Numerical Modeling of Distortion of Ti-6Al-4V Components Manufactured Using Laser Powder Bed Fusion

Abstract

The laser powder bed fusion (L-PBF) process is a powder-based additive manufacturing process that can manufacture complex metallic components. However, when the metallic components are fabricated with the L-PBF process, they frequently encounter the residual stress and distortion that occurs due to the cyclic of rapid heating and cooling. The distortion detrimentally impacts the dimensional and geometrical accuracy of final built parts in the L-PBF process. The purpose of this research was to explore and predict the distortion of Ti-6Al-4V components manufactured using the L-PBF process by using numerical modeling in Simufact Additive 2020 FP1 software. Firstly, the numerical model validation was conducted with the twin-cantilever beam part. Later, studies were carried out to examine the effect of component sizes and support-structure designs on the distortion of tibial component produced by the L-PBF process. The results of this research revealed a good agreement between the numerical model and experiment data. In addition, the platform was extended to predict the distortion in the tibial component. Large distortion arose near the interface between the tibial tray and support structure due to the different stiffness between the solid bulk and support structure. The distortion of the tibial component increased with increasing component size according to the surface area of the tibial tray, and with increasing thickness of the tibial tray. Furthermore, the support-structure design plays an important role in distortion reduction in the L-PBF process. For example, the maximum distortion of the tibial component was minimized up to 44% when a block support-structure design with a height of 2.5 mm was used instead of the lattice-based support. The present study provides useful information to help the medical sector to manufacture effective medical components and reduce the chance of part failure from cracking in the L-PBF process.

1. Introduction

Ti-6Al-4V is a type of titanium alloy that has been extensively used to manufacture medical components due to its low density, high strength-to-weight ratio, excellent corrosion resistance, good mechanical strength, and good biocompatibility [1,2]. Examples of medical components made with Ti-6Al-4V include bone scaffolds, hip prostheses, tibial components, and joint replacements [3]. The components have complex shapes with porous structures, and are generally implanted into the human body to replace a damaged biological structure. However, components with porous structure are mostly impossible to be produced by traditional processes. Hence, a laser powder bed fusion (L-PBF) process, which is a powder-based additive manufacturing method, is an emerging technology for manufacturing complex-geometry components [4,5,6]. This process can create three-dimensional components with full density by using a laser energy source to selectively melt the metal powder in a layer-by-layer addition [7,8,9]. However, when the metallic components are produced by L-PBF, they associate with complex thermal process due to the rapid heating and cooling cycles that result in large thermal gradients [10,11]. The occurrence of large thermal gradients during the L-PBF process gives rise to large thermal strain, leading to large thermal stress-induced distortion [10]. Moreover, the distortion in L-PBF is caused by the nonuniform expansion and contraction of melted and solidified layers during the building process [12]. The distortion, which is inevitable during the L-PBF process, is a major issue and a challenge to effectively producing the metal components with high geometric accuracy [8]. Nevertheless, the distortion can be alleviated by the determination of proper processing parameters, such as laser power (P), scanning speed (v), layer thickness (Lt), and scanning strategies [13,14]. Furthermore, the post-heat-treatment process, especially stress relief, is commonly used for stress relaxation and distortion mitigation upon the substrate removal [15,16].

In the last decade, the influence of processing parameters on distortion in the L-PBF process has been investigated by numerous researchers through experimental study and numerical modeling [17,18,19,20,21]. To numerically study the formation of residual stress and distortion in the L-PBF process, numerical models based on the finite-element method (FEM), including the thermomechanical model and the inherent strain method (ISM), are effective approaches. The thermomechanical model, which couples thermal and mechanical analysis, is usually applied to simulate the temperature histories that induced residual stress and distortion. Unfortunately, while this model can reveal complex relationships between process parameters and distortion, it is not suitable for macro- or part-level estimation. This is mainly due to drastic difference in spatial and temporal scales of the problem, leading to exceedingly high computational power requirements [22,23,24]. On the contrary, the inherent strain method (ISM) is a solely mechanical analysis, in which no thermal effects are calculated. It can be implemented to study and predict the residual stress and distortion in the macroscale and investigate the printing strategy optimization for minimizing residual stress and distortion in the L-PBF process [25]. Li et al. [26] developed a multiscale numerical modeling to predict part distortion of a twin-cantilever beam and found that the large tensile stress can appear on the top layers of the part with support on the solid substrate. Promoppatum and Yao [27] utilized numerical and experimental study to fully examine the influence of energy input and scanning length on residual stress reduction. The findings indicated that the residual stress was mainly dependent on the thermal histories and the surface temperature. Residual stress reduction can take place with higher surface temperatures generated from either high-energy input or small scanning length. Taufek et al. [28] analyzed the distortion of SS 316L SLM products using numerical modeling based on the inherent strain method (ISM). They concluded that a part built in a vertical direction can provide smaller residual stress than that of a part built in a horizontal direction. Xiaohui et al. [29] established the finite-element model to examine the influence of supporting structure design on residual stress in the overhanging structure of components in the SLM process of AlSi10Mg alloy. The results showed that when the contour-cone supporting structure was applied, the residual stresses in both X and Y directions were higher than those in contour-block and contour-entity supporting structures.

