distinct advantage of EBW lies in its ability to achieve robust welds with lower energy input and higher welding speeds. This reduced energy input translates to smaller heat-affected zones (HAZ) and minimal distortions, a notable contrast to processes with higher heat inputs like arc welding1).
Welding of stainless steels leads to issues such as sensitization and intergranular stress corrosion cracking. Sensitization occurs when stainless steel is exposed to temperatures between 450°C and 850°C, causing chromium carbide precipitation along grain boundaries, which depletes chromium and reduces corrosion resistance. Arc welding processes like gas tungsten arc welding (GTAW) and gas metal arc welding (GMAW) can exacerbate this due to HAZ and prolonged exposure to sensitization-prone temperatures3). In contrast, EBW with its highly localized heat input and rapid cooling minimizes the risk of sensitization, which is beneficial in preventing intergranular corrosion. Furthermore, EBW is typically conducted in a vacuum environment, which shields against oxidation, porosity, and other forms of contamination, ensuring pristine weld quality4).
Electron beam welding (EBW) is a highly efficient and precise welding method, increasingly adopted within the manufacturing chain and growing in importance across various industries such as aeronautical and aerospace sectors4-7). Currently, suitable welding parameters are typically defined based on experience and extensive testing trials on actual or similar materials and components. However, numerical simulations using finite element method (FEM) of the weld joint can significantly reduce the need for such testing trials, thereby saving time and cost associated with the welding process.
To perform a finite element based numerical analysis of EB welding, various parameters like, efficiency (ɳeff), and geometric parameter (top diameter dt, and bottom diameter db and depth of penetration H) of heat source model (HSM) are required as input. The details of the HSM are explained in Section 2. The selection of these parameters for analyzing EB weld joints is not straightforward, as these parameters depends up on material properties of work piece and welding input parameters like (accelerating voltage, beam current and focusing current and, beam focus quality). Therefore, calibration of the heat source parameters is essential for accurate analysis.
Many researchers use trial-and-error methods to calibrate these parameters based on measurements of the solidified fusion zone post-welding. Goncharov et al.8) and Kar et al.9) attempted to analytically predict the approximate shape of the fusion zone using heat transfer calculations. Giedt et al.10) developed a semi-empirical, dimensionless correlation to predict beam depth of penetration by incorporating input power, material thermal properties, welding velocity, and fusion zone geometry. However, Giedt et al.10) correlation relies on the width of the fusion zone to estimate penetration depth, introducing certain limitations. Koleva et al.11) predicted weld pool depth and width using a statistical model based on experimental data. Similarly, Kar et al.9) estimated the average beam spot diameter and bead depth of penetration for a parabolic HSM. The correlations provided by Koleva et al.11) and Kar et al.9) were obtained by minimizing the difference between predicted and experimental weld pool depths. These correlations can be useful for predicting beam depth of penetration when experimental data for a specific type of weld joint is unavailable. However, the challenge lies in establishing explicit relationships between geometric parameters (dt, db and H) and welding input parameters (such as accelerating voltage and beam current), in order to systematically calibrate the HSM parameters.
Most of the existing literature on the FEA used for EB welding focuses on thin-section components12,13). In contrast, many engineering components, particularly those in nuclear power plants where safety is paramount, involve thick sections that require a joint with minimal distortion and residual stresses. FEA of thick- section EBW encounter several challenges due to the complex physics involved in the process. For thin plates, the welding process is characterized by material melting uniformly within the welding pool throughout the thickness of the joint, resulting in a relatively consistent weld pool size through the thickness of the plates. Conversely, for thicker sections requiring deeper penetration, the high-power density of the electron beam leads to material vaporization, creating a “keyhole” that extends through the thickness of the specimen. The keyhole formation leads to taper weld bead geometry in thick section EB weld joint12).
