Thermally Induced Reduction of Hot Cracking Susceptibility in Electron Beam Welding of CuCr1Zr: A Numerical Approach

Electron Beam Welding (EBW) is a highly effective and accurate welding process that is being increasingly used in industrial work and is of growing importance in manufacturing. In the current study, solidication cracking in EBW of a CuCr1Zr cylindrical geometry was explored. To investigate and prevent occurrence of hot cracking, a thermomechanically coupled numerical model was developed using Finite Element Method (FEM). An additional heat source was considered, in order to inuence the resulting residual stress state, namely to minimize tensile stresses in the fusion zone during solidication. Hence, a methodical assessment of relevant parameters, such as the power, the diameter of the additional heat source and the distances between both heat sources was employed using Design of Experiments (DoE). It was found that for a particular parameter conguration, solidication cracking most likely could be averted.


Introduction
EBW is a fusion welding process where a narrow beam of electrons with high velocity is used to weld the two pieces of metals. The work pieces melt and partly evaporate as the kinetic energy of the electrons is transformed into heat upon interaction with the workpiece. The welding is usually carried out in vacuum to keep the energy density high [1]. Due to the high energy density, it can form a keyhole that results in deep and narrow welds. Thus, EBW remains indispensable in many aerospace, biomedical and mechanical applications namely due to a greater penetration depth, metallurgical purity of the weld, low heat input, small Heat Affect Zone (HAZ) and low susceptibility to deformation.
Over the past decade signi cant developments in modelling and simulation of the thermal cycle and the subsequent mechanical behaviour became very limited due to the inherent complexity of boundary conditions and the nonlinearity of material properties [2]. The development of numerical techniques like FEM has enabled researchers to overcome some practical di culties such as complex boundary conditions, arbitrary geometry, and temperature dependent material properties [3]. Despite the technological innovations in FEM, there are still some problems when it comes to EBW, especially hot cracking. In order to solve this problem, other authors have tried to apply different experimental techniques, for instance, applying auxiliary heat sources on both sides of the weld to produce thermal gradients, and cooling elements under the weld to generate a speci ed temperature eld altering the stresses occurring in the weld zone. However, such methodologies proved to be costly and time consuming [4], [5] creating the need for numerical studies which is exactly the goal of this work -a thorough numerical study on the hot cracking susceptibility in EBW of CuCr1Zr.

Material And Methods
Copper has outstanding electrical and heat transfer properties, moderately high-toughness and relatively high strength [6]. Many other common elements are alloyed with copper to improve certain material characteristics such as corrosion resistance or machinability commonly used in a wide range of industrial applications. In this speci c work, a copper alloy CuCr1Zr is used. This copper alloy contains small amount of chromium and zirconium which contribute to preserve excellent thermomechanical properties of copper at high temperature and increase wear resistance. Due to the very low solubility of Cr and Zr in copper, the thermal conductivity is still high. The excellent strength is attributed to the precipitation and particle-dispersion strengthening mechanisms [7]. Chemical composition is represented in Table 1. or heating sources to manipulate thermal gradients, generating a speci c temperature eld altering the stresses occurring in the weld zone. A comparison between the magnitude on the residual stresses can be seen in Figure 1, using LSND and conventional welding [5] These treatment processes are often either time-consuming, can end up increasing the cost and may also change the micro structures and mechanical properties [9]. A different alternative proposes a main welding beam while simultaneous multi-beam preheating on the side of the weld [10], [11]. Zhang et al. [10] found that the main advantages of this process over others, is that other processes need to employ complicated, heavy and costly additional facilities to generate heating or cooling source, while this method uses electromagnetic de ection to generate both welding and pre-heating beams, so there is no need to use auxiliary heating or cooling devices. Another advantage is the fact that due to EBW's characteristics; pre-heating beams can be generated as small as needed, making it possible to produce localized pre-heating areas with any geometric shapes and sizes. Based on these approaches a procedure is presented and investigated in this publication. In this procedure, a Thermally Induced Reduction of the Load (TIRL) is applied where an additional Secondary Heat Source (SHS) introduces compressive stresses during the solidi cation of the weld pool through a local treatment while the Main Heat Source (MHS) associated with the welding process causes melting. The vital parameters of the SHS: the angle, radial distance, power and representative dimension of the SHS are summarized in Figure  2.
One factor at a time (OFAT) method was applied to establish the value ranges for the aforementioned parameters. Subsequently, a systematic experimental approach using DoE was employed to assess statistical signi cance of the parameters and yield an optimum solution that avoids solidi cation cracking. A schematic representation of the methodology applied is presented in Figure 3.
The condition whether secondary melting was occurring was veri ed at all iterative steps. Initially, after having set a parameter window for the power applied, it was possible to adjust the radial distance that ensured maximum introduced compressive stresses whilst no melting caused by the SHS. This was performed successively until a parameter window was de ned for every parameter.

