Finite element analysis of temperature and residual stress profiles of porous cubic Ti-6Al-4V titanium alloy by electron beam melting

1. Introduction

Titanium alloy is widely used for the fabrication of orthopedic implant because of high mechanical strength, sound corrosion resistance, and good biocompatibility. However, the tremendous difference of elastic modulus between dense titanium alloy implants and bone creats the stress shielding phenomenon at the interface, which results in the premature failure of the implant [1]. Thus medical porous titanium alloy were developed and can be adjusted by the porosity to match the elastic modulus of bone tissue [2,3]. In addition, the three-dimensional opened networks structure would help the transportation of body fluid and nutrient, promoting the regeneration and repair of bone tissue and realizing the combined effects of structure, mechanics and bionics of porous titanium alloys [4,5]. Conventional fabrication processes, such as powder metallurgy [6], rapid prototype [7] and freeze-drying methods [8], are difficult to produce porous implants, with complex shape and desired porosity [9].

Electron beam melting (EBM) is an advanced additive manufacturing (AM) technique with the advantages of high flexible, rapid response on structure design, near net shape [10], and is an ideal choice for direct fabrication of three-dimensional parts with porous structures [11]. Significant research efforts have been required to fabricate biomedical porous titanium alloys with an emphasis on Ti-6Al-4V alloy by EBM [12,13], the relationship among microstructure, mechanical properties and EBM processing parameters [14,15], and the thermal [[16], [17], [18]] or stress behavior [[19], [20], [21]] by finite element method or other numerical simulation methods.

Luo and Zhao [22] reviewed and summarized the significance of finite element method in the connection of additive manufacturing issues such as material design, in-process monitoring and control, and process optimization. Pan et al. [23] simulated the build thickness dependent microstructure of EBM Ti-6Al-4V with thicknesses of 1 mm, 5 mm, 10 mm, 20 mm and found that cooling rates and thermal profiles during EBM process are favorable for the martensite formation and martensitic decomposition is faster in thicker samples. Huang et al. [24] investigated the effects of the linear energy density, volume shrinkage, scanning track length, hatch spacing and time interval between two neighboring tracks on the temperature distribution and molten pool dimensions. The rapid heating and cooling of EBM processes created residual stresses in as-built material, which has been predicted by theoretical modeling as well as experimentally verified. Romano et al. [25] compared the temperature distribution and melt geometry in laser and electron-beam melting processes containing titanium, stain-less steel, and aluminum powders. Vastola et al. [26] performed systematic finite element modeling of one-pass scanning to study the manufacturing parameters on the magnitude and distribution of residual stresses. Wu et al. [27] found that a reduction in residual stress is obtained by decreasing scan island size, increasing island to wall rotation to 45 deg., and increasing applied energy per unit length.

These numerical simulations of thermal or stress behaviours were mainly focused on the bulk materials during EBM of metal powders. There are limited researches on thermal and stress behaviours of porous titanium alloys during EBM. It is necessary to illustrate the complex thermo-mechanical behavior during the forming process for understanding the cyclic phase transformation and the integrated control of forming and performance. In this paper, establishment of three-dimensional finite element (FE) model of porous cubic Ti-6Al-4 V alloy during EBM based on solutions to heat transfer equations has been performed, and the thermal and stress behaviors, considering the non-linear temperature-dependent physical properties, were predicted with the aim of understanding the relationship of porous structure, process parameters and thermal or residual stress distribution.

2. Experimental

FE method framework was designed based on the ANASYS commercial software. The porous cubic FE models and electron beam scanning strategies are shown in Fig. 1. The pore size and strut of porous cubic structures are 1 mm × 1 mm and 0.2 mm, respectively. The sizes of single-layer and five-layer structures are 2.6 mm × 2.6 mm × 0.05 mm and 2.6 mm × 2.6 mm × 0.25 mm, respectively. Each layer is 0.05 mm. The substrate is 4.6 mm × 4.6 mm × 0.5 mm. The mesh size for powder bed and melt pool with 8-node hexahedron element is 0.025 mm × 0.025 mm × 0.025 mm and for the substrate with 10-node tetrahedron element is 0.2 mm × 0.2 mm × 0.2 mm. Two kinds of scanning strategies of electron beam shown in Fig. 1(c, d) were performed: annular and lateral scanning. The hatch spacing is 0.2 mm. Fig. 1(e, f) shows the points or scanning paths for analysis of thermal and stress behaviors.