In present study, we propose a heat source model based on the relation provided by Zuev14) and Goncharov et al.8) and its calibration of geometric parameters for welding of thick section components using EBW. This is aimed at addressing one of the key challenges encountered in finite element-based numerical simulations of EBW: the selection and calibration of the volumetric HSM. Austenitic stainless-steel plate (12 mm thick) weld joint in butt joint configuration has been used for calibration of HSM geometric parameters. To validate the calibrated heat source model parameters, bead-on-plate numerical simulations are performed, and the results regarding weld pool size are compared with experimental data. Subsequently, numerical analysis is employed to predict residual stress and displacement in welded plates, and these predictions are compared with existing literature on conventional weld joints. Through this approach, we aim to enhance the accuracy and reliability of numerical simulations in EBW processes.
2. Background of Heat Sources
The dynamic behaviour of electron beams, including their interaction with the work piece necessitates sophisticated volumetric HSM which should capture the complex heat transfer phenomena and phase changes occurring during welding.
Electron beam trajectories and their focusing play important role in determining the volumetric heat distribution and depth of penetration in the EBW of thick section components. The total heat injected into the work-piece during the welding process is defined by equation (1).
Where qeff is effective input power and Q(x,y,z) is the distribution function depending on the cartesian coordinates x, y and z of a point of the workpiece in 3D Space.
The kinetic energy of the electron beam gets converted into heat energy thereby melting the work- pieces. However, when the electron beam strikes the work-piece surface, it interacts with the electrons of the material leading to the emission of backscattered electrons (reflected), secondary electrons, and X-rays15) which causes loss of energy. Due to these losses, the efficiency of electron beam-based devices typically lies between 90-95%16). To account for this heat loss, heat absorption efficiency is considered, and the effective input power is given by equation (2)8).
Where Vacc and Ib are the accelerating voltage and the beam current, respectively and ɳeff is the heat absorption efficiency.
Various existing HSMs like, conical HSM with uniform power distribution17), a moving heat source with an ellipsoidal shape in the transverse cross-section18), a combination of surface and cylindrical volume heat sources model19), and a conical heat source with a Gaussian power distribution20) have been used by researchers for the heat distribution in EBW.
Among the various existing HSMs three-dimensional moving conical heat source with Gaussian distribution of power is widely used in the numerical simulation of EBW. This heat source includes the two parts, i. e., one is the Gaussian distribution radically which can give the good thermal distribution, another is linear distribution axially which can accurately describe the deep penetration during welding simulation. The mathematical model of power distribution is given by equation (3)20).
Where d 0 ( z ) = d t + d b − d t z b − z t ( z − z t )
In EBW, the size of keyhole formed in the material depends on various input parameters such as accelerating voltage (Vacc), beam current (Ib), focusing current (If), work distance, thermal properties of the material, welding speed, and more. Although these variables are not directly used in numerical analysis, but geometric parameters of HSM depends on these welding para- meters.
3. Modelling for Heat Source Calibration
Numerical simulation of EB welding of thick section components is not straight forward as arc welding. In arc welding of thick section, multiple beads are required to form joint therefore depth of penetration per bead is not important. However, in EBW, single bead forms the joint and therefore depth of penetration is very important. In arc welding, energy per unit length is used as an input parameter since the bead size is known in advance. In EB welding, quantification of energy per unit length or energy density requires estimation of the penetration depth achieved by the beam for a given geometry. All three-geometric parameters (dt, db and H) are inter related and sensitive to change in welding process parameters. For example, changing the beam spot diameter will lead to changes in depth of penetration and bottom diameter for a given beam power and work-piece. Hence, an interrelation between these geometric parameters in terms of welding process parameters is a needed.
Analytical and empirical relations have been developed for calculating the beam current to achieve the required depth of penetration14,22-24). Zuev14) provided the analytical correlation (equation (4)) for calculating the beam current necessary to achieve the desired depth of penetration in continuous welding based on the energy balance in the melting zone.
