1. Introduction
Currently, the industry is concentrated on new approaches in material management, whose main objective is to reduce the usage of raw materials by reusing materials from existing products, which is referred to as a circular economy^{[1]}. Repair is one of the common aspects of a circular economy, and its main goal is to extend the life of a product^{[1]}. Repair is more resourceefficient than recycling^{[2]} and is considered to be the most valueadded strategy in resource management^{[3]}.
The implementation of additive manufacturing (AM) technology for metallic parts is the best strategy to overcome limitations and difficulties related to manual repairing^{[2]}. The most commonly used AM technology for repairing is directed energy deposition (DED) technology that can restore damaged parts layer by layer^{[2]}.
Damage to valuable metallic parts occurs in the form of voids and cavities^{[3]}. The repair process includes the removal of the damaged region with subsequent redeposition. Several studies have been conducted to investigate the properties of deposited clads with different geometries. In Song et al.’s study, the trapezoidal groove of 304 stainless steel was repaired using 316 stainless steel powder by laser cladding with subsequent alloying of the surface by WC powders^{[4]}. Song et al. investigated the restoration of Vgrooves on an AISI1045 substrate using stainless steel powders and analyzed the properties of the deposited clad using numerous tests^{[5]}.
Experiments by Pinkerton et al. repairing H13 steel by H13 steel powders of triangular and square geometry slot demonstrated poor adhesion between the deposited material and the substrate on the sides of the squiregeometry groove owing to the lack of fusion^{[6]}.
Thermomechanical finite element analysis (FEA) was performed to investigate the postsolidification deposition characteristics of the model, such as residual stress distribution, dislocations, distortion, and final microstructure, which are practical for industrial applications^{[7]}.
In this study, residual stress characteristics when repairing different radius arcshaped parts by DED are analytically investigated. A total of eight models, of which four are inner radius ones and four are outer radius deposition ones with a corner angle of 45° were modeled. The heat source, thermal, and dimensional boundary conditions are described in the next section, and the results of the simulation are discussed in terms of the effective stress and 1^{st} principal stress distribution, and the results are summarized.
2. Finite Element Analysis
2.1 Heat Source Model and Thermal Losses
The metal powder in the DED process is blown through the nozzle onto the substrate or previously deposited layer and melted with a highly concentrated heat source such as a lasercreating melt pool, and the deposited region is built upon solidification of the melted material^{[47,810]}.
Because thermomechanical analysis depends on temperature histories, the definition of the heat source model is crucial^{[11]}. The real laser beam heat flux distribution was very close to the Gaussian distribution. The highest power density was at the center of the beam, and it exponentially decreased away from the center. At any plane perpendicular to the zaxis within the effective penetration depth (D_{0}), the heat flux can be described by Equation (1)^{[12]}:
where Q(x, y,z), Q_{0}, r, r_{i}, r_{e} , z_{e} and z_{i} are the volumetric heat flux at any point within the heat source volume, the maximum heat flux density at the center of the beam, the radius of beam at the top surface, the radius of beam at the penetration depth, the z coordinate of the heat source at the top surface, and the z coordinate of the heat source at the penetration depth respectively.
analysis involves longtime free air cooling as well as argon shielding. For more accurate simulation results, the average coefficient of the natural convection (h_{co nv} ) based on temperaturedependent properties of argon and air was implemented into the model. Convection coefficient was estimated by Equation (2)^{[13]}. The radiation coefficient h_{r ad} can be calculated using Equation (3)^{[13]}.
where NU_{L} , k, and L _{c} denote the average Nusselt number, thermal conductivity of the gas, and characteristic length of the specimen, respectively.
where ϵ, σ_{s}, T_{s} and T_{∞} are the emissivity of the material, the StefanBoltzmann constant, the surface temperature, and the ambient temperatures respectively.
For better efficiency of FEA, combined losses (${{q}^{\u2033}}_{loss}$) from convection and radiation can be interpreted as a single Equation (4)^{[13]}. The ambient and initial temperatures were set to 20°C. Fig. 1 shows the estimated equivalent coefficient of natural convection based on the temperaturedependent properties of air.
2.2 Analysis Model
The temperaturedependent material properties of AISI 1045 were assigned to the substrate and deposition models for more accurate FEA results. Temperaturedependent material properties of AISI1045 were obtained from JmatPro software^{[14]}. A total of eight models with eight beads for each model were created with deposition overlap assumed to be 33%. A schematic representation of FEA models is shown in Fig. 2, including revolve angle (θ), radius of curvature (R_{in}, R_{out}), and groove location and corner angle of the groove (β°). β° is equal to 45° for all models, while each model has a different θ° according to R_{in} and R_{out} so that each one has an approximately 30 mm outer bead. The value of θ for each radius of the model is shown in Fig. 3. Tables 1 and 2 list the bead lengths for each model.
To reduce the computational time and preserve the accuracy of the results, a fine mesh was generated only for deposition beads and the surrounding regions due to intensive heat flux and high temperatures in these regions. A biased mesh was mapped around the high heat flux region. Mesh generation and FEA were performed using a commercial software SYSWELD^{[12]}. The inner radius deposition model shown in Fig. 4 summarizes the dimensional and thermal boundary conditions of the models. The outer radius deposition model has the same dimensions, except that the deposition region is on the outer side.
This study focuses on the FEA model and numerical analysis. In the future, an explicit experiment will be required for further optimization and validation of the FEA model. The width (W) and height (H) of the bead and other parameters such as the power of the laser (P), scan speed (V), and efficiency (η) were taken from the experimental results of deposition of single bead AISI1045 on AISI1045 substrate using DED machine DABO300 (MAXROTEC) and shown in Table 3. Alternative direction deposition strategy was used for the analysis^{[9,10]}.
