Numerical analysis of the effect of the scan strategy on the residual stress in the multi-laser selective laser melting

The inevitably formed residual stress in the Selective Laser Melting (SLM) process leads to distortion, crack and even delamination of the workpiece. Single laser is commonly applied during SLM processing. However, its productivity is much lower than multiple lasers. In addition, the research of residual stress with multi-laser condition currently is limited in the open documents. In this paper, a three-dimensional (3D) thermo-mechanical model, with considerations of temperature dependent properties of Ti-6Al-4V, phase change and convective flow, is developed at first. Then, the numerical results of maximum temperature and dimensions of the molten pool are validated by available experimental data. Furthermore, a parametric study in regards to a series of scan strategies is investigated. According to the simulation results, the residual stress increases significantly when the laser number reaches four. The “two-zone technique” scan strategy decreases the equivalent residual stress by 10.6% compared to the successive scan strategy. With a shortening scan length, the residual stress first increases slightly, then decreases dramatically and attains the minimum when it is a quarter. Furthermore, for the multi- laser SLM process, carefully planning the scanning sequence and the sweeping direction to decrease heat concentration is beneficial in controlling the residual stress.


Introduction
Selective Laser Melting (SLM), as a promising Additive Manufacturing (AM) technology, is widely used in aerospace, biomedical, mold manufacturing and other industries in recent years [1][2][3]. The near-net-shape, geometrically complex and high-density metallic structures can be fabricated using this technology within a short design and manufacture cycle [4,5]. Due to the factor of rapid heating and cooling in the SLM processing, residual stresses are often inevitably generated in the deposited layers and substrate that leads to the distortion, crack and even delamination of the workpiece [6][7][8]. Therefore, it is of significance to investigate the mechanism of how SLM processing conditions affect the residual stress and then optimize the process parameters to control the residual stress. However, the experimental measurement of residual stress is expensive and time consuming. And the accuracy of the results depends on the dimension and precision of the workpiece and the accuracy of X-ray [9]. Moreover, the three-dimensional (3D) transient temperature field in the SLM processing cannot be obtained using thermocouple and infrared thermography [10,11]. The validated numerical simulation method is an effective alternative for evaluating the temperature field and the residual stress field.
Previously, researchers have developed numerical models to simulate the AM process [12,13]. Xinran et al. [14] developed a 3D transient model to investigate the thermo-mechanical response of Ti-6Al-4V part, but the accelerated heat transfer resulted from the convective flow within the molten pool was not taken into consideration. Mohammad et al. [15] and K. Ren et al. [16] increased the thermal conductivity of liquid metal to minimize temperature errors, while the convective flow is not considered. This leads to the calculated molten pool dimension inconsistent with the reality. A thermo-mechanical model coupled Finite Volume Method (FVM) with finite element method (FEM) was developed to predict temperature and residual stress distribution of the thin wall fabricated by laser cladding method [17]. Although the calculation results agree well with the experimental results, the thermal properties of powders are not considered in this simulation. Therefore, an integrated therm-mechanical model which taking consideration of reasonable heat source, convective flow, phase change, variable thermal and mechanical properties, is still in a great demand, in order to accurately simulate the SLM process.
The scan strategy is critical for the formation of residual stress in the final workpiece and researchers have investigated the influence of scan strategy on residual stresses [18][19][20][21]. Cheng et al. [22] analyzed the effects of six different scan patterns on material temperature distribution, stresses and deformations, and found that the component using 45 degrees inclined scanning method had the smallest residual stresses and distortions. Ali et al. [23] experimentally measured residual stresses in SLM Ti-6Al-4V parts and concluded that rotating scanning direction 90 degrees for each layer was conducive to reduce residual stress. Parry et al. [12] and Wu et al. [24] drew the same conclusion that reducing the length of the scan vector was effective in reducing residual stress.
However, Lu et al. [25] experimentally measured four SLM parts using 2 × 2 mm, 3 × 3 mm, 5 × 5 mm and 7 × 7 mm island scan strategies, and found that 3 × 3 mm scan length led to the highest residual stress. Thus, understanding the influence of scan strategy on the residual stress is still of great value.
In order to improve the productivity of the small-size SLM part, multiple lasers are introduced to decrease the time for laser scanning and depositing/removing powder [15]. With the multi-laser scan strategy development, the SLM process becomes dramatically complicated [26][27]. Unfortunately, the effect of multi-laser scan strategy on residual stresses has not been clearly revealed. Thus, in this paper, a thermo-mechanical model of the SLM process is presented to investigate the effect of scan strategy for single laser and multiple lasers on the residual stress, with consideration of temperature, phase dependent properties, phase change and convective flow. First, the

