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Experimental and numerical study on residual stress and geometric distortion in powder bed fusion process
Journal of Manufacturing Processes 46 (2019) 214–224
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Journal of Manufacturing Processes journal homepage: www.elsevier.com/locate/manpro
Experimental and numerical study on residual stress and geometric distortion in powder bed fusion process Tianyu Yua, Ming Lib,c, Austin Breauxb, Mukul Atrid, Suleiman Obeidatb, Chao Mab,c,
T
⁎
a
Dante Solutions, Inc., Cleveland, OH, USA Dept. of Engineering Technology & Industry Distribution, Texas A&M University, USA c Dept. of Industrial & Systems Engineering, Texas A&M University, USA d Ansys, Inc., Pune, Maharashtra, India b
ARTICLE INFO
ABSTRACT
Keywords: Powder bed fusion Additive manufacturing 3D printing Finite element analysis Residual stress Geometric distortion
The objective of the present work is to experimentally and numerically study the effect of the part size on the distortion of a part printed by a powder bed fusion process. A simplified thermo-mechanical finite element model was developed using a so-called superlayer concept (i.e., numerous layers deposited at once) to predict the geometric distortion induced by the thermal stress. A set of seven cylinders with different dimensions were modeled with this method. The same geometries were printed using a commercial selective laser melting machine and cut from the build plate using wire electrical discharge machining. The distortion of different cylinders was measured before and after the removal from the build plate using a coordinate measuring machine and compared with the simulation results. The experimental results showed the final distortion is largely contributed by the build plate removal process because it relaxes the thermal stress accumulated during the printing process. The modeling results show a high consistency with the experimental results, especially for the thicker parts. As the part thickness decreases, the prediction accuracy of the model decreases. When a superlayer thickness of 1 mm is used, the model can accurately predict the distortion for a cylinder with a diameter of 45 mm and a thickness of greater than 6 mm. This model provides an effective approach for geometric control of printed parts.
1. Introduction Additive manufacturing (AM) refers to a variety of methods that fabricate parts using a layer-by-layer strategy. AM provides the capability for producing parts of complex geometries without additional tooling. The most commonly used metal AM processes can be divided into two groups: 1) powder bed fusion (PBF) processes (e.g. selective laser melting, direct metal laser sintering, electron beam melting, and laser metal fusion) and 2) directed energy deposition (DED) processes (e.g. laser-engineered net shaping, laser metal deposition, direct metal deposition, and wire and arc additive manufacturing) [1,2]. PBF is advantageous over DED in the senses of geometric accuracy and surface finish. In a PBF process, either a laser or electron beam is used to fuse the feedstock powder which is pre-spread on the previous layer. Although AM technology is developing at a fast pace, there are still many issues to overcome. One of the most challenging issues is the residual stress and distortion due to the unique heating and cooling cycles. Because of the complexity of the PBF process, it is almost
impossible to calculate the process-induced residual stress and distortion explicitly. Additionally, the measurement (e.g., X-ray diffraction, neutron diffraction) of residual stress in AMed metal parts are very tedious and expensive, in a limited accuracy (off in the range of 70–100 MPa for X-ray diffraction [3]). Thus, to better understand how the residual stress and distortion are formed during the PBF process and control the parts quality, an efficient and effective numerical model is highly needed. Numerical simulation is a very powerful tool for assessing the process parameters effectiveness and predicting the manufacturing performance [4–9]. A decent amount of effort on modeling residual stress in AM was made recently [10–19], as listed in Table 1. For example, Alimardani et al. [10] developed a numerical method for modeling temperature field and thermal stress during DED. A four-track thin wall structure was modeled and experimentally verified by measuring the temperature and the deflection of the workpiece. Gu and He [11] and Hussein et al. [12] used a finite element model to investigate the temperature and stress fields of three and five tracks in PBF,
Corresponding author. E-mail addresses: [email protected] (T. Yu), [email protected] (M. Li), [email protected] (A. Breaux), [email protected] (M. Atri), [email protected] (S. Obeidat), [email protected] (C. Ma). ⁎
https://doi.org/10.1016/j.jmapro.2019.09.010 Received 28 September 2018; Received in revised form 30 July 2019; Accepted 12 September 2019 Available online 23 September 2019 1526-6125/ © 2019 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
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results. Section 5 provide a discussion on residual stress and distortion of the samples before and after removing the build plate. Conclusions are drawn in Section 6.
Table 1 Comparison of different modeling works on residual stress and distortion. Scale
Track Track Track Layer Layer Layer Part Part Part Part
Predicted stress
Predicted distortion
Validated distortion
Modeled build plate removal
Studied size effect
Reference
Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
Yes No No Yes Yes Yes Yes Yes Yes Yes
Yes No No No No No Yes Yes Yes Yes
No No No No No No No Yes Yes Yes
No No No No No No No No No Yes
[10] [11] [12] [13] [14] [15] [16] [17] [18] This work
2. Model description The commercial FEA software, ANSYS 18.2 Mechanical, was used on a workstation with 24 cores and 128 GB RAM. The model uses a transient thermo-mechanical solution based on the AM ACT 1.1 module. A thermal model is solved first without considering the mechanical behaviors (stress and distortion), resulting in a transient temperature field, which is then used as the input for the mechanical model to predict the thermal stress and distortion. 2.1. Assumptions and simplifications Some assumptions and simplifications were made to improve the modeling efficiency and productivity:
respectively. Li et al. [13] developed a multi-scale modeling approach to predict the distortions of a single layer under four different scanning strategies. Dai and Shaw [14] applied the powder-to-solid transition model to predict the generation of micro-scale residual stress and the distortion of two layers. Cheng et al. [15] simulated the residual stress and deformation under various scanning patterns using three layers. Most of the current modeling studies on AM are focused on incorporating high-fidelity process physics, addressing laser-matter interaction, and predicting effects of process parameters and scanning strategy. These models are computationally expensive, which makes them well-positioned for the cases of several spots [8], tracks [10–12], or layers [13–15], but inapplicable to industrial components of large and complex geometries. Seldom reported is the part-scale modeling of residual stress and distortion formation during the AM process. Zaeh and Branner [16] investigated the distortion and residual stress of a cantilever beam produced by a PBF process. They simplified the model by adding multiple layers (called superlayer in this paper) at once and applying a monolithic thermal load. The model was efficient enough to model the cantilever beam. However, the researchers did not model the effect of the build plate removal on the stress and the distortion. Build plate removal largely relaxes the residual stress and leads to a significant distortion [19]. Lack of consideration of the build plate removal in the model could be one of the reasons for the disagreement between the simulation and the experiment [16]. Neugebauer et al. [17] and Li et al. [18] used a similar approach (i.e., superlayer) to numerically study the distortion, achieving a reasonable accuracy with the consideration of the build plate removal. However, these researchers used only one part in their studies and did not vary the part size [16–18], which could have a significant effect on the distortion. The objective of this study is to experimentally and numerically study the effect of the part size on the distortion of a part printed by PBF. A part-scale model was developed for the prediction of distortion caused by the residual stress accumulated in the PBF process considering the effect of build plate removal. Finite element analysis (FEA) is used to predict the residual stress and distortion generated from the incremental printing process. A build plate is used to support and transfer heat from the printing parts. The build plate removal is necessary after finishing the PBF printing process, and this removal process is also simulated to predict the distortion induced by stress relaxation. In the current study, a set of seven cylinders is simulated in a transient thermo-mechanical model. SLM experiments are conducted to examine the results of the simulation. The modeling and experimental framework helps to enhance the fundamental understanding of the PBF part distortion behaviors, provide a better design routine, and support the development of suitable process strategies and component topologies. Section 2 defines the model setup. Section 3 describes the experimental methods, including the SLM process and distortion measurement details. Section 4 shows the simulation and experimental
1) Every twenty processing layers are represented by one superlayer; 2) Each superlayer is deposited on the previous superlayer at once; 3) The initial temperature of each superlayer after deposition is the melting temperature of the material (1371 °C for stainless steel 316 L in this study); 4) Thermal stress starts to accumluate at 800 ℃ (termed as reference temperature), above which the stress is consumed by grain growth and/or phase evolution. 2.2. Material, dimension, and meshing A printing process of seven cylinders of the same diameter (45 mm) but different thicknesses/height (2 mm, 4 mm, 6 mm, 11 mm, 16 mm, 21 mm and 31 mm) was simulated. As shown in Fig. 1, the cylinders were aligned evenly to reduce the interaction between each other. The material selected in this study is stainless steel 316 L because of its widespread applications. The thermophysical properties of 316 L are shown in Appendix (Table A1). The standard ANSYS element birth technique was used to generate the cylinder samples, simulating the additive nature of the process. The build plate and the seven cylinders are meshed with 178k nodes and 162k elements as shown in Fig. 1. The cylinders are meshed using sweep method with elements with a thickness of 0.5 mm and in-plane size of 1–1.8 mm. The element size for the build plate is 10 mm. 2.3. Simulation process Using ANSYS Mechanical, the thermal model is set up first to catch the thermal history during the printing and cooling process. The thermal data is then transferred to a stress model to calculate the residual stress and distortion level. The thermal model and stress model share the same material database, geometry, and mesh, although they are not directly coupled. After the seven cylinders are printed, they are allowed to cool down to room temperature. Afterward, the cylinders are removed from the build plate. The thermal model took 47 min and the stress model 5.5 h, indicating the high computational efficiency of the superlayer-based FEA model. 2.4. Governing equations The governing equation for the thermal model is the transient heat conduction equation,
c
T = t
where 215
2T
x i2
(1)
is the density, c is the specific heat, T is the temperature, t is
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Fig. 1. FEA Mesh for the printing parts with 178k nodes and 162k elements.
the time, x i is the spatial location, and is the thermal conductivity. After solving the thermal model, T of all nodes for different time frames will be stored and used as the input of the stress model. The governing equations used in the stress model are listed as follows. The constitutive equation in the elastic range is given by e ij
=
1+ E
ij
kk ij
E
+
T
coefficient of convective heat transfer, h 0 represents the gas convection coefficient on the top surface (50 W/(m2∙K) in this work), and h1 represents the powder convection coefficient on the interface between the powder and the part (10 W/(m2∙K) in this work). In the stress model, the bottom of the build plate is fixed.
(2)
ij
3. Experimental methods
where ij is the Cauchy stress, T is the increase in temperature, E is the Young’s modulus, is the Poisson’s ratio, is the coefficient of thermal expansion, and ij is the Kronecker delta function. ij is the elastic strain given by the strain-displacement relation, e ij
=(
uj ui + )/2 xj xi
A commercial selective laser melting (SLM) machine (Renishaw AM400) equipped with an inert gas system was used to fabricate the seven cylinders with the same configuration as in the model. Fig. 2 shows the cylinders built in this work. Stainless steel 316 L powder (316 L-0407) was supplied by Renishaw with a particle size of 15–45 μm. The process parameters used for printing are listed in Table 2. After printed and cooled down in the chamber, the samples were cut from the build plate by using wire electrical discharge machining (EDM). The EDM cutting direction is shown in Fig. 2. The distortion level of different samples was measured using a coordinate measuring machine (CMM) before and after removing the samples from the build plate. About 3900 data points were measured in each sample to reduce the uncertainty.
