Numerical Modeling of Thermal and Mechanical Behaviors in the Selective Laser

Sintering of Metals

Submitted in partial fulfillment of the requirement for

the degree of

Doctor of Philosophy

in

Mechanical Engineering

Patcharapit Promoppatum

B.S., Mechanical Engineering, King Mongkut’s University of Technology Thonburi

M.S., Mechanical Engineering, Carnegie Mellon University

Carnegie Mellon University

Department of Mechanical Engineering

Pittsburgh, Pennsylvania

April 2018

© Patcharapit Promoppatum, 2018 All Rights Reserve

iii

Acknowledgement

I feel extremely fortunate for the last five years that I have spent at Department of Mechanical

Engineering, Carnegie Mellon University (CMU) pursuing my Ph.D. education. I have met

numbers of supporting and inspiring people. Without them, I would have never been so close to

my degree, and my life in Pittsburgh would have never been as joyful as it has.

First and foremost, I would like to have a sincere gratitude for my advisor, Professor Shi-Chune

Yao. He has been a great example as an educator, researcher, and as a person. His continuous

care, optimism, enthusiasm, and support have been the major part of my educational success. I

will forever be grateful for time that I have worked under his advisory. He has also shaped me

with a critical-thinking attitude, and allowed me to have a freedom in directing the research’s

direction. There could not the greater honor as to be able to call myself his last Ph.D. student

over his 40 years’ tenure at CMU. Furthermore, rather than learning from him academically, it is

the privilege for me to closely witness a man who has lived his life happily and contributed

significantly to his students, society, and scientific community. This experience will forever

inspire me both in my career and personal life.

I would like to further extend my gratitude to committee members, Professor Burak Ozdoganlar,

Professor Sheng Shen, and Professor Bryan Webler for their support and comments, which

tremendously improved my studies. I want to thank Professor Ozdoganlar as his collaboration

with Professor Yao laid the initial foundation for my work in the additive manufacturing. I am also

benefitted from equipment in his laboratory. In addition, suggestions from Professor Webler

regarding residual stress measurement using X-ray and Neutron diffractions, as well as granting

me the access to his material preparation facility have played a very important role in the later

part of my study. My genuine appreciation to Professor Shen was from my very first semester at

CMU when I took his class, Direct Solar and Thermal Energy Conversion. At that time, as a

foreign student, I was lacking confidence in my capability to keep up with difficult content

presented in the class. Many times, after the class, he spent at least 30 minutes re-visiting the

materials he presented in the class for me. His willingness to help has positively motivated me

to work harder, and at the end of his class, I felt much more competent and believed that with

the positive working attitude, I am fully capable of excelling in a competitive environment at

CMU.

iv

In addition to my committee members, I want to express my gratefulness to Professor Anthony

D. Rollett and Professor P. Chris Pistorius as their expertise in material science and their

genuine care as a teacher have a significant contribution to the progress of my research.

Winning a modeling challenge has been the biggest achievement in my academic life at CMU,

and it was one of the most important factors that drew people’s interest when I was looking for a

position after graduation. This would not have happened without help from them. The

opportunity to work with them allows me to experience their working ethic, attention to details,

and joyfulness from doing a scientific research, which is seen to me as a clear epitome,

essential for being successful as teachers and researchers.

Apart from my research in additive manufacturing, I greatly benefitted from working on research

projects with Professor Dana Cupkova and Professor Venkat Viswanathan. Working with

Professor Cupkova and her architecture team was an eye-opening experience and taught me

how important it is to not put too much emphasis on the technical perspective but rather

considering a broader perspective and actual functionality. I always amazed by her creativity

and felt thankful for her invitation to the ACADIA conference, which was by far the most inspiring

conference I have attended.

I must also thank many of staff members from both Mechanical Engineering and Material

Science Departments for their technical and non-technical support in my study. I would like to

especially acknowledge Dr. Elizabeth Clark for her assistant in the X-ray diffraction, Dr. Sandra

Wolf and Todd Baer for their support in AM sample’s fabrication, William Pingitore for his help in

a sample preparation and for always being a very nice and supportive one, Edward R

Wojciechowski for his support and expertise in machining, and Chris Hertz for always being very

responsive to my many questions over these 5 years. Without these people, I could not imagine

how much more difficult it would have been to complete my education.

I always had a joy and grateful moments with previous students in Professor Yao’s group: Dr.