Physical model and assumption
After irradiated by laser beam, metal powders rapidly form into the molten pool. As the laser beam moving away, the molten pool cools down as a result of heat transfer including the heat conduction to deposited material and powder as well as the convection and radiation to the ambient atmosphere. Inside the molten pool, the Marangoni Effect promotes the flow of high temperature liquid metal to low temperature which accelerates the cooling process. The heat transfer, phase change, microstructure growth and more mechanisms result in the complexity of thermal behavior and mechanical condition. In this section, a transient thermo-mechanical numerical model is established to account for the complexity in the SLM process. In this model, the temperature and the liquid flow fields are first simulated in the thermal model, and then the results are imported to the mechanical model as the thermal load. Note that in this paper, the following four assumptions are made in order to simplify the numerical model: 1. The discrete powder bed is simplified as a homogeneous and continuous media. 2. Fluid in the molten pool is considered to be an incompressible flow of Newtonian fluid. 3. The surface of the deposited layer is considered to be flat. 4. Effects of strains induced by solid-state phase transformation are neglected.

Governing equation
In light of the thermal analysis, the Finite Volume Method (FVM) is adopted to calculate the temperature and flow fields. Based on the above assumptions, the equation of energy conservation is solved in this model, which can be expressed as: where ρ is the density, c p is the heat capacity, T is the temperature of martial, u is the velocity of the fluid flow, k is the thermal conductivity, ΔH refers to the latent heat of phase change, q l is heat loss on the top surface of the powder bed and Q is the volumetric Gaussian heat source. For a given element located at (x, y, z), the heat source Q(x, y, z) can be defined as: where A p is the absorption of the powder bed, P refers to the laser power, d is the distribution coefficient of laser beam, r 0 is the radius of the laser beam, (x l , y l ) is the horizontal location of the laser beam in the coordinate system of workpiece and I(z) refers to the exponential decay in z direction which is expressed as follows [28]: where A z is the attenuation coefficient, z t is the vertical location of the top surface of the processing layer.
To figure out the mechanical response in SLM process, the temperature field is imported into a quasi-static mechanical model based on the Finite Element Method (FEM). The total strain ε tot consists of elastic strain ε e , thermal strain ε th and plastic strain ε p shown as follows: A multilinear isotropic hardening model with Von-Mises yield criterion is used to calculate plastic strain. The elastic stress which is residual stress in the workpiece relative to elastic strain is expressed as:   where [D] is the elastic stiffness matrix. The thermal strain is defined as: where α th is the coefficient of thermal expansion and T ref is the reference temperature where thermal strain is assumed to be zero.

Boundary conditions
The temperature gradient on the surface of the molten pool contributes to the surface tension gradient, which induces Marangoni Effect. Therefore, the convective flow develops within the molten pool under the effect of Marangoni Effect, gravity and buoyancy. The Marangoni stress τ m can be defined as [29]: where τ mx and τ my are Marangoni stress in the × and y direction respectively, γ is the surface tension and dγ/dT is called the Marangoni coefficient.
In addition, the temperature-dependent convective and radiative heat flux which is the main heat loss on the top surface is considered in the physical model. The side surfaces of the part and substrate are considered to be heat insulated, and the bottom of the substrate is assumed to be at a fixed temperature which is equal to the ambient temperature. In addition, a fixed surface constraint is applied to the bottom of the substrate for mechanical analysis.

