Finite element analysis for residual stress of TB18 billet produced by laser directed energy deposition

In order to study the influence of process parameters on the residual stress of TB18 titanium alloy produced by laser directed energy deposition, a method combining numerical simulation with experimental verification was adopted. The distribution of residual stress in the deposited layer was obtained through experiments, and the influence of process parameters on the residual stress in the deposited layer was analyzed by finite element simulation. Finite element simulation is carried out for different cooling time and scanning strategy between layers. The results show that the residual stress of the deposited layer increases with the increase of the interlayer cooling time. By changing the scanning strategy, the peak residual stress can be reduced by 84.3% at most. This study provides guidance for selecting appropriate process parameters and reducing the residual stress of TB18 titanium alloy produced by laser directional energy deposition.


Introduction
Laser directed energy deposition (L-DED) technology is a metal additive manufacturing process that adds raw materials in the form of coaxial powder feeding, and then melts and accumulates layer by layer by point according to the preset path through high-power laser. The process has the advantages of short processing time, high forming dimension accuracy, flexible operation and small processing dimension limitation. It has broad application prospects in aviation, aerospace, automobile, ship and other fields [1][2][3][4]. L-DED additive manufacturing is a very complex process involving multi physical field coupling [5]. In the unsteady process of rapid heating and cooling, residual stress is easily formed in the deposited layer. Residual stress can be used to improve the fatigue strength of workpieces, but generally, residual stress will directly affect the deformation resistance to static and dynamic loads of additive manufacturing components, reduce the fatigue strength of materials, the resistance to stress corrosion of materials, and affect the dimensional stability of components [6][7][8][9][10][11]. Therefore, it is of great significance to master the residual stress distribution law of L-DED additive manufacturing components and reduce the residual stress in the cladding layer of L-DED additive manufacturing components for the application of L-DED process.
It is necessary to reduce the residual stress in the L-DED layer by optimizing the process parameters during the additive manufacturing process, which are mainly influenced by the laser directed energy deposition process parameters. Joaõ P M Cheloni et al [12] proposed a finite element model considering only the effect of temperature to simulate the complex temperature field generated by laser scanning along the deposition tracts and successive layers, and investigated the effect of different laser powers and scanning rates on the temperature field of the last fused layer, however, the temperature field and the stress field are interacting, and a more accurate temperature field distribution can be obtained by thermodynamic coupling analysis. Hongjian Zhao et al [13] focused on the residual stress distribution of additive manufacturing in gradient materials based on the finite element method. The results show that with the positive correlation between laser power and structural residual stresses. Their study material is gradient material, and due to the simple structure of the component only the relationship between power and gradient material stresses is considered. Blanka A Szost et al [14], by studying the final molded parts of arc-fed wire additive manufacturing and laser melting additive manufacturing processes, concluded that the residual stress distribution is similar for different additive manufacturing processes, and the stress variation during the additive process was not obtained due to the limitations of the additive process. Jun Cao et al [15] established a 3D fully coupled temperature-displacement finite element model of TC4 thick plates manufactured by electron beam additive using ABAQUS software to investigate the effect of preheating process on the deformation and residual stress distribution of the workpiece, and the results showed that at least two preheats are required to effectively reduce the residual stresses in additively manufactured components. The model and experiments of their study are based on multilayer single-pass deposition experiments, while the deposition process of actual additive parts is more complex.
In this paper, a new titanium alloy TB18 billet was prepared by coaxial powder feeding laser direct energy deposition equipment, and its stress distribution was tested by the small hole method. The finite element model of the new titanium alloy TB18 multilayer multi-pass fusion layer was established using finite element software, and the formation of residual stresses was simulated by the thermal coupling method, and the effects of different interlayer cooling times and scanning strategies on the residual stresses in the fusion layer of the additively manufactured components were analyzed to provide guidance for the reduction of residual stresses in the TB18 fusion layer.

Experimental materials
In this paper, TB18 powder is used as raw material and LMD-V coaxial powder feeding laser forming equipment is used. The laser is a YLS10000 fiber laser manufactured by IPG, with a laser focal length of 400 mm and a laser beam diameter of 6-8 mm. Argon is used as the protective atmosphere, TB18 plate is used as the deposition substrate, and the laser directed energy deposition parameters are: laser power 8kw, scanning speed 1000 mm s −1 , powder feeding rate 3.6 g s −1 , the diameter of laser beam is 7 mm, layer thickness 0.8 mm, and interlayer cooling time 120 s. The experimental part is shown in figure 1, and the geometry is 250 mm (x) × 40 mm (y) × 240 mm (z). 240 mm (z). Each deposited layer was scanned in a short-edge reciprocating manner with no rotation between layers.
The base plate has the same composition as the TB18 titanium alloy used in the powder, and its composition is shown in table 1.