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Fig. 1.   FE models of porous cubic structures: (a) single-layer FE model, (b) five-layer FE model, (c) annular scanning, (d) lateral scanning, (e, f) points a, b, c, d and P4, P5, P6 paths for analysis of temperature and stress behaviors. a₁‒a₅ represents the points from the first layer to the fifth layer at node a.

During EBM, the energy equation can be expressed by [17]:

$\rho c \frac{\partial T}{\partial t}=\frac{\partial}{\partial x}(k_{x} \frac{\partial T}{\partial y})+\frac{\partial}{\partial_{z}}(k_{x}\frac{\partial T}{\partial z})+Q$ (1)

where ρ is the density (kg/m³), C is the specific heat capacity (J/(kg K)), t is the time, k is the heat conductivity (W/(m K)), Q is the density of heat flux (W/kg). The electron beam was modeled as a Gaussian power distribution, and the heat flux q is given by [17]:

$q=\frac{2AP}{\pi \omega^{2}exp(\frac{-2r_{2}}{\omega^{2}})}$ (2)

where A is the powder absorptivity, ω is the beam spot radius, P is the electron beam power, and r is the radial distance from the center of the heat source.

Temperature boundary for initial thermal condition were kept at the fixed temperature of 730 °C (T₀) to reproduce the pre-heating step that preceded in actual EBM building. Considering the radiation and environmental convection, the boundary conditions of the temperature field during EBM are defined as [21]:

$k_{x}\frac{\partial T}{\partial x}n_{x}+k_{y}\frac{\partial T}{\partial y}n_{y}+k_{z}\frac{\partial t}{\partial z}+h(T-T_{0})+\sigma \varepsilon (T^{4}-T_{0}^{4})-Q=0$ (3)

where n is the vector along the normal direction of the surface, h is the natural convection coefficient (25 W/(m² K)), ε is the emissivity (0.35), and σ is the Stefan-Boltzmann constant taken as 5.67 × 10^-8 W/(m² K⁴).

The density of dense solid was taken as ρ_s=4440 kg/m³ and the powder density was calculated by ρ_p=(1-ϕ)ρ_s, where ϕ is the porosity (0.65). The thermal conductivity, thermal expansion coefficient, specific heat and latent heat of fusion are specified as non-linear temperature-dependent and taken from Refs. [25,28,29]. The solidus and liquidus temperatures are 1540 °C and 1649 °C, respectively. The detailed process parameters used in the FE analysis are shown in Table 1.

The “element birth and death” technique was introduced in the calculation and temperature field was calculated by ANSYS parametric design language. Then temperature field was applied as thermal loadings for mechanical analysis that was performed in the uncoupled way. The equation with thermo-mechanical coupled model for non-linear mechanical analysis can be given by [17]:

$\nabla \cdot \sigma=0$ (4)

where σ is the stress. Considering the elastoplastic behavior of Ti-6Al-4V alloy, the total strain consists of the elastic, plastic and thermal strains. The mechanical constitutive law for elastic deformation was expressed as σ = Cε_e, where C is the material stiffness tensor. Plasticity was modeled with isotropic hardening: ε_p = g (σ_y), where σ_y is the yield strength. The change of thermal strain was given by dε_T = αdT, where α is the thermal expansion coefficient and dT is the change in temperature during the time step.

3. Results and discussion

3.1. Thermal analysis

Fig. 2 shows the molten pools of porous cubic grids under annular and lateral scanning. Using the present process parameters, the width and depth of molten pool were about 211 μm, 81 μm under annular scanning and 207 μm, 72 μm under lateral scanning, respectively, which exceeds the hatching spacing (200 μm) and thickness of powder layer (50 μm). The maximum temperature can reach 2269 °C and 2197 °C under annular and lateral scanning, exceeding the melting point of the alloy. This guaranteed the overlaps between vertically deposited layers and horizontally adjacent tracks to obtain the sound metallic bonding.

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Fig. 2.   Width and depth of molten pool under annular (a) and lateral (b) scanning.

Fig. 3 shows the temperature distribution under annular and lateral paths after scanning for the single-layer FE model. The shape of “comet tail profile” for molten pool was not obvious, where this phenomenon was observed in many bulk materials [23]. Scanning speeding (200 mm/s) in our calculation is lower than that in references (1500-4000 mm/s) and there is enough time for the heat absorption of powder bed and heat dissipation along solidified areas. The temperature gradient in the front side of the moving molten pool is much larger due to low thermal conductivity of powders and is relatively smaller in the back side of molten pool due to the higher conductivity of solidified areas.