Material properties
To accurately predict the temperature field of workpiece, thermal properties of the metal in different phases should be considered, especially the thermal conductivity of powder and the heat capacity during phase change. The effective thermal conductivity of the powder bed mainly depends on the porosity. The equation of powder thermal conductivity k p can be expressed as [30]: where k g is the thermal conductivity of the ambient gas in the chamber, Φ is the porosity of the powder bed, k s is the thermal conductivity of solid metal and k r is the thermal conductivity of powder induced by radiation, which can be defined as: Fig. 6. Schematics of 18 scan strategies: (1) S14A, (2) S14B, (3) S14C, (4) S12A, (5) S12B, (6) S12C, (7) S12D, (8)S11A, (9) where F 0 is the radiation view factor which is taken as 1/3, D p is the average diameter of powder particles.
Due to the large amount of latent heat during phase change, the enthalpy of material H and sensible enthalpy h can be defined as: where h ref and T ref is the reference enthalpy and the corresponding temperature, θ is the liquid fraction which can be defined as: where T s and T l are the solidification and liquidation temperatures.
For the mechanical analysis, the elastic modulus, the yield strength,   Fig. 11. The cross-sectional temperature distributions of (a) S14A, (b) S12A and (c) S11A. the thermal expansion, the powder stress, the powder strain and liquid metal are assumed to be negligible. Therefore, the birth and death of element method is adopted, which can be used to clear the stresses or strains of deactivated elements [31].

Simulation overview
The Computational Fluid Dynamics (CFD) software FLUENT and FEM software ANSYS are used to calculate the temperature field and the stress field, respectively. The simulation process, as shown in Fig. 1, consists of thermal simulation, data conversion and mechanical simulation.
To improve the convergence and accuracy for both thermal and mechanical calculations, the powder bed and the contacted regions of substrate are meshed using hexahedral cells with fine resolution. The rest part of the substrate is meshed with coarse tetrahedral cells in order to save computing time. A User-defined Function (UDF), which is related to both time and temperature, is developed to define thermophysical properties, heat source and laser scan pattern. To distinguish the thermal properties between powder metal and liquid/solid metal, a 3D array is introduced to record the phases of all the powder bed elements. Once the temperature of the powder element reaches melting point, the liquid/solid metal property is assigned to it and the array is updated simultaneously.

Validation
The accuracy of the numerical simulation model is validated by experimental results from the literature [15,[32][33][34]. The molten pool dimensions or maximum temperatures of single-track Ti-6Al-4V with different scanning speeds are generally utilized for verification as shown in Fig. 2 [12,15,26,34].
Temperature-dependent thermal and mechanical properties of Ti-6Al-4V from the published literature are introduced in this simulation [35][36][37]. For the material in mushy zone, where the temperature is between solidus and liquidus temperature, the liquid fraction θ is introduced to express thermal properties. To restrict the movement of elements in FLUENT, the viscosity of the solid phase is set to be 1 N/ (m•s) which is extremely higher than that of the liquid phase.
The laser absorption of the melt pool is assumed to be temperature and phase independent and near-equal to that of solid and powder materials during the SLM process [15,34,35]. To determine the laser absorption and attenuation coefficient, simulations that operate SLM at the substrate Ti-6Al-4V without powder are performed firstly [32]. The SLM processing parameters are shown in Table 1. The laser power is fixed on 50 W and scanning speed varies between 100 mm/s, 200 mm/s and 300 mm/s. Since there is no pre-heat process, the initial temperature of the substrate, the powder bed and the ambient atmosphere are assumed as 300 K. As shown in Fig. 3, the simulation results are consistent with the experimental results under the values of laser absorption 0.47 and attenuation coefficient 5 × 10 4 . The molten pool dimension and the maximum temperature are both within the experimental range. The average error rate between the calculated results and the average experimental results is 1.5%. To further validate the numerical simulation model, simulations of SLM processes with 30 μm thick Ti-6Al-4V powder reported by Chen are performed and the results are compared with those of experiments [34]. The SLM processing parameters are shown in Table 1. By altering the scanning speed between 125 mm/s, 200 mm/s and 250 mm/s with constant laser power, the linear heat inputs range from 200 J/mm to 400 J/mm that are consistent with experimental conditions. As shown in Fig. 4, the simulation results agree well with the experimental results. The maximum relative error for the molten pool width and depth are 7.18% and 15.3%, respectively. Thus, it can be concluded that the calculated temperature field can be regarded as thermal load for the residual stress calculation.