Material properties
The temperature-related thermal properties of TB18 titanium alloy are obtained through experiments, linear interpolation and reasonable extrapolation. When the temperature is low, the temperature dependent properties of the material are measured experimentally. The linear expansion coefficient was measured using the ASTME 228 standard with a NETZSCH DIL 402 PC device at a temperature rise rate of 5°C min −1 . The ASTM-E1461 standard was used to test the thermal conductivity of the material using the LFA 467 HyperFlash device from NETZSCH. Young's modulus and Poisson's ratio are measured according to standard ASTM E1876-15 by RFDA HTVP 1750-C from IMCE, Belgium. The mechanical properties of the material are tested on Gleeble 3500 equipment. The specific heat capacity of the material was measured by using the ASTM-E1269 standard, using the STA 449 F3 Jupiter apparatus from NETZSCH, at a temperature rise rate of 10°C min −1 . At high temperatures, the temperature dependence of the material is difficult to measure directly, so an extrapolation method is used and the material parameters are shown in figure 2.

Hole drilling measurements
After the sample were cooled to room temperature, the residual stresses in the sample were measured by the hole-drilling method using an SRSS3-1 residual stress tester following the standard of ASTME837. The stress release coefficient was A = −0.07395 and B = −0.05411. the diameter of the borehole was 1.5 mm and the depth was 2 mm. the test location was shown in figure 3, and a total of 3 points were tested on the side of the deposited part, all units in the figure are millimeters (mm). The stress distribution pattern was analyzed by the directional stress test results.

Result
The results of the residual stress test after the sample has cooled to room temperature are shown in figure 4, where s 11 and s 33 represent the stresses along the x-direction and y-direction of the sample, respectively. The maximum s 11 stress of the sample is 99.0 MPa and the minimum value is 73.6 MPa. The maximum s 33 stress is 264.8 MPa and the minimum value is 63.7 MPa. Both s 11 and s 33 of the sample along the height direction decrease with the increase of the layer height. Based on the test results, it is clear that the residual stress decreases in the resolidified layers. This is the same conclusion reached in the literature [10] and [14], where the material deformation is limited thus leading to an increase in stresses due to the fact that the underlying material is most structurally constrained.

Finite element model an theories descriptions
In order to study the residual stress distribution and formation causes of laser directed energy deposition of TB18 multilayer multi-layer fusion layer, this paper simulates the deposition, cooling, and fixture relaxation processes, and establishes a 10-layer finite element model, the fusion layer is simplified as a rectangular body, and the model structure is shown in figure 5(a). In the deposition model, both the substrate and the deposition layer material are TB18 titanium alloy, the laser deposition site is located in the middle of the substrate, the substrate size is 105 mm × 125 mm × 12 mm, and the deposition layer size is 32 mm × 55 mm × 8 mm.
Previous finite element analyses of additive manufacturing have shown that a finite element mesh smaller than the radius of the laser spot at the deposition site can improve the accuracy and stability of the simulation results [16], and a reasonable enlargement of the mesh size away from the deposition site by using a static adaptive mesh can save computational costs [17][18][19][20]. In this paper, the mesh in the laser cladding layer is set as a fine mesh, as in figure 5(b), with the size of 2.5 mm × 1 mm × 0.8 mm, while the mesh away from the cladding layer is sparser as in figure 5(c), so as to meet the calculation accuracy while achieving the purpose of saving calculation time. Before the simulation started, the mesh type and mesh size were analyzed for convergence, and the selected mesh  type and mesh size ensured the accuracy and computational efficiency of the calculation results. In this finite element model, an 8-node trilinear displacement and temperature coupled integral cell (C3D8T) was used, and a total of 38,120 cells were generated.
At higher temperatures, it is assumed that the temperature-dependent properties are constant and the density of TB18 titanium alloy is 4.45 × 103 kg m −3 independent of temperature. The process parameters of laser directed energy deposition for the fabrication of TB18 titanium alloy were the same as before, with interlayer cooling times of 30 s, 60 s and 120 s. After deposition, cooling was carried out at room temperature (20℃) for 6 h. The laser light source model uses a double ellipsoidal model. The laser scanning strategy uses three methods, namely unidirectional deposition along the long edge, reciprocal deposition along the long edge and unidirectional deposition along the short edge, as shown in figure 6, with no rotation between layers.
The Model Change module that comes with the finite element analysis software ABAQUS can activate the cells sequentially. In this paper, the heat source is moved to a certain pass, then all the passes are activated to save the calculation time.
The transient temperature distribution T of the entire laser directed energy deposition process can be characterized by the partial differential equation (1): where Q is the power generated per unit volume of the workpiece, K is the thermal conductivity, T is the temperature, r is the density, and t is the time.
The initial temperature distribution of the workpiece T x y z , , , 0 ( )is denoted as: Where T 0 indicates that the ambient temperature is 20°C. The boundary conditions of the model temperature field consider the effect of convection and radiation pairs, and the control equation is where N is the normal direction of the surface, h c is the convective heat transfer coefficient, s is the Boltzmann constant, and l is the radiation coefficient. During laser directed energy deposition, the strain of the material changes with the temperature field. The total strain increment can be considered as the sum of the components of the elastic strain (e e ), plastic strain (e p ) and thermal strain (e th ). The total strain of the material (e) can be expressed as e e e e = + + 4 Mises where s Mises is the equivalent force, and s , 1 s , 2 and s 3 are the principal stresses.