In equation (4), Vw represents the welding speed (m/s), B average weld width, H depth of penetration and ηt represents thermal efficiency (indicating the proportion of input heat energy used in melting the material). In present analyses, an average value ηt =55% has been used8).
Sm represent the heat capacity (J/m3) corresponding to the melting temperature, calculated by the equation (5).
Where, ρ represents metal density (kg/m3); C represents specific heat capacity of metal (J/(kg·K); ΔTm = (Tm - To) i.e. increase from the initial temperature (To) to the melting point (Tm); Lm - specific heat of fusion(J/kg). For stainless steel, a value of Sm = 8.915 J/mm3 has been used8).
The cross-sectional area of the weld zone (Fm = B.H) can be obtained by equation (6) which is based on the effective input power (qeff) and welding velocity (VW)8),
Solving equations (2, 4 and 6), the top diameter of the conical heat source, dt can be expressed as equation (7),
There is a practical difficulty in use of Zuev14) model for determination of welding current because it requires mean weld width, depth of penetration and thermal efficiency of welding processes in advance. However, it provides a fair estimate of relation between beam spot diameter and beam depth of penetration assuming parabolic shape of heat source distribution. A relation between dt and db can be developed by mapping the truncated conical geometry (as used in equation (3)) with the parabolic heat source geometry. The mapping is achieved by equating the respective volumes of the heat source with the same top diameter, thereby eliminating the variable parameter db. The relation between the top and bottom diameters of the heat source is found to be as given by equation (8),
The parameters of conical hear source model (equation (3)) can be easily calibrated with the use of equations (7) and (8) for the numerical simulation of EB welding. Now, only two variables (dt and ηeff) need to be adjusted for a given set of EBW parameters to calibrate the HSM parameters. Multiple numerical analyses need to be performed by systematically varying the dt and ηeff to achieve this calibration.
4. Experimental Details
Two plates of size 87 mm in length and 50 mm in width and 12 mm thickness made of austenitic stainless-steel Type 304L have been joined using EBW process with fully clamped conditions. The nominal chemical compositions of the plate material are given in Table 1.
Table 1
Nominal chemical composition of plate materials (in wt. %)
| Element | C | Mn | S | P | Si | Cr | Ni | Fe |
|---|---|---|---|---|---|---|---|---|
| Plate | 0.03 max | 2.00 max | 0.03 max | 0.045 max | 0.75 max | 18.00-20.00 | 8.00-12.00 | Bal. |
The plates have been cleaned to remove oil, oxide scale, or other contaminants before assembling and welding. An accurate fit-up of the plates in butt joint configuration with clearance of less than 0.1 mm along the joining length have been ensured. The ‘run-on’ and ‘run-off’ tabs/plates are placed at the beginning and end of welding. The run-on tab allows the welding process to stabilize before it enters the main weld area, while the run-off tab ensures that the weld pool has fully solidified before the beam is stopped. These tabs have been removed after the welding. K-Type Therm- ocouples with a sensitivity of 41 μV/°C and an accuracy of ±2°C were affixed at the top surface of the plate near the weld to monitor temperature variations during welding. These thermocouples were strategically positioned at distances of 2 mm, 3 mm, 4 mm and 6 mm from the weld centerline. The butt weld joint configuration and the positions of the thermocouples relative to the weld centerline are illustrated in Fig. 2(a). The plate weld joint with thermocouples installed on it is shown in Fig. 2(b). The welding parameters used for joining the plates are detailed in Table 2.
Fig. 2
(a) Schematic of thermocouple locations with respect to weld center (b) Plate weld joint with thermocouples
Table 2
Parameters for plate welding
| Accelerating voltage (KV) | Beam current (mA) | Welding speed (mm/min) | Heat input (J/mm) |
|---|---|---|---|
| 76 | 85 | 1000 | 387.6 |
The temperature profiles recorded during the welding of plates at various locations from the weld centerline are illustrated in Fig. 3. The temperature gradually increased and reaching its peak value when the beam was closest to the thermocouple. Subsequently, the temperature to gradually decrease to ambient levels. It has been observed that the maximum temperature recorded by the thermocouple positioned at 2 mm from the weld centerline reached 950°C.