Numerical Implementation
The software Simufact Welding was used for the FEM simulations. It is designed for modeling and simulation of a wide range of thermal joining processes. Furthermore, it allows to model multiple heat treatment processes, variations of cooling and unclamping setups as well as mechanical loading of welded structures making it t for these types of thermal and thermomechanical simulations. The methodology applied for the numerical implementation was widely based on [12]- [14]. Firstly, the cylinder geometry was modeled using SolidWorks as can be seen in Figure 4.

Geometry Discretization
The detailed geometry is omitted from this publication due to con dentiality reasons. In terms of discretization of the domain, the simpli ed model built was an assembly of 4 components as is shown in Figure 5 which included a total of 97794 elements - Table 2. In the case of P1 and P4 hexahedral elements were chosen for its improved quality over tetrahedral elements. Tetrahedral elements were chosen for P2 and P3 due to their simple shapes, therefore easing computation for arbitrarily di cult volume and surface integrals in FEM, given that the quality for these parts was not considered as important [15]. The mesh was re ned where the heat source's path (P1) is located. The other components (P2, P3 and P4) have a coarse mesh. in Simufact Welding material library. Material's properties are described in Table 3.

Table 3 -Material Properties
The ow curves relating the effective plastic strain with the ow stress extracted from the Simufact library are presented in Figure 6.

Heat Source Model
The software uses the heat source models developed by Goldak et al. [2]. For heat distribution description in deep and narrow welds such as the ones produced by means of electron beam or laser beam, a conical heat source model seems to be more suitable. Heat source characteristics can be seen in in Equations (1) and (2).
The heat source model parameters: conical upper radius (r e ), conical lower radius (r j ), gaussian parameter (M), conical depth (d c = z i -z e ) were adapted using sensitivity analysis. Thermal simulations were conducted to calibrate process e ciency based on the experimental weld dimensions provided by the company. A short trajectory was chosen, the geometry used for the experiment was a at plate with 15mm of thickness, 400mm length and 150mm width and the process parameters considered for the experiment were 120 kV voltage, 12.5 mA beam current and a welding velocity of 20mm/sec An e ciency of 93.5% was attained and the corresponding weld dimensions numerical (case a.) vs. experimental (case b.) are shown in Figure 7.
A conical heat source was used with an upper diameter and lower diameter of 0,15 and 0,125, respectively, and a conical heat source depth of 1,565. Gaussian parameter was set to 3. The slight difference between numerical and experimental weld dimensions could be addressed due to slim approximations that were considered when performing sensitivity analysis for the heat source model.
Moreover, the process e ciency calibration was fully dependent on the aforementioned analysis.

Results And Discussion
Having successfully performed an assessment on the MHS such that all in uential heat source model parameters were known, it was possible to evaluate the stress distribution conducting thermomechanical simulations. The results for these simulations were exported to an Excel le and compared to the temperature dependent proof strength of the material. The condition for no hot cracking is given by: This criterion states that if the stresses were smaller than the admitted proof strength of the material, then hot cracking would not occur. The statistical software Minitab 19 was used to obtain an optimal solution for the SHS parameter con guration minimizing the stress to proof strength ratio.