Most previous studies on part-scale modeling that assessed the residual stress and distortion in the L-PBF process focused on simple shapes, such as thin-walled structures, cubic samples, canonical samples, and cantilever beams, among others [30].

Table 1. Previous studies of part-scale model in SLM process. Reprinted with permission from refs. [10,15,22,23,24,27,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45]. Copyright 2022 Elsevier.

No.	Part Geometry	Studies
1	Cantilever beam	[15,22,23,31,32,33,34,35,36] *
2	Cubic sample	[27,37,38]
3	Thin-walled part	[10,39,40]
4	Canonical sample	[24,32,41]
5	Complex shape parts	[42,43,44,45]

Noted: * Part geometry and support structure included.

presents the previous studies of part-scale modeling in the L-PBF process. It can be found that the part-scale modeling of distortion in complex shape parts produced by the L-PBF process have scarcely been developed and examined. Furthermore, studies that have reported the size effects on distortion of component produced by the L-PBF process are limited. Hence, it is necessary to examine the size effects on distortion of components fabricated using the L-PBF process to obtain final parts with high geometric accuracy.

Additionally, if there are overhanging surfaces in the components fabricated by L-PBF process, the support structure is generally required to support the overhanging surfaces and prevent part failure [29,46]. The support-structure design in the L-PBF process commonly involves numerous design parameters e.g., support types, support height, support radius, and support hatch distance. However, when the support structure is added during building process, the distortion will be significantly affected. Nevertheless, when an optimum support structure is selected, it may reduce or prevent the distortion of build components. Nonetheless, research on the effect of support-structure design parameters on distortion of build component in the L-PBF process is rare. Therefore, the effect of support-structure design on distortion was explored in the present research.

In this research, a macroscale modeling approach for the L-PBF process was proposed and implemented to investigate and predict the distortion of Ti-6Al-4V part manufactured by the L-PBF process. Firstly, the verification of the numerical model and mesh convergence analysis in this study were carried out with twin-cantilever beam parts. The details are explained in Section 3. Later, the effect of part sizes on the distortion of tibial component manufactured using the L-PBF process was investigated. In addition, the effect of support design on the distortion of tibial components was studied to mitigate the distortion. The numerical modeling was performed using the commercial software Simufact Additive 2020 FP1 (Simufact Engineering GmbH, Hamburg, Germany), based on the inherent strain method (ISM). The present numerical result was validated with the experiments reported by Siewert et al. [47]. The tibial component, an implant for the knee joint replacement, was selected as the build component in this study. The findings from this study are expected to serve as the guideline for medical sector to fabricate effective medical components with low geometric deviations.

2. Numerical Modeling

2.1. Mechanical Modeling

In mechanical analysis of the L-PBF process, the thermal histories obtained from the thermal analysis are subsequently applied to calculate the residual stress and distortion. A constitutive model for elastic (Hook’s law) and plastic behavior is defined in the built material. Nevertheless, the thermal and phase transformation effects are neglected for the mechanical analysis based on ISM. A constitutive model as shown in Equation (1) is applied for mechanical analysis in the L-PBF process. The value of inherent strain $\left\{{\epsilon }^{*}\right\}$ is defined as seen in Equation (2). Moreover, total strain including elastic, plastic, thermal, and phase transformation strains is described in Equation (3) [28]:

$$\sigma =\left[D\right]\left\{{\epsilon }^{elastic}\right\}$$

(1)

where $\sigma$ is Cauchy stress [MPa], D is the fourth order stiffness tensor,$\mathrm{and} {\epsilon }^{elastic}$ is elastic strain.