The welded plate has been sectioned across its thickness using a wire-cutting machine after completion of welding. The cross section of weld has been subjected to chemical etching with a solution composed of distilled water, nitric acid, and hydrochloric acid to reveal the weld pool size. Visual examination of the cross-section of the etched weld showed full penetration in the thickness direction, as depicted in Fig. 4. However, the width of the weld pool varied slightly across the thickness, displaying a teardrop shape near the upper surface and a bulge region at some depth. The average width of the weld pool was measured to be 1.7 mm, with a corresponding weld area of 20.4 mm², assuming the shape of the weld pool to be truncated conical. Notably, the width of the weld pool was observed to be maximum at the top and minimum at the bottom of the joint.
Additionally, bead-on-plate experiments have been performed to validate the developed correlation and heat source model, using various accelerating voltages, beam currents, and constant speeds. These experiments involved depositing a single bead with different welding parameters, as outlined in Table 3, onto a 12 mm thick plate composed of austenitic stainless-steel Type 304. Four numbers of beads were deposited on the plate, as illustrated in Fig. 5.
Table 3
Parameters of EB welding for bead on plate
The beads on the plate were sectioned across their thickness and chemically etched to identify the weld pool region. The weld pool sizes of all the beads are depicted in Fig. 6.
5. Numerical Simulations
The numerical simulations were performed for both the plate weld joint and the beads on plates using FE-based software. A transient coupled thermal and mechanical analysis was performed. The simulation process involved several steps, including the creation of the geometry of the plate to be welded, modeling and mesh generation, inputting material properties, selecting the heat source model, defining boundary and clamping conditions, specifying the weld path and reference line, among others. The temperature-dependent thermal and mechanical properties of the austenitic stainless steel used in the analysis are provided in Fig. 7 and 8.
The solidus and liquidus temperatures of the stainless steel used are 1400°C and 1455°C, respectively. The melting temperature, which is taken as the average of the solidus and liquidus temperatures, is 1428°C. Additionally, the latent heat of fusion is 268 kJ/kg14).
For the thermal analysis, an ambient temperature of 25°C was used. The temperature-dependent effective heat transfer coefficients for convection and radiation, heff , have been calculated using equation (9):
Where,
ε = Emissivity of material (taken as ε =0.8 for stainless-steel25)).
σ = Stefan’s Boltzmann’s Constant.
T = Actual temperature of element (K)
T∞ = Ambient Temperature (K)
h = heat transfer coefficient of convection (0.01 W/m2k for vacuum environment)
5.1 Numerical simulations of plate welding
The finite element mesh used for modeling of the plate weld joint, with dimensions of 87 mm in length, 50 mm in width, and 12 mm in thickness, is illustrated in Fig. 9. The final mesh configuration has been determined through a mesh convergence study involving various element sizes. A fine mesh with dimensions of 0.4 mm × 0.4 mm has been used in regions proximal to the weld pool and HAZ, while a coarser mesh measuring 5 mm × 5 mm in areas farther from the weld to minimize computational time. 3D quad-tetra elements have been used finite element model. Isotropic hardening material model has been used in mechanical analysis.
In the numerical simulation, a minimum boundary condition was employed to prevent rigid body motion. Specifically, the nodes across the thickness on one end surface of the plate were fully fixed (i.e., Ux=Uy=Uz=0). At the other end surface of the plate, fully clamped boundary conditions (Ux=Uy=Uz=0) were applied, consistent with the experimental conditions for plate welding. The mechanical boundary conditions utilized in plate welding are depicted in Fig. 10. For the bead- on-plate scenario, free boundary conditions were applied on the other end.