No secondary heat source
A thermomechanical simulation without the use of a secondary beam was employed and provided an overview on the residual stresses. This evaluation was performed on points selected aligned with the Xaxis and Y-axis (as is shown in Figure 8) given that the X-stress and Y-stresses at these points, respectively, are entirely equal to the radial stresses (that cause hot cracking). Furthermore, multiple points equally spaced in depth were considered to verify if cracking would, in fact, rstly emerge on the surface of the material.
During many trials it was veri ed that hot cracking occurred given that the stress applied on the selected points overcame the proof strength at the given temperature. Since the cracking was occurring rst in the surface, it was not necessary to check the stress values for the other points below the surface. Figure 9 illustrates the stress values over time on the previously de ned surface point. The risk for hot cracking was always during solidi cation, so the time window was designated accordingly.
To further understand the ratio between the residual stresses and the proof strength and to facilitate the evaluation on the incidence of hot cracking, the ratio was also mapped over time as can be seen in Figure  10. Hence, it can be acknowledged that cracking had occurred since the relation yielded a value greater than 1.

OFAT
Arbitrary values were assumed for an initial condition (d = 1mm, P = 50% and α = 0º, D was set to the same value as considered for the MHS to simplify rst simulations). It was concluded that utilizing a SHS with those characteristics too close to the MHS would produce a merge in the fusion zone, melting even more material than intended. To avoid melting by the SHS a sensitivity analysis using the OFAT approach was selected on the Power, keeping all other variables constant. By iteratively lowering the power, it was seen that for this angle and radial distance (α = 0º and d = 1mm), 2.5% was the maximum power that could be employed without secondary melting. After, an assessment keeping the power constant and α = 0º while adjusting the radial distance was executed, as is presented in Figure 11.
Subsequently, a study on the angle was performed. It was seen that the variation of the angles was very small (≈ 0°), so it is hereafter described as the tangential distance (dt) allowing for easier interpretation. The tangential distance had a clear impact on the results, the closer the heating sources were, the better the results. This made sense since in this case the cooling rate was very high so there was a need to quickly introduce compressive stresses. Finally, multiple tests were conducted for the surface diameter of the SHS, ranging from as low as D = 0.075mm up to as high as D = 5mm, but the results did not vary that much.