Effect of interlayer cooling time on residual stress
The thermal stress during deposition is related to the Young's modulus and linear expansion coefficient, which can be respectively expressed as where E is the Young's modulus, T is the temperature, and a is the Young's modulus. The change in thermal stress during deposition can be expressed by equation (9), where s th is the stress value in the deposited part and DT is the temperature variation. D E T , ( ) a DT ( ) are monotonic functions with respect to DT, so the larger DT is, the larger s th is in the deposited part. The melted material is not affected by thermal stresses, which are generated during the temperature gradient or the solidification and shrinkage of the melted material, so the reduction of thermal stresses also leads to a reduction of residual stresses [21]. The formation of residual stresses in the L-DED process can be explained by the temperature gradient mechanism and the cool-down mechanism. The temperature gradient mechanism is shown in figures 7(a), (b), where the material expands during the heating process but deforms elastically due to the restriction of the surrounding material. When the heating temperature is too high, the yield strength of the material decreases significantly, so the material deforms plastically, as shown in figure 7(a), because the expanding material bends in the direction away from the heat source. When the heat source is removed, the material gradually cools and the original heated expansion area begins to cool and shrink, causing tensile stresses in the upper layer of the material and compressive stresses in the lower layer, and bends toward the previously heated area, as shown in figure 7(b). The cool-down mechanism is shown in figures 7(c), (d). In the cooling mechanism, the upper material melts because of heating and the temperature is much higher than that of the lower material. When the material cools and solidifies, the upper material undergoes greater shrinkage deformation, which generates tensile stresses in the upper material and compressive stresses in the lower material.
In the study of residual stresses in additive manufacturing components, the stress s 11 along the deposition direction is usually considered as the most important material properties parameter [13]. With the second deposition path, the interlayer cooling time is 30 s, 60 s and 120 s respectively, and the distribution of stress s 11 along the midpoint of the side of the cladding after the deposited part is cooled to room temperature is shown in figure 8. Without changing other deposition parameters, the component stress s 11 varies in the same trend for different interlayer cooling times. In other words, the residual stress decreases in the resolidified layers. The peak s 11 stresses were 59.4 MPa, 161.6 MPa and 480.5 MPa for interlayer cooling times of 30 s, 60 s and 120 s, respectively. Different interlayer cooling times can lead to a difference of up to 87.6% in the peak s 11 stress, which shows that the effect of interlayer cooling time on the stresses in additively manufactured components is significant. The stress field clouds of the cladding at different interlayer cooling times are shown in figure 9 (Stress field in x-z plane of deposited layer), and the residual stress distribution patterns of the cladding sides at different three interlayer cooling times are the same. The maximum value of residual stress is located at the connection between the cladding and the base material, and the connection parts are all tensile stress, and the minimum value of residual stress is located at the top of the top cladding. When the interlayer cooling time is less than 120 s, the stress at the top middle position of the cladding side is compressive stress. When the interlayer cooling time is 30 s, the overall residual stress of the cladding is the smallest.
Equation 10 shows that the generation of stress in additive manufacturing is influenced by the change in temperature gradient during the deposition process, and a small temperature gradient during the deposition process results in low stress. The change of temperature field at the same position of cooling time between different layers in the L-DED process is shown in figure 10(a) (Intercepted temperature field nephogram of the x-z plane of the deposit near the heat source), the peak temperature during deposition are located at the center of the heat source, as shown by the black dashed line in the figure, the temperature of the two layers of material near the heat source is higher than the melting point of TB18 titanium alloy, which achieves the melting and metallurgical bonding of the powder. From the sparseness of isotherms near the heat source, it is known that the temperature change near the heat source at this location where the interlayer cooling temperature is 30 s is more  moderate. In order to observe the temperature change process, the coordinates of the midpoint of the first pass of the first layer were taken as (22.5,4,0.8) and the average value of its temperature gradient throughout the deposition process was observed, as shown in figure 10(b). The average temperature gradient of the deposition process gradually increases with the increase of the interlayer cooling time. At the same interlayer cooling time, the peak temperature increases and the temperature gradient decreases with the deposition process.
During the deposition process, heat accumulates in the cladding and the previously deposited cladding has a preheating effect on the subsequent cladding. When the heat source power is constant and the interlayer cooling time increases, the deposition temperature at the beginning of each layer decreases, as shown in figure 11, and the temperature gradient increases, so the residual stress increases. As the cooling time between layers decreases, the deposition temperature at the beginning of each layer increases further, the temperature gradient decreases further, and the residual stress in each cladding decreases.
From the simulation results, it can be seen that the residual stress in the cladding decreases with the decrease of the interlayer cooling time, and the stress distribution pattern is not affected by the interlayer cooling time, without changing the remaining process parameters. The residual stress decreases as the layer height of the cladding increases. When the interlayer cooling time is 30 s, the overall residual stress value of the cladding is the smallest, so the optimal interlayer cooling time is 30 s.