5.2 Calibration of heat source model
A conical HSM (represented by equation (3)) along with correlations given by equations (7) and (8) has been considered for numerical simulation of EBW. Here, two variables (dt and ηeff) need to be adjusted for a given set of EBW parameters of plate weld joint to calibrate the HSM parameters. Multiple numerical analyses have been performed by systematically varying the dt and ηeff to achieve this calibration. The dt and ηeff varied in the range of 1.2-1.6 mm and 80-95% respectively. The HSM parameters have been calibrated by comparing the numerical analyses and experimental results for the temperature profile and weld pool sizes of the plate weld joint. The numerical analyses and experimental results for temperature profiles and weld pool sizes have been compared for various dt and ηeff values, as well as their combinations. Specifically, the temperature profile at 2 mm from the weld center line was extracted from the numerical analyses for different dt and ηeff values and compared with experimental results. These comparisons are depicted in Fig. 11 to 13.
The comparison of temperature profiles in Fig. 11 shows that the cooling curve does not match the experimental data for dt = 1.2 mm and dt= 1.3 mm. Fig. 12 presents the temperature profile corresponding to dt = 1.4 mm. The temperatures are lower than the experimentally measured values for both ηeff = 85 % and ηeff = 85%. The cooling curve for ηeff = 90% nearly matches the experimental data, except for the peak temperature. The peak temperature cannot be used for comparison because the thermocouple is unable to capture it due to the rapid heating and cooling. The temperature for ηeff = 95% is higher than the experimentally measured value. As shown in Fig. 13, the temperature for dt = 1.5 mm and dt = 1.6 mm is also lower than the experimental data. The predicted temperature profile from numerical analysis compares well with the experimental data for the combination of dt = 1.4 mm and ηeff = 90%, as shown in Fig. 12. The predicted temperature profile also compares well with experimental data at 3 mm and 4 mm from the weld center line for dt = 1.4 mm and ηeff = 90%, as shown in Fig. 14 (a and b).
Fig. 12
Temperature profiles at 2 mm distance from the weld centerline for dt=1.4 mm and different ηeff
Fig. 14
Temperature profiles for dt =1.4 mm and ηeff =90%, (a) at 3 mm distance from the weld centerline (b) at 4 mm distance from the weld centerline
The good comparison of temperature profiles obtained from numerical analyses and experiments at different distances from the weld center line confirms the accuracy of the HSM parameters. The weld pool size, assessed from the chemically etched weld joint, is shown in Fig. 4. The weld pool sizes has been determined from numerical analyses considering the regions having temperatures above 1428°C (the melting point), as shown for varying dt (1.2-1.6 mm) and ηeff = 90% in Fig. 15(a-e). Furthermore, Fig. 16 illustrates the weld pool sizes (gray region) for a fixed dt of 1.4 mm and varying ηeff (80-95%), where the temperature exceeds the melting temperature of 1428°C.
Fig. 15
Comparison of weld pool sizes from numerical simulations with variation of dt and same ηeff = 90% at (a) dt =1.2, (b) dt =1.3, (c) dt =1.4, (d) dt =1.5, (e) dt =1.6
Fig. 16
Comparison of weld pool size using numerical simulation at dt = 1.4 mm with variation of ηeff (a) ηeff =80%, (b) ηeff =85%, (c) ηeff =90%, (d) ηeff =95%.
The numerical analysis results shown in Fig. 15 and 16 shows that geometric parameters of HSM are very sensitive to the formation of weld pool size of the EBW joint. As the heat source top diameter (dt) increases, the size of the melting zone initially increases and then reduces as shown in Fig. 15. This occurs because, at larger heat source diameters, the heat intensity becomes insufficient (due to the Gaussian distribution) to melt the material, resulting in a smaller melting zone. Fig. 16 shows that the extent of melting zone (in terms of width) increases with increase in ηeff. The formation of weld pool crown with increased width at the top in actual weld joint as shown in Fig. 4 has been observed whereas it is absent in the numerical simulations. This difference could be due to use of conical heat source in the numerical analyses, which does not capture the crown formation. Therefore, the weld pool width has been compared at a depth of 2.5 mm from the top, middle, and bottom of the weld pool. The neck formation of the crown vanishes near about 2.5 mm from the top. The predicted weld pool widths with dt = 1.4 mm, ηeff = 90% for all locations compare well with the experimental results. Thus, it can be inferred that the calibration parameters dt = 1.4 mm and ηeff =90% used in the HSM are suitable for numerical analyses.