(3)
where x i is spatial location, ui is the displacement. The linear momentum conservation equation is ji
xj
+ fi = 0
(4)
where fi is the body force per unit mass. The bilinear isotropic hardening is given by
d
ij
d
p ij
=d
e ij
+d
4. Results
p ij
3 Sij Sij Y ( p) < 0 2 3 Sij d kl 3 Sij 3 Sij Sij Y ( p) = 0 2K Y 2 Y 2
4.1. Simulated residual stress
0 =
The residual stress generated from the PBF process is critical to the
(5)
where ij is the total strain, ijp is the plastic strain, Sij (Sij = ij 3 kk ij ) is the deviatoric stress tensor, Y is the yield stress of the material in uniaxial tension, K is the tangent modulus, and p is the accumulated plastic strain magnitude. 1
2.5. Boundary conditions In the thermal model, for the element being built, the top surface is under convection with gas and radiation with the chamber, and the edge surface is assumed under convection with the unmelt powder:
T = h n (T t
Ts ) +
(T 4
Ts4 )
(6)
where T is the chamber gas temperature, Ts is the building layer temperature, is the emissivity to account for the radiation from the top surface, is the Stefan-Boltzmann constant, hn (n = 0, 1) is the
Fig. 2. Seven stainless steel cylinders printed by SLM. 216
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have higher tensile hoop and radial stress up to 282 MPa (Fig. 3(a) and (c)). With the increase of the thickness, the magnitude of the hoop stress decreases. The 21 mm sample is almost in a stress-free state, and the 31 mm sample shows a compressive state. After the build plate removal (Fig. 3(b) and (d)), stress relaxation reduces the hoop and radial stress of thin samples, and a compressive stress state shows up in the center area. Both the hoop and radial stress near the edge shows a non-uniform change compared to that near the center, and the stress near the edge shows more tension. For the 31 mm thickness cylinder, the hoop stress is in compression of −272 MPa near the center and −408 MPa on the edge; the radial stress is about −272 MPa near the center and −244 MPa on the edge. Insignificant change in the stress is observed in the 31 mm thick sample after removing the build plate, which shows the minimum effect from the build plate removal. Fig. 4 shows the hoop and radial stress of the bottom surfaces before and after the build plate removal. The bottom surfaces near the center tend to have low hoop and radial stress both before and after the build plate removal while the edges have much higher hoop and radial stress. This can be explained by the increase of stress intensity factor due to the geometric effect (cylinder bottom edge and build plate form a sharp corner) and rapid cooling induced residual stress accumulation. The thinner samples show less stress increase from the center to the edge. After the build plate removal (Fig. 4(b) and (d)), the hoop stress of the edge decreases by 200–500 MPa, and the maximum hoop stress is about 1050 MPa for the thicker samples. The maximum radial stress decreases by 100–600 MPa. Significant stress relaxation is observed in the in-
Table 2 Process parameters of SLM experiment. Parameter
Value
Layer thickness (μm) Hatch distance (μm) Point distance (μm) Exposure time (μs) Laser jump speed (m/s) Average laser scan speed (m/s) Recoater arm reset time (s)
50 110 60 80 5 0.65 8
distortion, performance, and reliability of the printed parts. Undesired residual stress can reduce the service life and even cause part failure in engineering applications. In this study, a cylindrical coordinate is used for each sample. Three stress components (hoop stress, radial stress, and axial stress) of the top and bottom surfaces of the cylinders along the radial direction and the center along the axial direction are investigated. Fig. 3 shows the hoop and radial stress of the top surfaces before and after the build plate removal. The hoop and radial stress of the center is generally higher than that of the edge before the build plate removal. As shown in Fig. 3(a) and (c), this center-edge stress difference is about 100 MPa for thin samples and 200 MPa for thick samples. However, after the build plate removal, the stress of the edge surpasses that of the center, indicating a non-uniform deformation. On the top surface before the build plate removal, thinner cylinders
Fig. 3. (a) Top surface hoop stress before build plate removal. (b) Top surface hoop stress after build plate removal. (c) Top surface radial stress before build plate removal. (d) Top surface radial stress after build plate removal. 217
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Fig. 4. (a) Bottom surface hoop stress before build plate removal. (b) Bottom surface hoop stress after build plate removal. (c) Bottom surface radial stress before build plate removal. (d) Bottom surface radial stress after build plate removal.
plane stress due to the build plate removal. The axial residual stress is directly associated with the axial deformation, so it is critical to investigate the axial stress distribution during the printing and the build plate removal processes. Fig. 5 shows the axial stress of the top surfaces before and after the build plate removal. The samples have almost zero axial stress in the center area (Fig. 5(a)). The edge has significantly higher stress of up to −270 MPa in compression. After the build plate removal (Fig. 5(b)), the axial stress stays almost unchanged, which means the plate removal induced axial distortion (to be presented) comes from the bottom part of the samples. Fig. 6 shows the axial stress of the bottom surfaces before and after the build plate removal. Before the build plate removal, the axial stress increases rapidly near the edge (Fig. 6(a)). The 2 mm sample shows the smallest stress increase of only up to 500 MPa. After the build plate removal (Fig. 6(b)), large stress relaxation occurs. The maximum axial stress decreases by 300–900 MPa, with the thinner samples releasing less stress. Stress relaxation in the bottom part of the sample causes the distortion during the build plate removal. In addition to the stress on the top surface and bottom surface along the radial direction, the stress at the center along the axial direction is also evaluated. Fig. 7 shows the hoop and radial stress along the axial direction before and after the build plate removal. Before the build plate removal, the in-plane stress has a peak of about 350 MPa near the
top surface. The stress curves of the thinner samples (2–11 mm) overlap with each other with the same trend. With the increase of the sample thickness, the stress curve shifts down, while the top surface hoop stress of 31 mm sample shows compressive stress of about −270 MPa. After the build plate removal, the hoop and radial stress near bottom surface of thicker samples (16–31 mm) increases to a similar level to the peak stress while thin samples show almost zero in-plane stress. The stress near the top surface drops to balance the bottom stress changes. Fig. 8 shows the axial stress along the axial direction before and after the build plate removal. Before the build plate removal, the stress near the top is almost zero. The thicker samples show compressive axial stress up to −300 MPa near the bottom. After the build plate removal, the stress shows a symmetric trend along the axial direction. It shows an almost plane-stress state ( axial = 0) near the top and bottom surfaces. 4.2. Measured and simulated distortion The modeling results of distortion contour are shown in Appendix (Fig. A1). The surface height data from CMM was processed to assess the distortion level. First, a mean plane is removed from the raw data to remove the tilt. The leveled surface has a raised edge because of a higher laser energy density at the border of the part as shown in Fig. 9(a). Therefore, the data at a radius (R) of larger than 20.5 mm is 218
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Fig. 5. (a) Top surface axial stress before build plate removal. (b) Top surface axial stress after build plate removal.
removed to eliminate the edge effect as shown in Fig. 9(b). The method used to determine the distortion level is shown in Fig. 9(c), which is given by D=Cave - cave
The comparison of the distortion level between the FEA and the experimental results is shown in Fig. 13. The black lines represent the distortion level before the build plate removal. Insignificant distortion (0–20 μm) is found. The red lines show increased distortion level after the build plate removal. The thinner cylinders have more significant distortion than thicker cylinders. The blue lines show the distortion contributed by the build plate removal, indicating the FEA result has a very good match with the experiment except for the thinner (2 and 4 mm) samples. This validates the excellent accuracy of the simplified FEA model for parts with a large thickness. The modeling results of thinner (2 and 4 mm) cylinders after the build plate removal do not match with the experimental results, whose reason requires further investigation. The cutting process causes a significant warpage for these two thin samples. This effect is less significant on thicker cylinders.