$$\left\{{\epsilon }^{*}\right\}=\left\{{\epsilon }^{total}\right\}-\left\{{\epsilon }^{elastic}\right\}$$

(2)

$$\left\{{\epsilon }^{total}\right\}=\left\{{\epsilon }^{elastic}\right\}+\left\{{\epsilon }^{plastic}\right\}+\left\{{\epsilon }^{thermal}\right\}+\left\{{\epsilon }^{phase}\right\}$$

(3)

where ${\epsilon }^{total}$ is total strain ${\epsilon }^{plastic}$ is plastic strain, ${\epsilon }^{thermal}$ is thermal strain, and ${\epsilon }^{phase}$ is phase transformation strain.

For the analysis of plastic strain in the Simufact additive software, the von Mises yield criteria and Prandtl–Reuss flow rule are used as shown in Equations (4) and (5) [25]:

$$f={\sigma }_{VM}-{\sigma }_{y }$$

(4)

$$d{\epsilon }^{plastic}=d\lambda \frac{\partial f}{\partial \sigma }$$

(5)

where f is yield function, ${\sigma }_{VM}$ is von Mises stress [MPa], ${\sigma }_y$ is yield stress [MPa], and dλ is plastic multiplier.

2.2. Numerical Modeling for Distortion Analysis

To carry out the macroscale numerical modeling for distortion analysis of the tibial component in Simufact Additive software (2020 FP1 version, Simufact Engineering GmbH, Hamburg, Germany), five steps are mainly involved. As shown in Figure 1, the sequence is as follows: (1) importing a model, (2) generating support structure, (3) creating voxel mesh, (4) calculating simulation, and (5) obtaining the simulation result of the as-built stage. After generating support structures, the material properties of Ti-6Al-4V as listed in

Table 2. Dimensions of tibial components used in this study.

Case Studies	Size **	Dimensions					Code Name
		Tray Thickness (T) (mm)	Length (L) (mm)	Width (W) (mm)	Height (H) (mm)	Tray Volume (mm3)
1	S	8	60	40	43	14,516	S-T8
2	M	8	66	45	43	18,097	M-T8
3	L	8	72	50	43	22,034	L-T8
4	M	4	66	45	39	9049	M-T4
5	M	6	66	45	41	13,573	M-T6
6	M	10	66	45	45	22,622	M-T10

** S = small, M = medium, and L = large.

were defined. The inherent strain values, layer parameters, and rotation parameters were also determined. The voxel element was created to analyze the distortion of the component. After the calculation process was completed, the numerical result of the tibial component’s distortion at the as-built stage can be obtained [42].

The Simufact Additive software, which is a scalable software solution for the metal additive manufacturing processes, requires the input of inherent strain values (${\epsilon }^{*}$) for calculation and prediction of the residual stress and part distortion. For the study, the inherent strains are acquired from the previous research of Siewert et al. [47]. Bidirectional scanning pattern is applied to build the twin-cantilever beam in the calibration process. The inherent strain values were calibrated by measuring cantilever distortion after cutting, and then the simulation model was calculated to match the experiment value. The inherent strain values in the X, Y, and Z directions are ${\epsilon }_x^{*}$ = −0.0047, ${\epsilon }_y^{*}$ = −0.0020, and ${\epsilon }_z^{*}$ = 0. The additional description of this calibration process can be found from [47]. Figure 2a shows the computational domain which consists of the build component, support structure, and substrate plate. Figure 2b shows the meshing of the computational domain.

The tibial components of all case studies were placed in the middle of the computational domain. The distance between the tibial component and substrate top surface of all cases was 5 mm. To explore the size effect on distortion in this study, rod support-structure design was used in all cases. The rod support radius was 0.3 mm, and the distance between each rod was 0.6 mm. The present study utilized voxel mesh number between 0.5 and 1.5 million elements. The initial temperature conditions were 298 K. The bottom of the substrate plate was set with zero displacement. As a result, the substrate is not allowed to move freely when the build components are produced. Figure 3 exhibits each size of the tibial component on the rod support. Table 2 presents the dimensions of the tibial components used in this study. The tibial component size S on the different support-structure designs is shown in Figure 4.

Figure 5 illustrates the patterns and dimensions of the support structure. The support volume for support-structure design with height of 5 mm case and height of 2.5 mm case were 1560 mm³ and 780 mm³ respectively.