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Fig. 3.   Temperature distribution under annular (a, b) and lateral (c, d) scanning at point a for single-layer FE models.

After electron beam source passing, it was found that the molten pool temperature under lateral scanning is lower than that under annular scanning. During EBM, the time for scanning one layer of powder under lateral path is 0.071 s, while it is 0.095 s under annular path. The struts underwent remelting under annular scanning and more energy was accumulated, leading to the higher molten pool temperature.

Fig. 4 shows the relationship of temperature, cooling rate and time under annular and lateral scanning for the single-layer grids. The molten pool temperature cooled down resulting in a positive cooling rate and the negative cooling rate represented the rise of temperature within molten pool. Under annular scanning, point a located in the middle of grids and underwent four times of electron beam scanning. Thus the molten pool temperatures have four peaks at point a: 1980 °C, 2137 °C, 1723 °C, 1539 °C. Under lateral scanning, point a underwent once electron beam scanning and there was only one peak temperature exceeding the melting point of Ti-6Al-4 V alloy. Other two peaks were affected by adjacent scanning tracks. After scanning, the maximum cooling rates for annular and lateral paths were 6.32 × 10⁵ °C/s, and 5.67 × 10⁵ °C/s, respectively.

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Fig. 4.   Correlation of temperature, cooling rate and time at point a under annular (a) and lateral (b) scanning for single-layer FE models.

In order to model the actual EBM process, the five-layer FE model was established and analyzed. Fig. 5 shows the temperature field of each layer under annular scanning. With increasing the powder layers, heat accumulation is becoming more significant. Compared to the first layer, the substrate temperature increased from 730 °C to 867 °C and molten pool temperature increased from 2082 °C to 2404 °C for the fifth layer due to the higher solid material thickness and the heat accumulation in the girds. The similar results were obtained under lateral scanning with less magnitude of temperature rise. For the first layer, the prediction for the maximum temperature is 2011 °C under annular scanning, while for the fifth layer it is 2187 °C.

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Fig. 5.   Temperature field distributions after annular scanning: (a) the first layer, (b) the second layer, (c) the third layer, (d) the fourth layer, (e) the fifth layer.

It is worth noting that the higher thermal capacity of the five-layer grids allowed them to retain more heat compared to the single-layer grids, because they have a larger mass. Moreover, the five-layer grids also have less time to cool down between successive scanning for each layer, suggesting that a lower cooling rate was involved in the five-layer grids as compared to the single-layer grids. Thus the five-layer grids have a higher average temperature than the single-layer ones.

The temperature distribution of point a after scanning are shown in Fig. 6. As scanning the fifth layer, the temperature of the fourth layer reached the melting point again, indicating the good metallurgical bonding between adjacent layers. With the increase of the distance from scanning track along build direction, the influence of thermal impact remarkably weakened, and with increasing time, temperature gradually decreased. The temperature under annular scanning is about 100 °C higher than that under lateral scanning. Temperature behavior of each layers under annular and lateral scanning in the five-layer grids were similar to that in the single-layer grids. For point a, there are four temperature peaks of each layer under annular scanning and one major temperature peak under lateral scanning. The temperature evolution shows a typical periodic behavior as each peak represents once electron beam scan.

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Fig. 6.   Relation of temperature and time at point a with 0.5 s after annular (a) and lateral (b) scanning for five-layer models.

Fig. 7 shows the temperature gradient along the P4, P5 and P6 paths just after annular and lateral scanning. The temperature gradient fluctuated greater under annular scanning than lateral scanning along the P4 and P5 paths and the larger temperature gradient was observed under annular scanning along the P6 path. In the five-layer models heat accumulation is remarkable with the increase of powder layers, and scanning time of each layer is longer for the annular scanning than lateral scanning. Thus annular scanning has the effect to deepen the heat affected zone to allow more time for heat conduction away from the melt pool and into the grids. As a result, thermal gradients occurred at a greater depth and exhibited the larger fluctuation compared to lateral scanning.

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Fig. 7.   Temperature gradient distributions along P4 (a), P5 (b), and P6 (c, d) paths just after scanning.

