Keywords

1 Introduction

3D printing is used widely in the manufacturing industry because of its advantage. It is applied in aerospace, biomedical, automotive, etc. The 3D printing process can be divided into two groups such as powder spreading and powder spraying method. Selective laser melting (SLM) is a powder spreading method using a laser source to melt a thin powder layer. SLM is developed in 1995 by German scientists [1].

Figure 1 shows the SLM process. During the SLM process, a product is formed by selectively melting layers of powder using a laser beam. The powder is heated and melted to form a liquid pool. Then the molten pool solidifies and cools down and starts to form the product. The building platform is lowered by the amount of layer thickness and a new layer is deposited. The process is repeated until the product is made. Because of its ability to heat and cool rapidly, the SLM is used commonly. However, several defects usually exist in an SLM part, such as part distortion and cracks. The printed parts are affected by key influence factors such as powder properties, printing process parameters, and machine characteristics.

Fig. 1
A schematic illustration of the S L M printing setup. The labeled parts are a laser source, optical system, deposition system, molten part, metal powder, build substrate, and build platform.

Model of SLM printing process

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To obtain the best quality in a printed part, key influence factors must be considered. However, during the SLM process, large thermal gradients and fast cooling exist that cause residual stress, which leads to a loss of part shape and other failures of the SLM part. The failure quality of parts results in manufacturing costs, as well as waste and scrap [2]. Therefore, all requirements and predictions for the process need to be fulfilled at the design stage to reduce failure and increase accuracy. To predict the quality at the design stage, a computational simulation is the best choice.

According to the previous research on this, the current methods to reduce deformation include [1, 2]:

  • Experiment: Achieve printing experiments to resolve the relationship between the printing parameters and deformation.

  • Simulation: Analysis to complete the residual stress and then predict the deformation.

The contribution of this research is to suggest the deformation calculation based on the inherent strain (IS) value. The IS values in x, y, z directions are determined by the heat-affected zone (HAZ). HAZ depends on the printing process parameters and material properties. This paper presents a method to calculate the HAZ in different plans according to various printing process parameters using Ti6Al4V alloy material with selective laser melting (SLM) printing method.

2 Calculation of Inherent Strain

Inherent strain resulting from welding is a combination of thermal strains \(\varepsilon_{thermal} , \) phase transfer strain \(\varepsilon_{phase}\), plastic strains \(\varepsilon_{plastic}\) and creep strain \(\varepsilon_{creep}\) [3, 4].

In SLM printing process, the total strain is calculated as follows:

$$ \varepsilon_{total} = \varepsilon_{elastic} + \varepsilon^{*} $$
(1)

in which \(\varepsilon_{elastic}\) is the elastic strain; \(\varepsilon^{*}\) is the inherent strain.

The \(\varepsilon^{*}\) value is calculated by considering other strains as follows:

$$ \varepsilon^{*} = \varepsilon_{thermal} + \varepsilon_{plastic} + \varepsilon_{phase} + \varepsilon_{creep} $$
(2)

During heating and cooling process, at the plastic stage the IS value in x, y, z directions can be defined as follows [5]:

$$ \varepsilon_{x}^{*} = \frac{{W_{x} }}{{F_{x} }}; \varepsilon_{y}^{*} = \frac{{W_{y} }}{{F_{y} }}; \varepsilon_{z}^{*} = \frac{{W_{z} }}{{F_{z} }} $$
(3)

In which Fx, Fy, and Fz are the heat effected zone (HAZ) areas where Wx, Wy, and Wz, respectively, are distributed. Wx, Wy, and Wz are calculated as follows:

$$ {\text{W}}_{{\text{x}}} \, = \,\xi .{\text{ q}}_{{\text{v}}} ;{\text{ W}}_{{\text{y}}} \, = \,{\text{W}}_{{\text{z}}} \, = \,{\text{K}}.{\text{ q}}_{{\text{v}}} $$
(4)

In which qv is the linear energy density (J/mm) [6]:

$$ {\text{q}}_{{\text{v}}} \, = \,\frac{{Q_{heat source} }}{u} $$
(5)

ξ and K (mm/J) values are used from the welding process due to the same heating and cooling process.