The material properties of Ti-6Al-4V alloys for the simulation were obtained from previous research of Siewert et al. [48] and the Simufact material database [49], as listed in

Table 3. Material properties of Ti-6Al-4V used for this study. Reprinted with permission from refs. [47,48]. Copyright 2022 Elsevier.

Material Properties	Values
Density	4.4 g/cm3
Young’s modulus	110 GPa
Yield strength	1030 MPa
Poisson’s ratio	0.31

.

Table 4. Process parameters used for this study. Reprinted with permission from ref. [47]. Copyright 2022 Elsevier.

Process Parameters	Values
Laser power	180 W
Scanning speed	1250 mm/s
Layer thickness	0.03 mm
Hatch spacing	0.105 mm

tabulates the process parameters used for this study.

3. Model Validation

Mesh convergence analysis was initially performed to evaluate the influence of the number of elements on the simulation results and to obtain the reliability of the numerical model. The twin-cantilever beam with dimensions of 7.5 mm (width) × 80 mm (length) × 13 mm (height) was used as a specimen, as illustrated in Figure 6a. The mesh convergence analysis was carried out by comparing the deflection measured at the middle point of the specimen, as illustrated in Figure 6b. The element sizes for the analysis were 0.6, 0.5, 0.4, 0.3, 0.25, and 0.2 mm. Each element size provided a total number of elements of 38,117, 56,691, 102,699, 224,901, 367,461, and 693,409, respectively. As illustrated in Figure 7, the simulation result can converge when the number of elements is greater than 100,000 elements. As a result, the element size of 0.25 mm was selected to further study the influence of part sizes and support-structure designs on the distortion of tibial component. In addition, to verify and obtain the accuracy of the numerical model implemented in this research, the result from numerical modeling in the present study was verified against the experiment from Siewert et al. [47]. Firstly, the twin-cantilever beam specimen as shown in Figure 6a has been printed with layer-by-layer addition. Then, the cantilever beam was removed from the substrate, where the deflection was seen due to the effect of stress relaxation. Figure 6b illustrates the comparison of the twin-cantilever beam distortion between the simulation result obtained from the present study and the experimental measurement from Siewert et al. [47]. It was found that the numerical result and the experiment result were in good agreement, with minor difference of 10.3%. Hence, the present numerical model can be used to investigate the distortion of the tibial component of the present study.

4. Results and Discussion

4.1. Distortion Distributions of Tibial Component

Figure 8 shows the total distortion distribution of the as-built S-T8 tibial component. The numerical results revealed that the large distortion occurred near the interface between the tibial tray and support structure. This is mainly due to the different stiffness of solid bulk and support structure. The maximum distortion arose around both edges of the long side of the tibial tray, which is marked in the red circles in Figure 8a. This region tended to be more susceptible to cracking. Meanwhile, the small distortion magnitude took place at the stem region and top surface of the tibial tray (blue region). In addition, it was noticed that the distortion magnitude gradually mitigated from the edge to the inside, as displayed in Figure 8c. Figure 8b amplifies the distortion at the edge region. It was observed that the distortion magnitude gradually decreased from the bottom surface at the interface to the top surface of the tibial tray because the constraint strength provided by the support structure decreased when the deposition height was increased. The black solid line in Figure 8b demonstrates the original shape of the tibial component.

Figure 9 exhibits the distortion distribution in each direction of the as-built S-T8 tibial component. The magnitude of distortion of the built component is represented by a color contour plot. The result revealed that the distortion distribution of the tibial component was different in each direction. According to the distortion distribution in the X-direction, the tibial component distorted inwardly at both edges of the tibial tray. The value of maximum distortion in the X-direction was around 0.34 mm, while the distortion value in the Y-direction was relatively less where its maximum magnitude was around 0.07 mm. It can be seen that the distortion magnitude in the Z-direction, which was along the build direction, was higher than that in the X- and Y-directions. The maximum distortion magnitude in the Z-direction was around 0.43 mm.

4.2. Effect of Part Sizes on the Distortion of Tibial Component

Figure 10 shows the total distortion of the as-built tibial component with different sizes. The distortion distributions of all cases were similar, but the distortion in each case was different.