3.2. Stress behavior

Fig. 8 shows the relation of Von Mises stresses of points a, b, c, d, e vs. time under annular and lateral scanning for the single-layer model. The thermal stress distributions during heating period were also shown in the figures. The variations of the stresses were complicated as the electron beam rapidly moved forth and back. During EBM process stage (0‒0.1 s), point a under annular scanning remelted by the electron beam and afterwards as the electron beam passed away, the high tensile stresses occurred. This resulted from the heated zone cooling down and the solidification shrinkage partially restrained by the plastic strain. When the electron beam passed over point a again, point a remelted and cooled down. Consequently, the second peak value of thermal stress is obtained. Point a experienced four remelting-solidifying cycles and thus has four peaks of thermal stress. The other points b, c, d, e experienced twice remelting-solidifying cycles and have two peaks of stress. Under lateral scanning, point a is subjected to the effect of moving electron beam and adjacent tracks, and the thermal stress exhibited two peaks.

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Fig. 8.   Von Mises stress distributions of points a, b, c, d, e under annular (a, b) and lateral (c, d) scanning and residual stress values (e, f) in single-layer models.

After cooling down to preheat temperature (730 °C), the stresses finally stabilized to a certain level and residual stresses were extracted by computing the stress profile shown in Fig. 8(e, f). The final residual stresses were found to be variant in different points. The maximum residual stress was obtained around point a either under annular scanning (353 MPa) or lateral scanning (341 MPa). The stress values of point b, c, d, e are similar but lower than that of point a. However, the stress distribution under lateral scanning is more homogeneous than that under annular scanning.

Fig. 9 shows the relation of Von Mises stresses vs time under annular and lateral scanning for the five-layer models. The stresses of point a were fluctuated strongly at the heating stage and ascended rapidly until the values turned to be finally constant after the grids cooled down. The Von Mises stresses of solidified layers were significantly affected by the subsequent scanning tracks. During EBM process, each new layer was deposited primarily in tension while forcing the underlying layer into compression and serving to relax underlying tensile stresses in the underlying layers [30]. The constraint of the connection to the base plate at the bottom of the layer resulted in primarily tensile stress again.

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Fig. 9.   Von Mises stress distributions at point a under annular (a) and lateral (b) scanning and residual stress values of points a, b, c, d, e (c, d) for five-layer models.

Comparing Fig. 9 (c, d) with Fig. 8 (e, f), we can find that the residual stresses for the five-layer models are lower than those for the single-layer models. The temperature of deposited grids repeatedly exceeded the stress relaxation temperature due to the remelting of adjacent scanning tracks and upper layers. This repeated stress relaxation would lead to a decrease in residual stresses.

As similar results to the single-layer grids, the lateral scanning process yielded a more uniform stress distribution than annular scanning. This difference is based on the different thermal history. The grids were melted on one aside without significant remelting of the struts under lateral scanning while four struts remelted twice under annular scanning, which led to more scanning time and more energies accumulated for each layer. Therefore, relatively homogeneous temperature distribution was obtained under lateral scanning and correspondingly uniform distribution of residual stress was observed. This is consistent with the results of temperature gradient distributions, as shown in Fig. 7.

Fig. 10 shows the Von Mises stresses distribution along the P6 path under annular and lateral scanning for the five-layer models. The magnitude of the stress decreases from the bottom to the top. The Von Mises stresses under lateral scanning are slightly higher than that under annular scanning. For example, the stresses of the first layers under annular and lateral scanning are 235 MPa and 249 MPa, respectively.

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Fig. 10.   Von Mises stress distributions along P6 path under annular and lateral scanning for five-layer models (a) and distortion behavior of porous Ti-6Al-4V prepared by EBM under lateral scanning (b).

As the electron beam scanned forth and back, the previously layers experienced heat-cooling cycles, inducing partial stress relief in the heat-affected zone (HAZ). The longer time remelting of former layers under annular scanning than lateral scanning led to the deeper depth of molten pool or HAZ in which the residual stress profiles extended into. This effect suggested that the partial relief of residual stresses under annular scanning is more pronounced along the P6 path. This would decrease the residual stresses under annular scanning.

With the increase of powder layers, the residual stresses decreased. During deposition of the first layer, cooling rate was high. As the build progressed, the powder layers got farther from the deposition, reducing the temperature gradient. As a result, the cooling rate was significantly lower. Consequently, the temperature gradients and cooling rates are smaller when depositing the following layers of the build. Therefore, Smith et al. [31] found that the residual stresses created by EBM mainly either occurred side of the melting layers outside of the HAZ or were concentrated in a region local to the melting layer. As more powder layers were deposited, the already solidified grids would experience many times of heating-cooling cycles in the HAZ. The intra-layer residual stresses induced by the deposition pattern persisted during deposition of subsequent layers, and superimposed with the existing tensile stress, resulting in the larger stresses of previous solidified layers. The large height of the grids in the build direction also meant that the first layer underwent more stress generating thermal cycles. As the girds cooled to the preheat temperature, the residual stress of the first layer is the largest and the fifth is the lowest. The actual distortion behavior of porous Ti-6Al-4V alloy prepared by EBM shown in Fig. 10(b) has proved the FE simulation results.