(7)
where D is the distortion level, and Cave and cave are the average heights in the circular band of 18.5 mm ≤ R ≤ 20.5 mm and in the inner circle of R ≤ 3.5 mm. Surface height data and distortion level for different samples before and after the build plate removal are shown in Figs. 10 and 11. The distortion level for all samples is shown in Fig. 12. It can be seen that before the build plate removal, the distortion is minimal and almost independent of the sample size. After the build plate removal, the thinner samples tend to distort more. The distortion magnitude increases from 20.9 μm to 183.8 μm with the decreasing sample thickness. The difference in distortion between before and after removal can be considered as the build plate removal induced distortion, which ranges from 6.5 μm to 169.3 μm. In the build plate removal process, the stress relaxation occurs on the bottom of the samples, and as a result, the stress state in the sample redistributes. The residual stress changes distort the samples, especially for thinner samples since it is a bottomup stress transfer.
5. Discussion 5.1. Stress relaxation induced distortion The current work helps to better understand the relationship between stress state and the distortion in the printing and the build plate removal processes. The axial distortion (Fig. 10) accumulates from the
Fig. 6. (a) Bottom surface axial stress before build plate removal. (b) Bottom surface axial stress after build plate removal. 219
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Fig. 7. (a) Through thickness hoop stress in the part center before build plate removal. (b) Through thickness hoop stress in the part center after build plate removal. (c) Through thickness radial stress in the part center before build plate removal. (d) Through thickness radial stress in the part center after build plate removal.
Fig. 8. (a) Through thickness axial stress in the part center before build plate removal. (b) Through thickness axial stress in the part center after build plate removal.
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Fig. 9. (a) CMM measured raw data (b) Processed data after removal of mean plane and edge (c) Criterion for distortion assessment.
stress redistribution of the lower part. The hoop and radial stress in the center of bottom surface changes slightly after the build plate removal while the stress of the edge decreases by about 30% (Fig. 4). The axial stress in the center of the bottom surface shows a stress-free state after removal while the stress of the edge decreases by about 50% (Fig. 6). These stress relaxation near the edge of the bottom surface contributes to the distortion changes upon the build plate removal. Most AM parts are fabricated on a build plate, and the removal is a necessary step in the manufacturing process. The findings provide insights for predicting and controlling distortion in the AM process with consideration of the build plate removal. A smart design of support structure could reduce this build plate removal induced distortion to a certain level, especially for thin structures, which will be investigated in the future work. Thick parts, such as the 31 mm sample in this study, are almost free of build plate removal induced distortion. This provides a guidance for parts thickness design to minimize the distortion level.
investigating. If such a thin structure needs to be printed, either a support structure or a different building direction should be considered. 5.3. Advantages, limitations and future work The part-scale FEA modeling is very efficient and effective for residual stress and distortion study with a low computation load and a good accuracy. It provides guidance for parts design and optimization to achieve desired geometry and residual stress, which are required in the industry. Both residual stress and distortion generated in PBF process are highly dependent on geometry or aspect ratio, and this study provides insights on how these geometric factors affect the parts. Some of the limitations of this framework are as follows. The current work focuses on the process design and modeling without considering the melting pool evolution and solidification. The superlayer in the model consists of multiple physical layers, and the accuracy can be improved using finer mesh but with an increase in the computational load. The build plate removal uses a simplified element death technique, and the incremental cutting process is not considered. Future work includes measurement of residual stress, design of different support structures, modeling build plate or support removal, and extending to different materials (e.g., Inconel) and more complex geometries.
5.2. Warping of thin cylinder due to build plate removal The experimental and numerical distortion results agree with each other for the thin (2 mm and 4 mm) cylinders before the build plate removal but disagree after the removal. This is because of these two cylinders are so thin that they warp easily due to the loss of structural confinement once removed from the build plate. The removal process was simulated by deactivating the build plate elements at once, which is different from the gradual wire EDM cutting process. However, it is believed that the disagreement between experimental and numerical distortion results is not due to the difference between the model setup and the actual experimental conditions because the warping direction is perpendicular to the cutting direction. The warping must have happened due to other reasons, which is an interesting topic worth further
6. Conclusions The current work investigated the residual stress induced distortion before and after the build plate removal in PBF process through both numerical and experimental approaches. Due to the cyclic heating and cooling process, residual stress is accumulated and distributed in the printed parts. The model successfully predicts the distortion level for
Fig. 10. CMM measured distortion contour for different samples before and after build plate removal.
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Fig. 11. Thin samples warping during the cutting process.
Fig. 13. A comparison between FEA and experimental distortion results.
Fig. 12. CMM measured distortion for different samples before and after build plate removal.
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6–31 mm thick samples. The build plate removal process induces stress relaxation on the parts and thus distortion, and especially the thin ones. The build plate removal process had a minimal effect on the distortion
of the 31 mm thick sample but resulted in a distortion level as high as 170 μm on the 2 mm thick sample. The cutting effect on thin parts needs to be further invstigated.
Appendix A
Fig. A1. FEA distortion results before and after build plate removal.