Scan strategy
The Ti-6Al-4V SLM process with multiple tracks and two layers is simulated using the validated numerical model. The dimensions of the powder bed and the substrate are 5 × 5 × 0.1 mm 3 and 8 × 8 × 1 mm 3 respectively. As reported by Yadroitsev, the optimal scanning speed is 120 mm/s with 50 W laser and 50 μm powder layer thickness [32]. According to the simulation result, shown in Fig. 5, the depth of the melt pool is 68.4 μm which is higher than the powder layer thickness. It means that the powder bed can be fully penetrated. In addition, to achieve high relative density, the hatch spacing is selected as 100 μm that is shorter than the width of the intersection of the melt pool and the substrate (107 μm). The other simulation parameters are shown in Table 1.
There are 18 simulations with various scan strategies performed in this paper where the first and second layers are fabricated in the same scan strategy. The schematics of these scan strategies with different numbers of lasers, scan lengths, scanning directions, sweeping directions and sequences are shown in Fig. 6. The direction of the arrow represents the scanning direction of the laser, and the sweeping direction is perpendicular to the scanning direction and points to the ellipsis. Different colors and line styles of arrows indicate sequences in the simulation. Taking case S14A as an example, the 'S' stands for 'scan strategy'; the '1′ means that the number of lasers is 1; the '4′ represents that the length of scan track is 4 times of the minimum scan length; the 'A' is used to distinguish other factors, such as scanning direction, sweeping direction and scanning sequence.

Results and discussion
The powder bed is rapidly heated by the laser beam and the heat transfer within the material is relatively slow, hence a steep temperature gradient is developed in the workpiece leading to the formation of the molten pool. Under the effect of plastic deformation during the heating and cooling process, residual stresses develop in the heated zone around the melton pool. Meanwhile, for the molten pool (fusion zone), no residual stress or plastic strain develops in melting state due to the fluidity of molten metal. However, the solidification and the shrinkage of the molten pool lead to tensile stress as temperature drops down. These two mechanisms determine the formation of residual stress in the workpiece [7].
To detail the effect of scan strategy on residual stress, the transversal stress σ x , the longitudinal stress σ y and the equivalent stress σ e of 64 points are investigated, which are uniformly distributed on the top surface of each layer. As shown in Fig. 7, the average values of σ x , σ y and σ e have been calculated for all the cases.
In general, residual stresses of the second layer are slightly lower than those of the first layer. As the horizontal dimension of the deposited layer is smaller than that of the substrate, the first deposited layer is easier to deform than the substrate [14]. As a result, less plastic strains and residual stresses are developed in the second layer. Since residual stress trend of the second layer is similar to the first layer, the discussions below are mainly focused on the results of first layer.

Two-zone technique and scanning direction
Since residual stresses of S14C are lower than those of S14A or S14B in both longitudinal and transversal directions, as shown in Fig. 7 (a), it can be concluded that "two-zone technique" scan strategy effective in lowering the residual stress. In the first sweep of S14C, the deposited tracks have fewer constraints from adjacent tracks because of large hatch space. This results in a relatively low residual stress. During the second sweep, scan tracks are surrounded by solidified materials with higher thermal conductivity, the heat accumulation and the dimension of molten pool will decrease. As shown in Fig. 8, the width of molten pool in the second sweep of S14C (125.5 μm) is 14.2% shorter than that of S14A (146.3 μm), and the depth of S14C (53.5 μm) is 8% shorter than that of S14A (58.2 μm). Since larger molten pool results in more volume shrinkage after solidification, the residual stress in S14A is higher than that in S14C in the second sweep. Furthermore, it can be seen that altering scanning directions has little effect on the average residual stress, since the average σ e stress of S14B is similar to that of S14A.

Number of lasers
Comparing cases with same scan patterns, excluding case S22A, there are no significant differences in average σ e stresses when the number of lasers increases from 1 to 2. This is due to the fact that lasers are relatively far away in the double-laser scanning process that avoids the heat concentration effect. However, as shown in Fig. 7 (a), the average longitudinal stress of case S22A is 13.3% higher than that of case S12A. Since the scanning vectors of the two lasers are collinear, scan tracks merge on the longitudinal direction in S22A. As a result, both the head of the upper scan track and the end of the bottom one are obstructed in shrinkage that induces high residual stresses. The residual stress distribution is continuous at scan track joints in S22A ( Fig. 9 (b)), which is different from that of S12A ( Fig. 9 (a)). This effect is equivalent to an increase in the scan track which is called the scan track extension effect (STEE) in this article.
The results in Fig. 7 (a) show that the residual stress becomes significantly high when the number of lasers comes to 4, for instance, the average equivalent residual stress of S42B is 9.0% higher than that of S22B. Taking S22A and S24A as examples, the cross-sectional temperature distributions of S24A and S44A are shown in Fig. 10. It can be seen that the heated zone round the molten pool of S24A ( Fig. 10 (a)) is smaller than that of S44A ( Fig. 10 (b)) due to the shorter relative distance between lasers in S44A. Thus, the average longitudinal stress of S44A is 9.8% higher than that of S24A.