5.3 Numerical simulations of beads on plate
The numerical simulations of bead-on-plate deposited for various combinations of welding parameters (Table 3), named as Bead-1, Bead-2, Bead-3, and Bead-4, have been carried out using the calibrated heat source input parameters. The heat source parameters used in the numerical simulations are presented in Table 4. In Table 4, the parameters dt and db are obtained from eq. 7 and eq. 8. The experimentally measured and numerically evaluated weld pool sizes for these beads are shown in Fig. 17-20.
Table 4
Heat source parameters used in numerical simulation of beads on plate
Fig. 17
Comparison of weld pool size for bead-1, (a) experiment (b) Numerical based on temperature contour
Fig. 18
Comparison of weld pool size for bead-2, (a) experiment (b) Numerical based on temperature contour
Fig. 19
Comparison of weld pool size for bead-3, (a) experiment, (b) Numerical based on temperature contour
Fig. 20
Comparison of weld pool size for bead-4, (a) experiment, (b) Numerical based on temperature contour
Fig. 17-20 show that the weld pool sizes for all beads obtained from numerical analyses compare reasonably well with the experimental results. The bulging of the weld pool at the middle of the depth has been observed in experiment due to overfocusing of the beam. In an overfocused beam, the electron beam is focused below the top surface of the plate, resulting in a larger melting zone in that region. This bulging effect cannot be accurately represented in numerical simulations using a conical heat source. The numerical simulations of the pipe weld joint and their validation have been carried out using this calibrated HSM in26). This further validates the numerical analysis procedure and the HSM.
6. Evaluation of Residual Stresses and Distortion in Plate Welds
After validating the heat source model and finite element procedure, a mechanical analysis of the plate weld joint prepared by EBW was performed using the heat source geometric parameters dt = 1.4 mm and ηeff = 90%. The boundary conditions used are shown in Fig. 10. The high temperature during welding causes the metals to expand. As the weld pool solidifies, the weld metal and its surrounding metals attempt to contract. However, this contraction is restrained by the colder base metal, resulting in locked-in residual stresses within the weld joint. These residual stresses gradually increase as the weld solidifies and cools to room temperature. The Von-mises residual stress contour plot in the plate EBW obtained from the numerical analysis is shown in Fig. 21.
Fig. 21 illustrates that the von Mises residual stresses are notably higher in the HAZ. Additionally, Fig. 22 depicts the transverse and longitudinal residual stresses at the weld center along the thickness direction. The reference distance zero is set at the root of the plate weld. Longitudinal residual stresses are compressive across the plate thickness at the weld center, while transverse residual stress is compressive at both the top and root surfaces, with tensile stress evident in the middle section of the plate.
The residual stresses at the top and root surfaces of the plate weld are further detailed in Fig. 23 and 24, respectively. Observations reveal that transverse residual stresses at the weld top are compressive at the weld center, transitioning to tensile at a distance of 4 mm from the weld center. Initially, the weld pool, in its liquid state, exhibits no stress at the melting point. However, as the welding heat source moves away, the molten weld pool begins to solidify, inducing shrinkage and resulting in compressive residual stress in the transverse direction within the weld. Concurrently, the solidified weld starts to stretch the adjacent base metal, generating tensile residual stress in the base metal in the transverse direction. In contrast, longitudinal residual stresses at the weld top are initially tensile at the weld center, peaking at a distance of 4 mm before gradually decreasing and becoming compressive at 11 mm from the weld center. In the longitudinal direction, as the molten weld pool solidifies with the movement of the welding heat source, the solidified weld bead stretches the adjacent metal, causing tensile stress at the weld center. The transverse and longitudinal stress values at the top of the weld center are (-)118 MPa and 120 MPa, respectively, with peak values reaching 165 MPa and 170 MPa at the top surface, as depicted in Fig. 23. Along the fixed ends of the plate, longitudinal stresses are zero, while transverse stress measures 75 MPa in tension.