The results indicated that the distortion of the tibial component tends to increase with larger size according to the surface area of the tibial tray. This is due to the shrinkage of the component that was approximately proportional to the part size and the influence of constraining strength between the built component and support structure [39,49]. As shown in Figure 11, the maximum total distortions of the tibial component of S-T8, M-T8, and L-T8 were 0.54 mm, 0.63 mm, and 0.67 mm, respectively. It can be noticed that when the component size was increased from small size (S) with the tibial tray surface area of 1815 mm² to medium size (M) with the tibial tray surface area of 2262 mm² and large size (L) with the tibial tray surface area of 2754 mm², the maximum total distortion was increased by 16.7% and 24.1% respectively. It was also discovered that the distortion in the Z-direction is more dominant than that in the X- and Y-directions. A similar result was also reported by Shi et al. [50].

Figure 12 exhibits the total distortion at the as-built stage of medium-size tibial components with different tray thicknesses. The numerical result showed that the distortion distributions of all cases were similar. The distortion of the tibial component increases with the increasing thickness of the tibial tray, but has an insignificant effect on the higher thickness component.

Figure 13 illustrates the maximum distortion of the as-built tibial component with different tray thickness. The maximum total distortion value of the tibial component fabricated with different thicknesses of the tibial tray was 0.45 mm for M-T4, 0.53 mm for M-T6, 0.63 mm for M-T8, and 0.64 mm for M-T10. It was noted that when the thickness of the tray increased from 4 mm (M-T4) with tray volume of 9049 mm³ to 6 mm (M-T6) with tray volume of 13,573 mm³, 8 mm (M-T8) with tray volume of 18,097 mm³ and 10 mm (M-T10) with tray volume of 22,622 mm³, the maximum total distortion value of the tibial component was increased by 17.3%, 39.4%, and 41.6% respectively. This is because of the decrease in constraint strength with an increase in the thickness of the tibial tray.

4.3. Effect of Support-Structure Design on the Distortion of Tibial Component

A further study is the investigation of the effect of support-structure design on the distortion of the tibial component. It can be seen that the support-structure design plays an important role in the reduction of distortion in Figure 14. When the height of the support was decreased, the lower distortion of the tibial component and less material usage can be obtained. However, the difficulty of the build part and support removal should be noted. Meanwhile, usage of the proper design of the support structure, such as block structure instead of the lattice-based, or rod structure, provided a more effective distortion reduction than decreasing the support height, as illustrated in Figure 14c,d. The case with block support-structure design with the height of 2.5 mm has a minimum total distortion. This is due to the influence of support-structure stiffness.

Figure 15 illustrates the maximum total distortion value of the S-T8 tibial component fabricated under different support-structure designs. The result showed that the maximum distortion value of the S-T8 tibial component was reduced by 14.8% when the height of the rod support structure was lowered from 5 to 2.5 mm. While changing the support design from rod structure to block structure, the maximum distortion value of the S-T8 tibial component was reduced by 29.6% with the same height of 5mm, and 44.4% with the lower height of 2.5 mm.

Nonetheless, the support structure must be eliminated after the printing and cutting process. Generally, the support removal process can be done manually. The support structure in the L-PBF process was commonly eliminated from the solid substrate plate by the metal-cutting methods, such as wire EDM and milling [51]. Therefore, an optimum support-structure design that can be easily removed from the substrate plate should be carried out for reduction of the support removal time in subsequent work.

5. Conclusions

The present study investigated the effect of part sizes and support-structure designs on the distortion of the tibial component fabricated by the L-PBF process using macroscale numerical modeling with the inherent strain method (ISM), in which the thermal effect was not considered. Numerical modeling was conducted with Simufact Additive 2020 FP1 software. The main results of the present study can be summarized as follows:

• The large distortion takes place near the interface between the tibial tray and support structure because of the different stiffness between solid bulk and support structure. The maximum distortion occurs around both edges of the long side of the tibial tray. In addition, the distortion value in the Z-direction was greater than that of the X- and Y-directions.
• The distortion of the tibial component increases with the larger size according to the surface area of the tibial tray. The thickness of the tibial tray tends to increase the distortion of the component, but has less effect on the higher thickness component.
• The support-structure design plays a significant role in distortion reduction. Decreasing the height of the support gives lower distortion and less material usage, while the difficulty of the part removal should be considered. Applying an appropriate design of the support, such as block structure instead of rod structure, gives a more effective reduction of the distortion than decreasing the support height.