Dongzhi Guo, Xinran Zhao, Akshay Iyer, and Dudong Feng. Thank you for your friendship,

support, and discussion over my research, which have aided me well during my study. I also

appreciated my friendship with Recep Onler. Recep was one of the people lending his hands to

me every time I needed help. His selfless personality makes him a likable one to his colleagues,

and I cannot thank him enough for his support in my study.

v

I acknowledged the financial support from the Royal Thai Government, which covered the tuition

fee and living expenses over the course of my study. I would not have been able to live and

study abroad without help from the Thai government. I also wanted to thank a grant from

AmericaMakes (FA8650-12-2-7230), which was among the first grants to fund our study in the

additive manufacturing. It was also such an honor to be awarded the John and Claire Bertucci

Fellowship, in which my thankfulness was toward John and Claire as well as the fellowship

committee for their recognition in the importance and potential contribution of my research.

While I enjoyed my research and study at CMU, I must admit that many people outside a school

had significant influence on my everyday life, and my interaction with them made my time in

Pittsburgh very much memorable. I would like to thank my Thai friends for their care, laughter,

and delicious meal they cooked. This small and heartwarming community always made me feel

welcome. In addition, my involvement with Pittsburgh Steeline and Pittsburgh Steelers was one

of the coolest experience I had in my life. I always look forward to performing with my drumming

fellows on a game day every Sunday. There will never be any other experiences that can

replace drumming for 65,000 enthusiastic Steelers fans at the Heinz Field. Not to mention that I

met my significant other, Attitra Lelahuta, because of one of the events with Steeline, in which

she has played a very supportive role over the last year of my study. Thank you Steeline!

Last but not least, I have a sincere gratitude for my family, who lives on the other side of the

globe, for allowing me to be away from home for many years to fulfill my dream in getting the

higher education in the United States. I also wanted to thank my two younger brothers for doing

a great job in taking care of family and letting me be worry-free while living and studying in the

United States.

vi

Abstract

The selective laser sintering (SLS) process or the additive manufacturing (AM) enables the

construction of a three-dimensional object through melting and solidification of metal powder.

The primary advantage of AM over the conventional process is providing the manufacturing

flexibility, especially for highly complicated products. The quality of AM products depends upon

various processing parameters such as laser power, laser scanning velocity, laser scanning

pattern, layer thickness, and hatch spacing. The improper selection of these parameters would

lead to parts with defects, severe distortion, and even cracking. I herein perform the numerical

and experimental analysis to investigate the interplay between processing parameters and the

defect generation.

The analysis aims to resolve issues at two different scales, micro-scale and product-scale. At

the micro-scale, while the numerical model is developed to investigate the interaction of the

laser and materials in the AM process, its advantages and disadvantages compared to an

analytical approach (Rosenthal’s equation), which provides a quicker thermal solution, are

thoroughly studied. Additionally, numerical results have been verified by series of experiments.

Based on the analysis, it is found that the simultaneous consideration of multiple processing

parameters could be achieved using the energy density. Moreover, together with existing

criteria, a processing window is numerically developed as a guideline for AM users to avoid

common defects at this scale including the lack of fusion, balling effect, and over-melting.

Thermal results at a micro-scale are extended as an input to determine the residual stress

initiation in AM products. The effect of energy density and substrate temperature on a residual

stress magnitude is explored. Results show that the stress magnitude within a layer is a strong

function of the substrate temperature, where a higher substrate temperature results in a lower

stress. Moreover, the stress formation due to a layer’s addition is studied, in which the stress

relaxation at locations away from a top surface is observed.

Nevertheless, even though the micro-scale analysis can resolve some common defects in AM, it

is not capable of predicting product-scale responses such as residual stress development and

entire product’s distortion. As a result, the multiscale modeling platform is developed for the

numerical investigation at the product level. Three thermal models at various scales are

interactively used to yield an effective thermal development calculation at a product-scale. In

vii

addition, the influence of the multiple layers, energy densities and scanning patterns on the

residual stress formation has been addressed, which leads to the prediction of the residual

stress development during the fabrication. The distortion of products due to the residual stress

can be described by the product-scale model. Furthermore, among many processing

parameters, the energy input and the scanning length are found to be important factors, which

could be controlled to achieve the residual stress reduction in AM products. An optimal choice of

a scanning length and energy input can reduce an as-built residual stress magnitude by almost

half of typically encountered values. Ultimately, the present work aims to illustrate the

integration of the computational method as tools to provide manufacturing qualification for part

production by the AM process.

viii

Table of Content

Acknowledgement ..................................................................................................................... iii

Abstract ...................................................................................................................................... vi

Table of Content ...................................................................................................................... viii

List of Tables ............................................................................................................................. xi