Effect of scanning strategy on residual stresses
Three different scanning strategies were used, in which the inter-layer cooling time is 120 s, and the remaining process parameters are maintained constant, as a way to investigate the effect of different scanning strategies on the residual stresses in the components build by laser directed energy deposition. The stresses under different  additive scanning strategies are shown in figure 12. The three scanning strategies have the same stress state distribution, showing a 'compression-tension-compression' stress state, and the middle tensile stress is larger and the two sides are smaller. The peak stress magnitude of the additive part changes as the scanning strategy changes. The peak tensile stress value of the first scanning strategy is 107.5 MPa, the peak tensile stress value of the second scanning strategy is 41.3 MPa, and the peak tensile stress value of the third scanning strategy is 263.6 MPa. Compared with the first scanning strategy and the third scanning strategy, the use of the second reciprocal scanning strategy can reduce the peak stress by 61.6% and 84.3%, respectively, and the stress distribution is smoother along the selected study path.
The stress distribution in the top layer of the fused layer for the three scanning strategies is shown in figure 13. According to the stress field cloud, when the scanning direction is along the long side, the stress field is shown in figures 13(a), (b), and the residual stress distribution law is basically the same for the first and the second scanning methods. When the scanning direction is changed to along the short side, the stress field is shown in figure (c), and the residual stress distribution pattern of the cladding changes, and the peak residual stress in the middle part increases significantly.
The temperature field distributions at different moments of the 10th layer for the three scanning strategies are shown in figure 14. According to the temperature field cloud map and can be seen, the front of the heat source isotherm distribution is dense, the temperature gradient is large, indicating that the heat is more  concentrated. The temperature distribution double ellipsoidal heat source distribution pattern is the same, and changing the scanning strategy does not affect the temperature distribution pattern near the heat source. The overall temperature increases with the deposition process. When the scan is along the long side, the temperature field distribution is basically the same between the unidirectional scan and the middle moment in the reciprocal scan, as shown in figures 14(b), (e), but the isotherm distribution is sparse and the temperature gradient is smaller in the reciprocal scan. When the scan becomes along the short side, the temperature field distribution at the middle moment is shown in figure 14(h), and the isotherm distribution is denser than the other two paths, which means that the temperature gradient is large.
Under either scanning strategy, the middle part of the member has difficulty in dissipating heat and is more likely to generate heat accumulation compared to the edge position of the member, which leads to an increase in temperature gradient and causes an increase in stress, which is the same as the results of Hongjian Zhao [13] et al. The reciprocal scanning strategy makes the whole deposition process more continuous and the temperature gradient is smaller compared to the unidirectional scanning strategy, so the overall residual stress is smaller and the stress variation is smoother compared to the unidirectional scanning strategy. Deposition along the long side can alleviate the serious heat accumulation problem in the middle of the member during deposition along the short side, reduce the temperature gradient in the deposition process, reduce the thermal stress in the deposition process, and thus alleviate the stress concentration phenomenon in the middle of the member.
In combination with the above discussion, laser directed energy deposition for fabrication of TB18 titanium alloy with scanning strategy 2 results in parts with minimal peak tensile stress.