Table A1 Thermophysical properties of stainless steel 316 L. Temperature (oC)
Density (kg∙m−3)
Young’s Modulus (GPa)
Poisson’s Ratio
Yield Strength (MPa)
Tangent Modulus (MPa)
Coefficient of Thermal Expansion
Thermal Conductivity (W∙m−1∙K−1)
Specific Heat (J∙kg−1∙K−1)
26.85 126.85 226.85 326.85 426.85 526.85 726.85 826.85 926.85 1026.9 1126.9 1226.9 1372
7954 7910 7864 7818 7771 7723 7624 7574 7523 7471 7419 7365 7127
194.7 190 184.5 178.2 171 161.7 141 127.3 112.5 95 73 51 51
0.25 0.23 0.285 0.319 0.322 0.305 0.291 0.24 0.24 0.24 0.24 0.24 0.24
_ 217.4 188.8 165.2 154.9 144.7 124.2 111.1 73.6 36 18.4 15 15
_ 2023 1765 1531 1363 1195 858.6 693.6 558.2 423 293.4 265 265
1.48E-05 1.56E-05 1.63E-05 1.69E-05 1.74E-05 1.79E-05 1.87E-05 1.90E-05 1.93E-05 1.95E-05 1.96E-05 1.98E-05 1.99E-05
13.44 15.16 16.8 18.36 19.87 21.39 24.06 25.46 26.74 28.02 29.32 30.61 32.41
498.73 512.12 525.51 538.48 551.87 565.26 591.62 605.01 618.4 631.78 644.75 658.14 671.53
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Journal of Manufacturing Processes journal homepage: www.elsevier.com/locate/manpro
Experimental and numerical study on residual stress and geometric distortion in powder bed fusion process Tianyu Yua, Ming Lib,c, Austin Breauxb, Mukul Atrid, Suleiman Obeidatb, Chao Mab,c,
T
⁎
a
Dante Solutions, Inc., Cleveland, OH, USA Dept. of Engineering Technology & Industry Distribution, Texas A&M University, USA c Dept. of Industrial & Systems Engineering, Texas A&M University, USA d Ansys, Inc., Pune, Maharashtra, India b
ARTICLE INFO
ABSTRACT
Keywords: Powder bed fusion Additive manufacturing 3D printing Finite element analysis Residual stress Geometric distortion
The objective of the present work is to experimentally and numerically study the effect of the part size on the distortion of a part printed by a powder bed fusion process. A simplified thermo-mechanical finite element model was developed using a so-called superlayer concept (i.e., numerous layers deposited at once) to predict the geometric distortion induced by the thermal stress. A set of seven cylinders with different dimensions were modeled with this method. The same geometries were printed using a commercial selective laser melting machine and cut from the build plate using wire electrical discharge machining. The distortion of different cylinders was measured before and after the removal from the build plate using a coordinate measuring machine and compared with the simulation results. The experimental results showed the final distortion is largely contributed by the build plate removal process because it relaxes the thermal stress accumulated during the printing process. The modeling results show a high consistency with the experimental results, especially for the thicker parts. As the part thickness decreases, the prediction accuracy of the model decreases. When a superlayer thickness of 1 mm is used, the model can accurately predict the distortion for a cylinder with a diameter of 45 mm and a thickness of greater than 6 mm. This model provides an effective approach for geometric control of printed parts.
1. Introduction Additive manufacturing (AM) refers to a variety of methods that fabricate parts using a layer-by-layer strategy. AM provides the capability for producing parts of complex geometries without additional tooling. The most commonly used metal AM processes can be divided into two groups: 1) powder bed fusion (PBF) processes (e.g. selective laser melting, direct metal laser sintering, electron beam melting, and laser metal fusion) and 2) directed energy deposition (DED) processes (e.g. laser-engineered net shaping, laser metal deposition, direct metal deposition, and wire and arc additive manufacturing) [1,2]. PBF is advantageous over DED in the senses of geometric accuracy and surface finish. In a PBF process, either a laser or electron beam is used to fuse the feedstock powder which is pre-spread on the previous layer. Although AM technology is developing at a fast pace, there are still many issues to overcome. One of the most challenging issues is the residual stress and distortion due to the unique heating and cooling cycles. Because of the complexity of the PBF process, it is almost
impossible to calculate the process-induced residual stress and distortion explicitly. Additionally, the measurement (e.g., X-ray diffraction, neutron diffraction) of residual stress in AMed metal parts are very tedious and expensive, in a limited accuracy (off in the range of 70–100 MPa for X-ray diffraction [3]). Thus, to better understand how the residual stress and distortion are formed during the PBF process and control the parts quality, an efficient and effective numerical model is highly needed. Numerical simulation is a very powerful tool for assessing the process parameters effectiveness and predicting the manufacturing performance [4–9]. A decent amount of effort on modeling residual stress in AM was made recently [10–19], as listed in Table 1. For example, Alimardani et al. [10] developed a numerical method for modeling temperature field and thermal stress during DED. A four-track thin wall structure was modeled and experimentally verified by measuring the temperature and the deflection of the workpiece. Gu and He [11] and Hussein et al. [12] used a finite element model to investigate the temperature and stress fields of three and five tracks in PBF,
Corresponding author. E-mail addresses: [email protected] (T. Yu), [email protected] (M. Li), [email protected] (A. Breaux), [email protected] (M. Atri), [email protected] (S. Obeidat), [email protected] (C. Ma). ⁎
https://doi.org/10.1016/j.jmapro.2019.09.010 Received 28 September 2018; Received in revised form 30 July 2019; Accepted 12 September 2019 Available online 23 September 2019 1526-6125/ © 2019 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.
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results. Section 5 provide a discussion on residual stress and distortion of the samples before and after removing the build plate. Conclusions are drawn in Section 6.
Table 1 Comparison of different modeling works on residual stress and distortion. Scale
Track Track Track Layer Layer Layer Part Part Part Part
Predicted stress
Predicted distortion
Validated distortion
Modeled build plate removal
Studied size effect
Reference
Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes
Yes No No Yes Yes Yes Yes Yes Yes Yes
Yes No No No No No Yes Yes Yes Yes
No No No No No No No Yes Yes Yes
No No No No No No No No No Yes
[10] [11] [12] [13] [14] [15] [16] [17] [18] This work
2. Model description The commercial FEA software, ANSYS 18.2 Mechanical, was used on a workstation with 24 cores and 128 GB RAM. The model uses a transient thermo-mechanical solution based on the AM ACT 1.1 module. A thermal model is solved first without considering the mechanical behaviors (stress and distortion), resulting in a transient temperature field, which is then used as the input for the mechanical model to predict the thermal stress and distortion. 2.1. Assumptions and simplifications Some assumptions and simplifications were made to improve the modeling efficiency and productivity:
respectively. Li et al. [13] developed a multi-scale modeling approach to predict the distortions of a single layer under four different scanning strategies. Dai and Shaw [14] applied the powder-to-solid transition model to predict the generation of micro-scale residual stress and the distortion of two layers. Cheng et al. [15] simulated the residual stress and deformation under various scanning patterns using three layers. Most of the current modeling studies on AM are focused on incorporating high-fidelity process physics, addressing laser-matter interaction, and predicting effects of process parameters and scanning strategy. These models are computationally expensive, which makes them well-positioned for the cases of several spots [8], tracks [10–12], or layers [13–15], but inapplicable to industrial components of large and complex geometries. Seldom reported is the part-scale modeling of residual stress and distortion formation during the AM process. Zaeh and Branner [16] investigated the distortion and residual stress of a cantilever beam produced by a PBF process. They simplified the model by adding multiple layers (called superlayer in this paper) at once and applying a monolithic thermal load. The model was efficient enough to model the cantilever beam. However, the researchers did not model the effect of the build plate removal on the stress and the distortion. Build plate removal largely relaxes the residual stress and leads to a significant distortion [19]. Lack of consideration of the build plate removal in the model could be one of the reasons for the disagreement between the simulation and the experiment [16]. Neugebauer et al. [17] and Li et al. [18] used a similar approach (i.e., superlayer) to numerically study the distortion, achieving a reasonable accuracy with the consideration of the build plate removal. However, these researchers used only one part in their studies and did not vary the part size [16–18], which could have a significant effect on the distortion. The objective of this study is to experimentally and numerically study the effect of the part size on the distortion of a part printed by PBF. A part-scale model was developed for the prediction of distortion caused by the residual stress accumulated in the PBF process considering the effect of build plate removal. Finite element analysis (FEA) is used to predict the residual stress and distortion generated from the incremental printing process. A build plate is used to support and transfer heat from the printing parts. The build plate removal is necessary after finishing the PBF printing process, and this removal process is also simulated to predict the distortion induced by stress relaxation. In the current study, a set of seven cylinders is simulated in a transient thermo-mechanical model. SLM experiments are conducted to examine the results of the simulation. The modeling and experimental framework helps to enhance the fundamental understanding of the PBF part distortion behaviors, provide a better design routine, and support the development of suitable process strategies and component topologies. Section 2 defines the model setup. Section 3 describes the experimental methods, including the SLM process and distortion measurement details. Section 4 shows the simulation and experimental
1) Every twenty processing layers are represented by one superlayer; 2) Each superlayer is deposited on the previous superlayer at once; 3) The initial temperature of each superlayer after deposition is the melting temperature of the material (1371 °C for stainless steel 316 L in this study); 4) Thermal stress starts to accumluate at 800 ℃ (termed as reference temperature), above which the stress is consumed by grain growth and/or phase evolution. 2.2. Material, dimension, and meshing A printing process of seven cylinders of the same diameter (45 mm) but different thicknesses/height (2 mm, 4 mm, 6 mm, 11 mm, 16 mm, 21 mm and 31 mm) was simulated. As shown in Fig. 1, the cylinders were aligned evenly to reduce the interaction between each other. The material selected in this study is stainless steel 316 L because of its widespread applications. The thermophysical properties of 316 L are shown in Appendix (Table A1). The standard ANSYS element birth technique was used to generate the cylinder samples, simulating the additive nature of the process. The build plate and the seven cylinders are meshed with 178k nodes and 162k elements as shown in Fig. 1. The cylinders are meshed using sweep method with elements with a thickness of 0.5 mm and in-plane size of 1–1.8 mm. The element size for the build plate is 10 mm. 2.3. Simulation process Using ANSYS Mechanical, the thermal model is set up first to catch the thermal history during the printing and cooling process. The thermal data is then transferred to a stress model to calculate the residual stress and distortion level. The thermal model and stress model share the same material database, geometry, and mesh, although they are not directly coupled. After the seven cylinders are printed, they are allowed to cool down to room temperature. Afterward, the cylinders are removed from the build plate. The thermal model took 47 min and the stress model 5.5 h, indicating the high computational efficiency of the superlayer-based FEA model. 2.4. Governing equations The governing equation for the thermal model is the transient heat conduction equation,
c
T = t
where 215
2T
x i2
(1)
is the density, c is the specific heat, T is the temperature, t is
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Fig. 1. FEA Mesh for the printing parts with 178k nodes and 162k elements.
the time, x i is the spatial location, and is the thermal conductivity. After solving the thermal model, T of all nodes for different time frames will be stored and used as the input of the stress model. The governing equations used in the stress model are listed as follows. The constitutive equation in the elastic range is given by e ij
=
1+ E
ij
kk ij
E
+
T
coefficient of convective heat transfer, h 0 represents the gas convection coefficient on the top surface (50 W/(m2∙K) in this work), and h1 represents the powder convection coefficient on the interface between the powder and the part (10 W/(m2∙K) in this work). In the stress model, the bottom of the build plate is fixed.
(2)
ij
3. Experimental methods
where ij is the Cauchy stress, T is the increase in temperature, E is the Young’s modulus, is the Poisson’s ratio, is the coefficient of thermal expansion, and ij is the Kronecker delta function. ij is the elastic strain given by the strain-displacement relation, e ij
=(
uj ui + )/2 xj xi
A commercial selective laser melting (SLM) machine (Renishaw AM400) equipped with an inert gas system was used to fabricate the seven cylinders with the same configuration as in the model. Fig. 2 shows the cylinders built in this work. Stainless steel 316 L powder (316 L-0407) was supplied by Renishaw with a particle size of 15–45 μm. The process parameters used for printing are listed in Table 2. After printed and cooled down in the chamber, the samples were cut from the build plate by using wire electrical discharge machining (EDM). The EDM cutting direction is shown in Fig. 2. The distortion level of different samples was measured using a coordinate measuring machine (CMM) before and after removing the samples from the build plate. About 3900 data points were measured in each sample to reduce the uncertainty.