List of Figures ........................................................................................................................... xii

List of Symbols ...................................................................................................................... xviii

1. Introduction ......................................................................................................................... 1

1.1 Selective laser melting .................................................................................................. 1

1.2 Melt pool modeling in the SLM process ........................................................................ 4

1.3 Defects in SLM products and processing window ........................................................ 6

1.4 Residual stress in SLM products .................................................................................. 9

1.5 Product-scale simulation ............................................................................................. 11

1.6 Controlling of residual stress from scanning patterns ................................................. 13

1.7 Objectives ................................................................................................................... 16

2. Experimental approach .................................................................................................... 18

3. Analytical and numerical modeling ................................................................................ 21

3.1 Analytical solution for thermal result ........................................................................... 21

3.2 Numerical solution for thermal result .......................................................................... 22

3.2.1 Governing equation and boundary condition ...................................................... 22

3.2.2 Material thermophysical properties ..................................................................... 24

3.3 Point thermal model .................................................................................................... 28

3.4 Region thermal model ................................................................................................. 29

3.5 Product thermal model ................................................................................................ 31

3.6 Mechanical modeling .................................................................................................. 32

3.7 Multi-scale modeling ................................................................................................... 34

4. Comprehensive comparison between analytical and numerical solution for thermal

prediction .................................................................................................................................. 36

4.1 Melt pool configuration from experimental, numerical, and analytical results ............. 36

ix

4.2 Thermal predictions: Temperature gradient, cooling rate, and solidification rate ....... 39

4.3 Microstructural prediction: Solidification map and primary dendrite arm spacing

(PDAS) ................................................................................................................................... 44

4.4 Conclusions ................................................................................................................ 48

5. Numerical-based processing window ............................................................................ 49

5.1 Designing experiment ................................................................................................. 49

5.2 Melt pool characteristics ............................................................................................. 50

5.3 Lack of fusion under scanning tracks ......................................................................... 53

5.4 Influence of the energy deposition density ................................................................. 54

5.5 Balling effect ............................................................................................................... 57

5.6 Comprehensive processing window ........................................................................... 59

5.7 Conclusions ................................................................................................................ 62

6. Residual stress development in single and multiple layers ......................................... 63

6.1 Overview and modeling strategy ................................................................................ 63

6.2 Temperature profile and residual stress within a plane .............................................. 64

6.3 Effect of solid substrate temperature on the residual stress ....................................... 66

6.4 Residual stress development in multiple layers .......................................................... 67

6.5 Conclusions ................................................................................................................ 70

7. Multiscale modeling of thermal and microstructural behaviors with the experimental

validation in a product-scale ................................................................................................... 71

7.1 Experimental samples ................................................................................................ 71

7.2 Thermal measurement setup and sample preparation ............................................... 73

7.3 Large-scale thermal development .............................................................................. 75

7.4 Small-scale thermal development with the experimental result comparison .............. 77

7.5 Melt pool size comparison .......................................................................................... 81

7.6 Microstructural predictions: precipitates, columnar grains and primary dendrite arm

spacing (PDAS) ...................................................................................................................... 82

7.6.1 Precipitate formation ........................................................................................... 82

7.6.2 Solidification map ................................................................................................ 84

7.6.3 Primary dendrite arm spacing (PDAS) ................................................................ 85

7.7 Conclusions ................................................................................................................ 89

x

8. Multiscale modeling of the residual stress development with the experimental

validation in a product-scale ................................................................................................... 90

8.1 Determination of an actual solid substrate temperature ............................................. 90

8.2 Numerical results of a residual stress development in Ti-6Al-4V block ...................... 92

8.3 Comparison between numerical prediction and residual stress measurement from X-

ray diffraction for the as-built stress in Ti-6Al-4V block .......................................................... 94

8.4 Conclusions ................................................................................................................ 96

9. Influence of scanning lengths on residual stress reduction ........................................ 97

9.1 Designing experiment ................................................................................................. 97

9.2 Processing map of scanning length and energy input ................................................ 98

9.3 Bending curvature from bridge structure .................................................................... 99

9.4 Residual stress from X-ray diffraction ....................................................................... 102

9.5 Conclusions .............................................................................................................. 104

10. Summarization ................................................................................................................ 105

11. Recommendation for future work ................................................................................. 107

Appendix A: Publications and Presentations ...................................................................... 108

References .............................................................................................................................. 109

xi

List of Tables

Table 1: Current commercial materials directly processed by AM, by AM process category from

Ref. [2] .................................................................................................................................. 2

Table 2: Typical processing parameters used for Inconel 718 product in EOS M280 from Ref. [8]

.............................................................................................................................................. 4

Table 3: Technical data of EOS M290 ...................................................................................... 19

Table 4: Room temperature thermal properties of Inconel 718 used in the Rosenthal equation.

