Volumetric three-dimensional heat source model with Gaussian distribution is adopted for this study. The flux model for FEAs are assumed to be Eqs. (1), (2), and (3) [10].

During deposition process the surface temperature is different across the specimen. The difference in surface temperature causes variation in heat losses due to buoyancy forces of air. Shielding gas, such as Argon (Ar) also contributes to heat losses due to forced convection. In order to improve accuracy of computations temperature dependent properties of Air and Ar are used to estimate coefficients of natural convection. Temperature dependent properties of air are show on Fig. 2. Estimation of coefficient of convection is accomplished by Eq. (4) [6,11].

Fig. 2 
Temperature dependent properties of air

Considering the position of surface Nusselt number can be calculated by Eqs. (5), (6), and (7) [6,11].

The Rayleigh number and temperature of air are estimated from Eqs. (8), and (9) [6,11].

The initial and ambient temperature are set to 20^oC. Estimated natural convection coefficient for each surface of the substrate are illustrated on the graph on Fig. 3.

Fig. 3 
Coefficients of natural convection

Inconel 718 and AISI 1045 are chosen as a powder materials of powder and substrate, respectively. Corresponding temperature dependent material properties are assigned to the analysis materials. The data of temperature dependent properties of materials is obtained from SysWeld and JmatPro softwares [12,13].

Total of 9 models were created according to combination of the width and the length of the deposited region. Table 1 shows dimensions of deposited region. The dimensions and thermal boundary conditions are shown in Fig. 4. Heat loss induced by convection is considered for all surfaces except for clamp area. During deposition process heat loss due to forced convection is applied on top surface.

Fig. 4 
Boundary conditions of FE models

The model creation and analysis are performed by SysWeld software. The region of the specimen near deposition region and around the heat affected zone (HAZ) is designed with fine mesh structure. The region is affected by high heat flux and rapid temperature changes. Hence, fine mesh structure is essential in the vicinity of the deposited region to improve accuracy of FE analysis. Mesh structure of the models is shown on Fig. 5.

Fig. 5 
Mesh structure of FE models

Deposition of three layers of Inconel 718 powder over AISI 1045 carbon steel is analyzed in the present work. Parameters of FEAs are cited from research work of Kim, et al [10]. Deposition conditions are shown in Table 2 [10].

The analysis results include 40 minutes cooling process after deposition process. All models reach temperature around 30^oC at the end of cooling stage. This temperature is sufficient to properly evaluate thermo-mechanical characteristics of the specimen after deposition. Fig. 6(a) shows global effective stress distribution of each model in ISO view. From Fig. 6(a) can be observed that the stress is concentrated along the corner between substrate and deposited bead. Specifically, along the corner of starting and ending location of each bead. Typically, effective stress is distributed in the surrounding area of deposited bead.

Fig. 6 
Global effective stress distributions for different area of the deposited region (After cooling)

Estimated effective stresses on overall surfaces are in a range between 600-700 MPa. The stress distribution is almost similar irrespective of the deposited area. From this result, it is revealed that surface stress distribution pattern is not significantly affected by deposition width and length.

For the evaluation of inside stress, the cross-section along the center line is selected, as can be seen in Fig. 7(a). The mid-section is adopted to obtain residuals stress distribution in steady state conditions. Most of the deposited bead build under steady state conditions. This means similar post-process thermo-mechanical characteristics can be observed inside the most of the specimen. Hence, evaluation of stresses in the mid-section can give insights on the effects of deposition process on mechanical properties of the build part. Fig 7(b) shows effective stress distribution in the vicinity of deposited clad. Stresses are distributed in the heat affected zone and surrounding area. For all cases stress concentration region is under deposited bead and on top surface of the substrate. The excessive stresses are under the first and the last deposited bead. Increased length of deposition caused decreased residual stresses for the case of W = 6, and 10 mm. The location of the maximum effective stress is almost similar except for the case of W = 10, and L = 20 mm.

Fig. 7 
Effective stress distributions in the vicinity of deposited region for different area of the deposited region (After cooling)

The graph in Fig. 8 shows maximum effective stresses in cross-section X’-X. For the case of W = 6 and 10 mm, the lowest maximum effective stress is estimated at L = 20 while the highest maximum effective stress is predicted at L = 30 mm. The maximum effective stress decreases when the width of the deposited region increases. The maximum effective stress is almost identical irrespective of the deposited length for the case of W = 8 mm. But overall, maximum stresses in case of 8 width are in between W = 6, and W = 10 mm. This result suggests that overall maximum stresses tend to get lower with increasing the width of the deposited region. The maximum effective stress is greatly influenced by the deposition length for the case of W- 6 mm. From this result, it is noted that the width and length of deposited region slightly affect residual stress distribution pattern and maximum stress location. However, the magnitude of the maximum effective stress can be influenced by both the width and the length of the deposited region. In addition, it is noted that the maximum effective stress is sensitively changed when the deposition length is reduced.

Fig. 8 
Maximum effective stress

According to the results of the analysis the displacement in all the models is upward at the free end of the substrate, as shown in Fig. 9. The results on Fig. 9 are magnified 15 times for better visual representation.

Fig. 9 
Deformed shape and displacement in section X'-X

From Fig. 9, it is noted that both increased width and increased length resulted in greater displacement. Points A and B are the edges of the substrate. A is point at the starting side of deposition. Each model has maximum displacement at one of these points. Maximum displacement for each case can be observed on the graph on Fig 10. For each width of the deposited region, increase in length shows almost linear increase in displacement. This is in contrast to the maximum residual stress. The stress-area relation is not as direct as in the case of displacement. During the deposition process by DED two types of stresses occur due to shrinkage of solidification and cooling of deposited metal [14]. These stresses are transverse (T_s) and longitudinal (L_s) directions, as shown in Fig. 11 [14]. Transverse stresses act perpendicularly towards direction of deposition. Longitudinal stresses are acting along the deposition direction. The predicted upward displacement at the free end of substrate is primarily caused by longitudinal stresses. Increased width and length of deposition region facilitate growth of cumulative effect of longitudinal stresses. The analysis results showed greater stresses in the vicinity of deposited region of smaller width.

Fig. 10 
Maximum displacement

Fig. 11 
Transverse and longitudinal stress caused by the shrinkage of metal

However the of the effect of increased area of deposition clad on thermal residual stresses is considerably more significant. From this result it can be concluded that maximum displacement is highly affected by the area of deposition region.

Distortion is represented as a difference in maximum displacement between opposite edges of the substrate, as shown in Eq. (10). The maximum distortion is estimated by subtracting distortion at point B from displacement at point A of Fig. 9.

Point A is located at the side closer to starting point of deposition and point B is closer to the side of the ending side of deposition. Fig. 12 shows predicted maximum distortions for different deposition area. The distortion pattern is changed for the case of W = 6, and 8, while that is identical for the case of W = 10 mm. From the Fig. 12, it is noted that the distortion increases when the deposited area augments except for W = 8, and L = 30 mm.

Fig. 12 
Maximum distortion

From these results, it is revealed that the distortion behavior of small width of deposition has less distortion but it is difficult to predict when change in length of deposited bead is applied. The larger width of deposition shows larger distortion however it also exhibits more predictable behavior. Similarly to welding process, predictable behavior of metal substrate can be used to mitigate negative effects of deposition process during preprocessing and preparation stage [14].