(3)
where x i is spatial location, ui is the displacement. The linear momentum conservation equation is ji
xj
+ fi = 0
(4)
where fi is the body force per unit mass. The bilinear isotropic hardening is given by
d
ij
d
p ij
=d
e ij
+d
4. Results
p ij
3 Sij Sij Y ( p) < 0 2 3 Sij d kl 3 Sij 3 Sij Sij Y ( p) = 0 2K Y 2 Y 2
4.1. Simulated residual stress
0 =
The residual stress generated from the PBF process is critical to the
(5)
where ij is the total strain, ijp is the plastic strain, Sij (Sij = ij 3 kk ij ) is the deviatoric stress tensor, Y is the yield stress of the material in uniaxial tension, K is the tangent modulus, and p is the accumulated plastic strain magnitude. 1
2.5. Boundary conditions In the thermal model, for the element being built, the top surface is under convection with gas and radiation with the chamber, and the edge surface is assumed under convection with the unmelt powder:
T = h n (T t
Ts ) +
(T 4
Ts4 )
(6)
where T is the chamber gas temperature, Ts is the building layer temperature, is the emissivity to account for the radiation from the top surface, is the Stefan-Boltzmann constant, hn (n = 0, 1) is the
Fig. 2. Seven stainless steel cylinders printed by SLM. 216
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have higher tensile hoop and radial stress up to 282 MPa (Fig. 3(a) and (c)). With the increase of the thickness, the magnitude of the hoop stress decreases. The 21 mm sample is almost in a stress-free state, and the 31 mm sample shows a compressive state. After the build plate removal (Fig. 3(b) and (d)), stress relaxation reduces the hoop and radial stress of thin samples, and a compressive stress state shows up in the center area. Both the hoop and radial stress near the edge shows a non-uniform change compared to that near the center, and the stress near the edge shows more tension. For the 31 mm thickness cylinder, the hoop stress is in compression of −272 MPa near the center and −408 MPa on the edge; the radial stress is about −272 MPa near the center and −244 MPa on the edge. Insignificant change in the stress is observed in the 31 mm thick sample after removing the build plate, which shows the minimum effect from the build plate removal. Fig. 4 shows the hoop and radial stress of the bottom surfaces before and after the build plate removal. The bottom surfaces near the center tend to have low hoop and radial stress both before and after the build plate removal while the edges have much higher hoop and radial stress. This can be explained by the increase of stress intensity factor due to the geometric effect (cylinder bottom edge and build plate form a sharp corner) and rapid cooling induced residual stress accumulation. The thinner samples show less stress increase from the center to the edge. After the build plate removal (Fig. 4(b) and (d)), the hoop stress of the edge decreases by 200–500 MPa, and the maximum hoop stress is about 1050 MPa for the thicker samples. The maximum radial stress decreases by 100–600 MPa. Significant stress relaxation is observed in the in-
Table 2 Process parameters of SLM experiment. Parameter
Value
Layer thickness (μm) Hatch distance (μm) Point distance (μm) Exposure time (μs) Laser jump speed (m/s) Average laser scan speed (m/s) Recoater arm reset time (s)
50 110 60 80 5 0.65 8
distortion, performance, and reliability of the printed parts. Undesired residual stress can reduce the service life and even cause part failure in engineering applications. In this study, a cylindrical coordinate is used for each sample. Three stress components (hoop stress, radial stress, and axial stress) of the top and bottom surfaces of the cylinders along the radial direction and the center along the axial direction are investigated. Fig. 3 shows the hoop and radial stress of the top surfaces before and after the build plate removal. The hoop and radial stress of the center is generally higher than that of the edge before the build plate removal. As shown in Fig. 3(a) and (c), this center-edge stress difference is about 100 MPa for thin samples and 200 MPa for thick samples. However, after the build plate removal, the stress of the edge surpasses that of the center, indicating a non-uniform deformation. On the top surface before the build plate removal, thinner cylinders
Fig. 3. (a) Top surface hoop stress before build plate removal. (b) Top surface hoop stress after build plate removal. (c) Top surface radial stress before build plate removal. (d) Top surface radial stress after build plate removal. 217
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Fig. 4. (a) Bottom surface hoop stress before build plate removal. (b) Bottom surface hoop stress after build plate removal. (c) Bottom surface radial stress before build plate removal. (d) Bottom surface radial stress after build plate removal.
plane stress due to the build plate removal. The axial residual stress is directly associated with the axial deformation, so it is critical to investigate the axial stress distribution during the printing and the build plate removal processes. Fig. 5 shows the axial stress of the top surfaces before and after the build plate removal. The samples have almost zero axial stress in the center area (Fig. 5(a)). The edge has significantly higher stress of up to −270 MPa in compression. After the build plate removal (Fig. 5(b)), the axial stress stays almost unchanged, which means the plate removal induced axial distortion (to be presented) comes from the bottom part of the samples. Fig. 6 shows the axial stress of the bottom surfaces before and after the build plate removal. Before the build plate removal, the axial stress increases rapidly near the edge (Fig. 6(a)). The 2 mm sample shows the smallest stress increase of only up to 500 MPa. After the build plate removal (Fig. 6(b)), large stress relaxation occurs. The maximum axial stress decreases by 300–900 MPa, with the thinner samples releasing less stress. Stress relaxation in the bottom part of the sample causes the distortion during the build plate removal. In addition to the stress on the top surface and bottom surface along the radial direction, the stress at the center along the axial direction is also evaluated. Fig. 7 shows the hoop and radial stress along the axial direction before and after the build plate removal. Before the build plate removal, the in-plane stress has a peak of about 350 MPa near the
top surface. The stress curves of the thinner samples (2–11 mm) overlap with each other with the same trend. With the increase of the sample thickness, the stress curve shifts down, while the top surface hoop stress of 31 mm sample shows compressive stress of about −270 MPa. After the build plate removal, the hoop and radial stress near bottom surface of thicker samples (16–31 mm) increases to a similar level to the peak stress while thin samples show almost zero in-plane stress. The stress near the top surface drops to balance the bottom stress changes. Fig. 8 shows the axial stress along the axial direction before and after the build plate removal. Before the build plate removal, the stress near the top is almost zero. The thicker samples show compressive axial stress up to −300 MPa near the bottom. After the build plate removal, the stress shows a symmetric trend along the axial direction. It shows an almost plane-stress state ( axial = 0) near the top and bottom surfaces. 4.2. Measured and simulated distortion The modeling results of distortion contour are shown in Appendix (Fig. A1). The surface height data from CMM was processed to assess the distortion level. First, a mean plane is removed from the raw data to remove the tilt. The leveled surface has a raised edge because of a higher laser energy density at the border of the part as shown in Fig. 9(a). Therefore, the data at a radius (R) of larger than 20.5 mm is 218
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Fig. 5. (a) Top surface axial stress before build plate removal. (b) Top surface axial stress after build plate removal.
removed to eliminate the edge effect as shown in Fig. 9(b). The method used to determine the distortion level is shown in Fig. 9(c), which is given by D=Cave - cave
The comparison of the distortion level between the FEA and the experimental results is shown in Fig. 13. The black lines represent the distortion level before the build plate removal. Insignificant distortion (0–20 μm) is found. The red lines show increased distortion level after the build plate removal. The thinner cylinders have more significant distortion than thicker cylinders. The blue lines show the distortion contributed by the build plate removal, indicating the FEA result has a very good match with the experiment except for the thinner (2 and 4 mm) samples. This validates the excellent accuracy of the simplified FEA model for parts with a large thickness. The modeling results of thinner (2 and 4 mm) cylinders after the build plate removal do not match with the experimental results, whose reason requires further investigation. The cutting process causes a significant warpage for these two thin samples. This effect is less significant on thicker cylinders.