Scan length
With the scan length decreasing for the single laser SLM process, the average longitudinal stress increases slightly from 632.3 MPa (S14A) to 661.4 MPa (S12A) and then drops dramatically to 575.2 MPa (S11A). Also, the multi-laser process has the same trend. In order to investigate the effect of scan length on residual stress, S14A, S12A and S11A are taken as examples. Since the scan length decreases, the constraints on the scan track during the cooling process reduce that effectively mitigates residual stresses. In addition, it can be seen by the cross-sectional temperature distributions of S14A, S12A and S11A which are shown in Fig. 11 that the temperature gradient decreases with the scan length. However, reducing scan length leads to a larger dimension of molten pool, resulting in larger scale shrinkage and higher residual stress for fusion zone.
Based on the above analysis, it can be found that the changes of constraints on scan tracks, temperature gradient and molten pool sizes resulted from varying scan length have opposite effects on residual stresses. This explains the phenomenon that the longitudinal residual stress has no direct correlation with the scan length.
On the transversal direction, it is clear that the transversal residual stress increases with the decrease of scan length, since average σ x stresses of S14A, S12A and S11A increase successively. The fact is that the heated zone perpendicular to scan vector increases with the decrease of time interval that induces more plastic deformations and higher residual stresses.

Scanning sequence and sweeping direction
From Fig. 7 (a), there are no obvious differences between residual stresses of S12A, S12B and S12C. It can be concluded that the scanning sequence has little effect on residual stress for the single laser process. However, residual stress is sensitive to scanning sequence under the condition of multi-laser scanning. The average σ e stress of S22A is 11.9% higher than that of S22B and the average σ e stress of S41A is 22.8% higher than that of S41B. This is because scanning vectors are collinear in both S22A and S41A that results in the STEE. Therefore, it is significant for multi-laser SLM process to reduce residual stress by carefully planning scanning sequence to mitigate the STEE between islands. Although 4 lasers have been employed, the average σ e stress of S41B is close to that of cases with double-laser scan strategies.
From results of S12A, S12D, S42A and S42B, it can be indicated that varying sweeping direction is beneficial for residual stresses to be evenly distributed in both longitudinal and transversal directions. As shown in Fig. 7 (a), there is no big difference in the average Von Mises stress between S12A and S12D. This indicates that varying sweeping direction for the single laser SLM process has little effect on the magnitude of residual stress. However, the average σ e stress of S42B is 4.1% lower than S42A. Hence, it is significant to apply appropriate sweeping directions to reduce residual stresses for the multi-laser scan strategy. Furthermore, the mergence of melted regions between upper and bottom islands in S42B, as shown in Fig. 12, promotes constraints on scan tracks. As a result, the distributions of longitudinal and transversal residual stresses are not rotated symmetric, as shown in Fig. 13, and the longitudinal average stress is 6.6% higher than transversal one. Through varying sweeping directions to mitigate merging effect among the islands, the average σ e stress of S42C is reduced by 2.2%.

Conclusion
A 3D thermo-mechanical model where the temperature and phase dependent properties of Ti-6Al-4V, phase change and convective flow are considered is developed to simulate the SLM process. The numerical model is first validated by experimental results and then used to calculate temperature fields and stress fields with different scan strategies. The major findings are summarized below: (1) The "two-zone technique" scan strategy has shown a reduction in the residual stress by 10.6% in comparison to the traditional successive scan strategy. (2) The residual stress is significantly higher as the number of lasers increases to four, which is due to more heat input. (3) The residual stress is sensitive to the scan length for both the singlelaser scan strategy and the multi-laser scan strategy, but there is no consistent correlation between them. With the scan length decreasing in the single laser SLM process, the average longitudinal stress first increases slightly by 4.6% and then decreases dramatically by 13.0%. (4) For the multi-laser SLM process, the scanning sequence and the sweeping direction are two crucial factors in controlling residual stress. After changing the scanning sequence of S41A, the average equivalent stress of S41B reduces by 19.0%. Meanwhile, by altering the sweeping direction of S42A, the average equivalent stress of S42B decreases by 6.2%.