In order to decrease the warped distortion of porous grids, the change of power during EBM was carried out and the distributions of stress are shown in Fig. 11. In the first and second layers, the applied power was 100 W. When the following layers was scanned, the power decreased to 80 W. As a result, the molten pool temperature would decrease and temperature gradient would become flattened. Although the Von Mises stresses are higher, the distributions are more homogeneous. The porous Ti-6Al-4V alloy by EBM through changing the power is shown in Fig.11(b), revealing that there is no obvious distortion. In our experience, the distortion can be also controlled by increasing the scanning speed, but it is much difficult to operate and the roughness of porous grids would increase.

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Fig. 11.   Von Mises stress distributions along P6 path under annular and lateral scanning through changing the power from 100 W to 80 W for five-layer models (a) and the corresponding porous Ti-6Al-4V alloy grids (b).

Fig. 12 shows the stress profiles along the P4 path when temperature cooled down to preheat temperature. The stress distributions showed anisotropy, where the stresses in X and Y plane were higher than those in Z direction. The stress values were 0-274 MPa in X direction, 6.0-276 MPa in Y direction and from -12 to 13 MPa in Z direction.

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Fig. 12.   Stress distributions along the P4 path for five-layer models after cooling to 730 °C: (a) X direction, (b) Y direction, (c) Z direction.

It was worth noting that the residual stresses were mainly concentrated on the nodes and the stress component perpendicular to the strut was close to zero. Along the P4 path of porous cubic grids, the solidified area was subjected to restraint from the previous area, and the strut was a free boundary since the area around the strut was porous.

Fig. 13 shows the Von Mises stress distributions along the P4 and P5 paths for the five-layer models. The average stresses under lateral scanning are larger than annular scanning, which is in agreement with the results in Figs. 9‒12 .

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Fig. 13.   Von Mises stress distributions along P4 (a) and P5 (b) paths for five-layer models.

It is noteworthy that the maximum stresses occurred around point a, either for annular scanning or lateral scanning. In regard to point a, there was restraint force from the neighboring four struts. At the EBM process stage, as the electron beam scanned the struts, the compressive stress would arise at the center of moving electron beam. The central node a is a hot spot and accumulated the largest energies, as shown in Fig. 7(a, b). After cooling down, the heat dissipated along the struts and depth. Due to the constraints of the struts, the tensile stresses would occur at the central node and the maximum stress appeared around the hot spot.

4. Conclusions

The temperature field and stress behaviors of porous cubic grids based on the single-layer and five-layer models have been studied. The following conclusions can be drawn:

(1) The overlaps of neighboring tracks and good metallurgical bonding of adjacent layers can be achieved by EBM parameters. The scanning time for annular path is longer than lateral path and more energies would be accumulated, leading to higher molten pool temperature under annular scanning.

(2) The higher thermal capacity of the five-layer grids compared to the single-layer grids led to the higher average temperature. Along build direction, the larger temperature gradient was observed under annular scanning and relatively homogeneous temperature distribution can be obtained along scanning direction under lateral scanning.

(3) Thermal stress fluctuated strongly with EBM process and afterwards cooling to preheat temperature the values tended to be constant. The repeated stress relaxed in the five-layer grids by remelting of adjacent tracks and layers, resulting in the lower residual stress than the single-layer grids.

(4) Compared to the annular scanning, the lateral scanning process yielded a more uniform stresses distribution and relatively larger average stress.

(5) In the five-layer models the heating-cooling cycles of the HAZ caused by the forth and back of electron beam brought in the larger residual tensile stresses for the previously solidified layers and thus the largest residual stresses were found at the first layer. Changing the process parameters during EBM can result in the more uniform distribution of stress and then decrease the distortion of porous grids. The stress distributions showed anisotropy and the maximum residual stress occurred around the central node.

Acknowledgment

The work was financially supported by the Natural Science Foundation of Shandong Province, China (No. ZR2019MEM012), the Major Scientific and Technological Innovation Program of Shandong Province, China (No. 2019JZZY010325), the Key Research Program of Frontier Sciences, CAS (No. QYZDJ-SSW-JSC031-02) and the National Natural Science Foundation of China (No. 51871220).

Reference

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