(7)
where D is the distortion level, and Cave and cave are the average heights in the circular band of 18.5 mm ≤ R ≤ 20.5 mm and in the inner circle of R ≤ 3.5 mm. Surface height data and distortion level for different samples before and after the build plate removal are shown in Figs. 10 and 11. The distortion level for all samples is shown in Fig. 12. It can be seen that before the build plate removal, the distortion is minimal and almost independent of the sample size. After the build plate removal, the thinner samples tend to distort more. The distortion magnitude increases from 20.9 μm to 183.8 μm with the decreasing sample thickness. The difference in distortion between before and after removal can be considered as the build plate removal induced distortion, which ranges from 6.5 μm to 169.3 μm. In the build plate removal process, the stress relaxation occurs on the bottom of the samples, and as a result, the stress state in the sample redistributes. The residual stress changes distort the samples, especially for thinner samples since it is a bottomup stress transfer.
5. Discussion 5.1. Stress relaxation induced distortion The current work helps to better understand the relationship between stress state and the distortion in the printing and the build plate removal processes. The axial distortion (Fig. 10) accumulates from the
Fig. 6. (a) Bottom surface axial stress before build plate removal. (b) Bottom surface axial stress after build plate removal. 219
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Fig. 7. (a) Through thickness hoop stress in the part center before build plate removal. (b) Through thickness hoop stress in the part center after build plate removal. (c) Through thickness radial stress in the part center before build plate removal. (d) Through thickness radial stress in the part center after build plate removal.
Fig. 8. (a) Through thickness axial stress in the part center before build plate removal. (b) Through thickness axial stress in the part center after build plate removal.
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Fig. 9. (a) CMM measured raw data (b) Processed data after removal of mean plane and edge (c) Criterion for distortion assessment.
stress redistribution of the lower part. The hoop and radial stress in the center of bottom surface changes slightly after the build plate removal while the stress of the edge decreases by about 30% (Fig. 4). The axial stress in the center of the bottom surface shows a stress-free state after removal while the stress of the edge decreases by about 50% (Fig. 6). These stress relaxation near the edge of the bottom surface contributes to the distortion changes upon the build plate removal. Most AM parts are fabricated on a build plate, and the removal is a necessary step in the manufacturing process. The findings provide insights for predicting and controlling distortion in the AM process with consideration of the build plate removal. A smart design of support structure could reduce this build plate removal induced distortion to a certain level, especially for thin structures, which will be investigated in the future work. Thick parts, such as the 31 mm sample in this study, are almost free of build plate removal induced distortion. This provides a guidance for parts thickness design to minimize the distortion level.
investigating. If such a thin structure needs to be printed, either a support structure or a different building direction should be considered. 5.3. Advantages, limitations and future work The part-scale FEA modeling is very efficient and effective for residual stress and distortion study with a low computation load and a good accuracy. It provides guidance for parts design and optimization to achieve desired geometry and residual stress, which are required in the industry. Both residual stress and distortion generated in PBF process are highly dependent on geometry or aspect ratio, and this study provides insights on how these geometric factors affect the parts. Some of the limitations of this framework are as follows. The current work focuses on the process design and modeling without considering the melting pool evolution and solidification. The superlayer in the model consists of multiple physical layers, and the accuracy can be improved using finer mesh but with an increase in the computational load. The build plate removal uses a simplified element death technique, and the incremental cutting process is not considered. Future work includes measurement of residual stress, design of different support structures, modeling build plate or support removal, and extending to different materials (e.g., Inconel) and more complex geometries.
5.2. Warping of thin cylinder due to build plate removal The experimental and numerical distortion results agree with each other for the thin (2 mm and 4 mm) cylinders before the build plate removal but disagree after the removal. This is because of these two cylinders are so thin that they warp easily due to the loss of structural confinement once removed from the build plate. The removal process was simulated by deactivating the build plate elements at once, which is different from the gradual wire EDM cutting process. However, it is believed that the disagreement between experimental and numerical distortion results is not due to the difference between the model setup and the actual experimental conditions because the warping direction is perpendicular to the cutting direction. The warping must have happened due to other reasons, which is an interesting topic worth further
6. Conclusions The current work investigated the residual stress induced distortion before and after the build plate removal in PBF process through both numerical and experimental approaches. Due to the cyclic heating and cooling process, residual stress is accumulated and distributed in the printed parts. The model successfully predicts the distortion level for
Fig. 10. CMM measured distortion contour for different samples before and after build plate removal.
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Fig. 11. Thin samples warping during the cutting process.
Fig. 13. A comparison between FEA and experimental distortion results.
Fig. 12. CMM measured distortion for different samples before and after build plate removal.
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6–31 mm thick samples. The build plate removal process induces stress relaxation on the parts and thus distortion, and especially the thin ones. The build plate removal process had a minimal effect on the distortion
of the 31 mm thick sample but resulted in a distortion level as high as 170 μm on the 2 mm thick sample. The cutting effect on thin parts needs to be further invstigated.
Appendix A
Fig. A1. FEA distortion results before and after build plate removal.
Table A1 Thermophysical properties of stainless steel 316 L. Temperature (oC)
Density (kg∙m−3)
Young’s Modulus (GPa)
Poisson’s Ratio
Yield Strength (MPa)
Tangent Modulus (MPa)
Coefficient of Thermal Expansion
Thermal Conductivity (W∙m−1∙K−1)
Specific Heat (J∙kg−1∙K−1)
26.85 126.85 226.85 326.85 426.85 526.85 726.85 826.85 926.85 1026.9 1126.9 1226.9 1372
7954 7910 7864 7818 7771 7723 7624 7574 7523 7471 7419 7365 7127
194.7 190 184.5 178.2 171 161.7 141 127.3 112.5 95 73 51 51
0.25 0.23 0.285 0.319 0.322 0.305 0.291 0.24 0.24 0.24 0.24 0.24 0.24
_ 217.4 188.8 165.2 154.9 144.7 124.2 111.1 73.6 36 18.4 15 15
_ 2023 1765 1531 1363 1195 858.6 693.6 558.2 423 293.4 265 265
1.48E-05 1.56E-05 1.63E-05 1.69E-05 1.74E-05 1.79E-05 1.87E-05 1.90E-05 1.93E-05 1.95E-05 1.96E-05 1.98E-05 1.99E-05
13.44 15.16 16.8 18.36 19.87 21.39 24.06 25.46 26.74 28.02 29.32 30.61 32.41
498.73 512.12 525.51 538.48 551.87 565.26 591.62 605.01 618.4 631.78 644.75 658.14 671.53
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