At the weld root, transverse residual stresses exhibit compression at the weld center, transitioning to tension at a distance of 4 mm from the weld center. Meanwhile, longitudinal residual stresses at the weld root are initially tensile at the weld center, reaching maximum values at 3 mm before gradually decreasing and becoming compressive at 4 mm from the weld center. The peak values of transverse and longitudinal stress at the root of the weld center are (-)170 MPa and 48 MPa, respectively, as illustrated in Fig. 24. Along the fixed ends of the plate, longitudinal residual stresses are zero. During EBW, the weld metal undergoes thermal expansion during heating and contraction during cooling. This non-uniform straining occurs due to sharp temperature gradients, structural constraints, and variations in material properties with temperature throughout the welding process. As a result, complex strains develop in the weld, leading to distortion in the plate. Various factors contribute to this distortion, including restraint, total heat input, weld metal volume, and associated cooling rate. Welding-induced transverse deformations can lead to significant distortion, affecting the dimensional accuracy and functionality of the welded component. Measuring these deformations allows for better control and mitigation of distortion during and after the welding process. Displacement contours in the transverse direction are depicted in Fig. 25. In the numerical analysis, fixed end boundary conditions were applied, resulting in zero distortion at the ends of the plate.
The transverse displacement contour indicates a positive displacement on the left side of the plate and a negative displacement on the right side, suggesting material movement toward the weld center, indicative of shrinkage in the transverse direction. Shrinkage is more pronounced at the top surface of the plate compared to the bottom. The maximum displacement occurs near the weld center. The maximum transverse shrinkage is calculated as the difference between the peak transverse displacement values on both plates, which are 0.30 mm.
7. Discussion
The plate weld joint was meticulously prepared and numerically simulated under actual welding parameters and conditions. Results from both experiments and analyses have been extensively discussed in preceding sections, focusing on weld pool size, distortion, and residual stresses, comparing them with experimental findings and existing literature. While the correlations between heat source and welding parameters, alongside the heat source model, were validated using thermal profiles and weld pool size, some disparities were noted, particularly concerning the reinforcement of the weld at the top. Employing a composite heat source model in numerical simulations might yield more comparable results. Despite this discrepancy, the width across the thickness of the weld exhibited good agreement, typically within a variation of 15%, consistent with findings from other researchers16,27,28). In addition to the heat source model, the heat absorption efficiency proved crucial in predicting temperatures and weld pool size accurately. A heat absorption efficiency of 90% yielded comparable results, aligning with other researchers’ findings in the 90-95% range11). Achieving accurate numerical simulations in EB welding necessitates replicating the thermal effect produced by the beam in the keyhole. Therefore, accurately estimating the keyhole’s size is paramount, ensuring the heat source mirrors the keyhole’s heat distribution pattern. The correlation between the heat source diameter and welding input parameters is vital for precise simulation of EB weld joints. Through this study, a heat source top spot diameter of 1.4 mm was identified as a suitable parameter for predicting temperature and weld pool size in plate weld joints.