3. Results and Discussions
3.1 Residual Stress Distribution of Repaired Inner Radius Side
Small volume depositions of three layers with eight beads in each model were analyzed. The results of the thermomechanical analysis in SYSWELD provide data on the postsolidification deposition characteristics of the models, including the stress distribution, displacement, and final microstructure. Fig. 5 shows the typical displacement at the end of cooling for the deposition of the inner radius side. Maximum displacement was observed at the edges of the deposition. This is because the displacement in that region was not restrained as much. In addition, destructive tests for residual stress evaluation are based on strain release by cutting metal specimens, indicating that the stressed regions of metal are constrained from dislocations^{[15]}. Hence, regions with a high displacement show lower residual stresses. According to the FEA results, the displacement is widely spread throughout the volume of the models at the sides and tends to decrease toward the center. This indicates higher stresses at the center region; hence, the center section was selected to analyze the stress distribution.
The results of the analysis show an almost symmetrical distribution of the effective stress and 1^{st} principal stress in all FEA models at the end of cooling. This is due to zero stress initial conditions and the use of an alternative direction deposition strategy ^{[10,13]}. From the observation of residual stresses for the cut through the center section for the distribution of concentrated effective stress and 1^{st} principal tensile stress, a similar pattern is observed irrespective of the radius of deposition, as shown in Figs. 6 and 7, respectively. Concentrated residual stresses were observed under the deposition corner and extended from there until the outer surface below the deposition beads. Furthermore, the stresses spread in the overlap of the beads in the first layer and along the inclination at the corner.
The compressive 1^{st} principle stress is at a certain distance from the deposition region, as demonstrated in Fig 7. Tensile and residual stresses must balance each other in equilibrium. A similar location of maximum effective stress (σ_{v,max}) under the 1^{st} bead is applicable for all models except the 40mm radius model, which has the maximum effective stress located on the surface under the deposition region, as shown in Fig. 6.
3.2 Residual Stress Distribution of Repaired Outer Radius Side
Fig. 8 shows the typical displacement distribution for the outer radius deposition case. Similarly, a higher displacement indicates higher stresses at the center. Hence, cutting through the center section was used for the analysis of stress distribution. A similar pattern for the effective and 1^{st} principal stresses as in the inner radius side deposition observed in the outer radius side deposition, as shown in Figs. 9 and 10.
Highly concentrated stress were observed under the deposition beads and in the overlap of beads in the first layer. Starting from the corner until the outer surface was below the deposition region. However, for the outer radius deposition case, the overlap region and inclination at the corner were affected to a considerably lesser extent. Location of maximum principal stress (σ _{v,max}) is similar for all models except the 60 mm radius model, which has maximum stresses at the corner. The magnitude of stress at the corner of the groove is high for all models; however, for a larger radius, the tensile 1^{st} principal stress is predominant at the location under the overlap of the first layer.
3.3 Maximum Stress
At the end of 1 h of cooling, the temperatures of the models were at 35 °C37 °C. The results of the simulation show that the deposited region underwent transformation to the martensitic phase. This is caused by heat dissipation from the deposited beads to the rest of the body through conduction, which contributed to the quenching effect^{[6]}. The yield strength or fracture strength in the case of brittle martensite was 1,804 MPa at 50 °C^{[14]}. The range of maximum 1^{st} principal stress was 1,4161,498 MPa for R_{in} and 1,2551,321 MPa for R_{out}. The effective stress ranges are between 1,1911,234 MPa and 1,0911,126 MPa for R_{in} and R_{out}, respectively.
While a direct relation between radius size and thermal residual stresses was not observed, the smallest radii of analysis demonstrated exceptional results. In the outer radius side deposition case, the thermal residual stresses are considerably lower. In the case of a 60mm radius, the lowest stresses are observed.
However, in the inner radius deposition case, the smallest radius model of 40 mm showed the highest stresses. Therefore, a further investigation of residual stress for the reparation of the inner radius side by DED for small radius parts is necessary. All models show both maximum effective and 1^{st} principal stresses lower than the yield strength, as shown in Figs. 11 and 12. From this result, it can be concluded that cracks or fractures are not likely to occur.
4. Conclusion
In this study, we analytically investigated the residual stress characteristics according to the radius of deposition and corner location during the repairing process by DED. From the results of the analysis, the following conclusions can be drawn:

1. When the inner radius of the metal part is repaired by DED, the location of the maximum stress concentration appears closer to the surface under the deposition clad for small radius parts.

2. After repairing the outer side of the round part, the maximum thermal residual stress is at the corner of the groove when the radius of the part is small. However, a small radius tends to exhibit lower stresses overall.

3. The maximum residual stresses at the end of cooling were considerably higher for the inner side deposition parts. Although the residual stress is in the range lower than the yield strength for all models, further investigation on the magnitudes of residual stress after repairs is needed for parts smaller than 40 mm.

4. The metal fracture behavior is influenced by residual stresses and can be the reason for fractures at considerably low applied stresses. Other negative effects of residual stresses include fatigue failure and lower corrosion resistance^{[15]}. In future works, other repair strategies need to be considered, such as different inclination angles and, instead of sharp corners, the effect of grooves with different radius filets can be investigated.