3 Numerical Model for SLM Process

For determining the temperature distribution during SLM process, the mathematical model of the heat transfer is as follows [7]:

$$ \rho C\frac{\partial T}{{\partial t}} + \rho C{\text{u }}\nabla T = \nabla \left( {k\nabla T} \right) + Q_{heat source} $$
(6)

in which T is temperature; ρ, C, k, and u are density, thermal capacity, thermal conductivity factor and laser velocity, respectively. Q is defined as follows:

$$ Q_{heat source} = Q_{required power} + Q_{conduction} $$
(7)
$$ Q_{required power} = Q_{delivered laser} - Q_{convection} - Q_{radiation} - Q_{conduction} $$
$$ Q_{delivered laser } = Q_{ } \frac{6\sqrt 3 }{{ab\pi \sqrt \pi }}e^{{\left[ {\frac{{ - 3z^{2} }}{{a^{2} }} + \frac{{ - 3y^{2} }}{{b^{2} }}} \right]}} \left[ {\frac{{f_{f} }}{{C_{f} }}e^{{\frac{{ - 3x^{2} }}{{C_{f}^{2} }}}} + \frac{{f_{r} }}{{C_{r} }}e^{{\frac{{ - 3x^{2} }}{{C_{r}^{2} }}}} } \right] $$

The power distribution is given by the moving Goldak’s double-ellipsoid heat source model as shown in Fig. 2 [8]. The model parameters are explained in Table 1.

Fig. 2
A Goldak's double-ellipsoid heat source model on a cuboidal material, with 3-D coordinates marked at a point on the upper surface. The printing direction is along the x-axis. The peak for heat input distribution rises along the z-axis. The parameters C r, C f, a, and b are marked.

The Goldak’s heat source model

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Table 1 SLM process parameters for Ti6Al4V material
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4 Results and Discussion

To determine the HAZ during the SLM process, we use the data from Table 1 as the input data using Comsol tool. Figures 3, 4 and 5 show the method to calculate the HAZ in Fx, Fy, Fz plans.

Fig. 3
A heat map for H A Z in the F x plane of the material, along the z and x axes. The center portion of the upper edge with length e and width c has the highest temperature, followed by an average temperature in an adjacent portion of length x and width d. The rest of the plane has a low temperature.

Calculation of the HAZ in Fx plane

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Fig. 4
A heat map for H A Z in the F y plane of the material. A patch on the right center of the plane with length e and width 2 a has the highest temperature, surrounded by a region of average temperature, with length x and width 2 a plus b towards the left. The rest of the plane has a low temperature.

Calculation of the HAZ in Fy plane

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Fig. 5
A 3-D model of a cuboidal material with the F z plane indicated. A heat map for H A Z in the F z plane is on the right. A semicircular region on the top center with radius c and width a has the highest temperature, and is surrounded by a region of average temperature with radius d and width b.

Calculation of the HAZ in Fz plane

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$$ F_{x} = \left( {x + e} \right).d - e.c] $$
(8)

With d = 0.153 mm; c = 0.09 mm; e = 1.97 mm; and x = 2.34 mm, we have Fx = 0.482 mm2. With ξ = 1.57 10–3 mm3/J; qv/h2 = 104.52 J/mm3 for Ti6Al4V and h = 0.045 mm, we have:

$$ \varepsilon_{inh - x} = - 0.000937 $$
$$ F_{y} = 2\left( {x + e} \right).\left( {a + b} \right) - 2a.e] $$
(9)

With a = 0.19 mm; b = 0.102 mm; e = 1.97 mm; and x = 2.34 mm, we have Fy = 1.275 mm2.

With K = 0.58 10–3 mm3/J; qv/h2 = 104.52 J/mm3 for Ti6Al4V and h = 0.045 mm, we have:

$$ \varepsilon_{inh - y} = - 0.0000963 $$
$$ F_{z} = 2\left[ {\left( {a + b} \right).d - a.c} \right] $$
(10)

a = 0.19 mm; b = 0.106 mm; d = 0.153 mm; and c = 0.09 mm, we have FZ = 0.0564 mm2. With K = 0.58 10–3 mm3/J; qv/h2 = 104.52 J/mm3 for Ti6Al4V and h = 0.045 mm, we have:

$$ \varepsilon_{inh - z} = - 0.00217 $$

With the same processing parameters such as the beam width, velocity, power, and material, we used Simufact™ tool for predicting the inherent strain. The predicted results are shown in Fig. 6. In comparison with the proposed method, the biggest IS value is in the Z direction, this is correct for both simulation and experiment [2, 7].

Fig. 6
2 screenshots of panels from the Simufact software application window. The panel for inherent strains has values of epsilon x, epsilon y, and epsilon z. The estimation panel has values of beam width, velocity, power, efficiency, and material.

Calculation of the inherent strain by using Simufact

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5 Conclusions

In this study, we propose to calculate IS value based on printing process parameters and HAZ area. To calculate the HAZ area during SLM printing process, the heat transfer equation was established. Then, Comsol™ was used to determine the temperature distribution in the printed part. From the temperature distribution, the HAZ surfaces were calculated. The IS values are used to calculate the deformation of the printed part in all directions. From that, the reasonable printing process parameters are determined to reduce distortion when carrying out the actual printing process. The proposed IS calculation results are reliable when compared with the IS values calculated by Simufact™ software under the same conditions and printing method.