Residual stresses and distortion in the plate weld joint were estimated through numerical analysis, depicted in Fig. 21-24. Notably, transverse residual stresses were found to be compressive, while longitudinal residual stresses were predominantly tensile at the top surface of the plate near the weld center. The maximum Von- mises residual stress recorded was 205 MPa, as illustrated in Fig. 21. Comparatively, Zhang Hong et al.20) reported peak Von-mises stress in the center of a weld on a 14 mm thick plate of polycrystalline nickel alloy. The elevated stresses observed in the nickel alloy could be attributed to its high thermal conductivity, resulting in a larger weld pool and HAZ compared to SS304L. However, despite these differences, the location of the peak Von-mises residual stress (Fig. 21) aligns closely with the findings reported by Zhang Hong et al.20), indicating a similar distribution pattern near the weld center.
Comparing the residual stress distribution patterns on the top surface of the plate with findings reported by Kumari et al.13), Xia et al.27), and Ragavendran et al.29) reveals notable similarities. For instance, Xia et al.26) reported peak transverse residual stresses of 170 MPa at the top surface of a weld joint on a 50 mm thick plate of 316L material, a value approximately consistent with the present analysis, as depicted in Fig. 23. Furthermore, the evolution of compressive transverse residual stresses at the top of the plate weld center, transitioning to tensile stresses at a distance from the weld center, aligns with the observations by Xia et al.27), as illustrated in Fig. 23. Additionally, the peak longitudinal residual stresses away from the weld center, as reported by Xia et al.27), also correspond to the present findings shown in Fig. 23. It’s worth noting that the values of peak longitudinal stresses reported by Xia et al.27) are higher compared to the present analysis (Fig. 23) due to the larger size of the weld pool resulting from the higher thickness of the material.
The comparison with the study by Ragavendran et al.29) reveals that longitudinal residual stresses in a 5.6 mm thick SS 316LN plate weld from multi-pass TIG welding are notably higher compared to the EBW plate weld examined in this study. However, both exhibit a similar pattern of tensile stresses at the weld center transitioning to compressive stresses away from it. This disparity in residual stresses can be attributed to the larger weld pool volume inherent in TIG welding compared to EBW. Similarly, the analysis by Kumari et al.13) on 2.5 mm thick plates EB weld made of Inconel 718 aligns with our findings, showing a peak in longitudinal residual stress away from the weld center, consistent with the observations depicted in Fig. 23.
The present study demonstrates the feasibility of accurately simulating thick EBW joints using finite element methods, offering a valuable tool for optimizing weld parameters to minimize residual stresses and distortions.
8. Conclusions
Experimental and numerical analyses have been carried out with an aim to develop a robust numerical analysis procedure for EBW. The key conclusions drawn from this study are as follows:
1) A calibration procedure for volumetric heat source model parameters has been developed for the numerical simulation of electron beam welding, especially for thick section weld joints.
2) The selection of heat source model parameters in the numerical simulation of the EBW joint is very sensitive to the formation of weld pool size. The geometric parameters of the heat source model significantly affect the formation of weld pool shape and heat absorption efficiency is responsible for the extent of the melting zone (in terms of width).
3) The calibration of the heat source has been done by comparing temperature profiles and weld pool shape (depth and width) obtained from numerical simulations and experimental results. Extensive analyses determined that a welding heat absorption efficiency of 90% and a conical heat source diameter of 1.4 mm are suitable parameters for use in the numerical simulation of the weld joint for 12 mm thick plate weld joint.
4) The comparable results of weld pool shape obtained from numerical simulations and experiments for the beads on plate showed that the calibrated HSM can be used to simulate the EBW in thick section weld joints with reasonable accuracy. Weld pool crown formation at the top can not be captured by the conical heat source model adopted in the present simulations.
5) Transverse and longitudinal residual stresses in plate weld joints were estimated via numerical analysis. The transverse and longitudinal stress values at the top of the weld center were found to be (-)118 MPa and 120 MPa, respectively, while at the root of the weld joint, they were (-)170 MPa and 48 MPa, respectively.
These findings underscore the efficacy of the proposed numerical analysis procedure in accurately simulating EBW processes and evaluating critical parameters such as temperature profiles, weld pool sizes, and residual stresses.
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