Design of Experiment (DoE)
Having gathered understanding of the impact of each variable on the outcome of the simulations, the next step would be to build a parameter window for the DoE. A full factorial design (FFD) was chosen due to its reliability and capacity of giving information on how each factor is correlated with each other [16], [17]. A 3-level full factorial DoE with 4 factors was initially chosen, Table 4. Thus, a total number of 81 simulations were conducted to perform the statistical analysis. To improve this rst trial, Analysis of Variance (ANOVA) was conducted from which resulted the identi cation and elimination of irrelevant factors and levels [18]- [20]. Hence, D was dropped and a reworked DoE with 3 factors at 2 levels was carried out, as can be seen in Table 5. The tangential distance levels were considered at levels dt = 3.5mm and dt = 3mm and the radial distance levels were d = 2mm and d = 2.5mm since these yielded the best results. The power levels were increased to 45% and 50% because increasing the power seemed to produce better results. Nevertheless, increasing it even more would lead to melting by the SHS, so these values were adapted to match both the distances properly. The statistical results are presented in Table 6. Despite tting satisfactorily to the provided simulation results (reasonable R 2 ), the current model yielded a poor predicted R 2 . The modi ed nal DoE was further enhanced from the analysis of the Pareto charts of standardized effects for a signi cance level of 5%. Therefore, all the unnecessary factors and interactions were also removed (interaction between tangential distance and both radial distance and power -AC and BC -as is presented in Figure 12), and the model obtained had predicted R 2 of 99,55%. The statistical results are presented in Table 7. The relations between the response and two other factors were extracted and evaluated through contour charts using the software Minitab 19 [19] and are displayed in Figure 13. Figure 13a) exhibits the connection between the radial distance of both heat sources and the power applied while retaining the tangential distance constant at 3mm. The results improved with a relative parabolic increase of power and radial distance -increasing power required a greater distance between both heat sources; whereas a lower power would allow a lesser distance between both heat sources to replicate similar results. This makes sense given that when the temperature is near the melting temperature, the results are worsened. Case b) relates the tangential distance between heat sources and the Power. Conducting experiments with a combination of smaller tangential distances and power applied proven to be advantageous. It can also be concluded that the tangential distance had a bigger impact in comparison to the power. Lastly, case c) presents the effect on the response, considering the radial and tangential distance of both heat sources. The results for this case were extracted keeping the power at a constant value of 50%. Comparably, to the case b), the radial distance displayed superior in uence for lower values of the tangential distance. The response is given by the following regression equation (1).
The response optimizer in Minitab allows to identify the combination of input variables that optimize a single response. It calculates an individual desirability for each response and weights each by the importance. These values are combined to determine the composite, or overall, desirability of the response system [19], [21]. In this case, equation (1) was to be minimized. The results showed signi cant improvement when compared to the rst trials, there were multiple feasible solutions from which hot cracking could be mitigated as seen in Table 8. Promising parameter combinations were from 1 to 5. Numerical validation was conducted to gauge the suitability of the statistical model, as a result condition 1 was selected for a nal testing as can be seen in Figure 14. The weld dimensions were respected and the peak temperature of the SHS at distinct times throughout the simulation was 1020ºC. Hot cracking also did not seem to occur (stress/proof strength<1) under these circumstances, so a solution was found and therefore, con rming that the TIRL method is appropriate to avoid hot cracking in CuCr1Zr as can be seen in Figure 15.

Summary And Conclusion
The main conclusions that arise from the work accomplished in the context of this publication are presented and labeled according to the topics addressed.
Results improved using the OFAT and DoE procedures, but it is unclear that this solution is a global optimal (principally due to the rapid cooling rate of this particular Cu-alloy); The in uence of each parameter individually and amongst each other was measured based on the stress level variation. The tangential distance (or angle) had the highest impact in introducing compressive stresses during solidi cation while the diameter of the SHS had the least effect; Smaller distances between heat sources or higher power for the SHS could produce secondary fusion of the material (and possible blend of fusion zones); Longer distances between heat sources were proven to be inconsequential, given that solidi cation for this speci c alloy occurred rapidly; The work performed con rms that numerically it is possible to reproduce electron beam welds in CuCr1Zr whilst preventing the incidence of hot cracking using the TIRL method.
Declarations a. Funding (information that explains whether and by whom the research was supported) The project was funded by Dobeneck-Technologie-Stiftung b. Con icts of interest/Competing interests (include appropriate disclosures) The assembly dimensions as well as setup details were omitted from the publication due to con dentiality reasons for the entity.
c. Availability of data and material (data transparency) Not applicable (con dentiality) d. Code availability (software application or custom code) Not applicable (con dentiality) e. Ethics approval (include appropriate approvals or waivers) Compliance with Ethical Standards f. Consent to participate (include appropriate statements) The entity has consented to the submission of the case report for submission to the journal. g. Consent for publication (include appropriate statements).
The participant has consented to the submission of the case report to the journal. h. Authors' contributions (optional: please review the submission guidelines from the journal whether statements are mandatory) Not applicable [21] H. Singh, V. Kumar, and J. Kapoor, "Multi-response optimization of WEDM process parameters during the fabrication of microchannels for industrial applications," Mater. Today Proc., Jul. 2020, doi: 10.1016/j.matpr.2020.06.134. Figure 1 Residual Stresses comparison using LSND vs. conventional welding [5]   Flowchart depicting the methodology implemented to de ne the working range of process parameters        Ratio between stress and proof strength over time for different radial distances using OFAT. Dashed line represents case of hot cracking Figure 12 Pareto chart for the standardized response of the modi ed DoE for 5% signi cance level Peak temperature at different times