............................................................................................................................................ 22

Table 5: Material properties of the Inconel 718 used for predicting the PDAS. ......................... 46

Table 6: Geometrical and processing parameters for Test A and B .......................................... 50

Table 7: Comparison the presence of lack of fusion between numerical and experimental results

(at constant scanning speed of 1357 mm/s) ....................................................................... 54

Table 8: Combination of the laser power and scanning velocity, which yields the identical energy

density ................................................................................................................................ 55

Table 9: Determination of the ball effects under various processing parameters ...................... 59

Table 10: Predicted average residual stress in a layer based on various initial temperature .... 67

Table 11. Predicted layer temperature increase from room temperature when the point of

interest is built ..................................................................................................................... 77

Table 12 Inconel 718 physical properties used for KF model [13] ............................................. 86

Table 13: Processing parameters of Ti-6Al-4V part for a residual stress prediction validation .. 91

Table 14: Elastic constants for the residual stress calculation from XRD [98] ........................... 95

Table 15: Processing parameters .............................................................................................. 98

xii

List of Figures

Figure 1: A typical layout of the SLM system .........................................................................................3

Figure 2: Exemplary metal parts fabricated by SLM/EBM: (a) aircraft bracket (approx. 150 mm

width), by Airbus; (b) acetibular cup for human hip implant (approx. 75 mm diameter), by

Arcam; (c) injection mold tool with internal cooling passages (approx. 200 mm width), by

Sodick. The parts are post-processed, including support removal, machining, and

polishing, in addition to the SLM/EBM process as the construction method. Pictures are

taken from Ref. [7] ..............................................................................................................................3

Figure 3: Plan view of a Rosenthal plot of the melt pool boundary, calculated for Inconel 718

with absorbed power of 142 W and V=0.96 m/s. The point heat source is at the intersection

between horizontal and vertical axes ..............................................................................................5

Figure 4: Simulation results of formation process of nonuniform single track (cross section view).

The black dashed curve represents the bottom boundary of melted region in the substrate.

The arrows are velocity vectors. Images are taken from Ref. [19]. ............................................6

Figure 5: Single track process map for the first layer of SS grade 316L (~25 µm) powder.

Powder layer thickness is 50 µm from Ref. [26] ............................................................................8

Figure 6: Panels (a) and (b) show the color map of the s11 and s22 stress components,

respectively, at a vertical midplane cutting through the simulation domain. Panels (c) and

(d) show the line plot of the same stress components as previously mentioned, along the

vertical line shown in panel (a). Images taken from Ref. [48] ...................................................11

Figure 7: Summarization for the efficient predictive model of part distortion and residual stress in

selective laser melting .Images taken from Ref. [54] ..................................................................12

Figure 8: Consideration of different scanning strategies. Images taken from Ref. [55] .................14

Figure 9: Max S11 and S22 stress comparison for all cases. Image taken from Ref. [55] ...........15

Figure 10: Overview of the computational scheme .............................................................................17

Figure 11: DMLS Machine EOS M290. (Source: EOS) ......................................................................18

Figure 12: A) Size distribution of EOS IN718 powder and B) Morphology of powder particles

from SEM ...........................................................................................................................................19

Figure 13: Process sequence of SLM process, which consists of preparing the CAD model,

assigning printing parameters and creating the sliced file, and uploading prepared file to

the AM machine for fabrication .......................................................................................................20

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Figure 14: A) Thermal boundary conditions: i. Laser heat input on the top surface ii. Heat losses

due to convection and radiation iii. Insulated walls iv. Constant temperature at bottom, B)

Bulk 3D geometry considered in the FE calculation. ..................................................................24

Figure 15: Thermophysical properties of Ti-6Al-4V at solid, liquid, and powder conditions: A)

Thermal conductivity, B) Density, and C) Specific heat .............................................................25

Figure 16: Absorptivity of solid, powder, and liquid of Ti-6Al-4V as a function of temperature,

where αs denotes the solid’s absorptivity, αeff the effective absorptivity of powder, αl the

liquid absorptivity, and Tm ................................................................................................................27

Figure 17: Thermophysical properties for Inconel 718 as functions of temperature from Ref. [14]:

A) density; B) specific heat; C) thermal conductivity; D) emissivity. .........................................28

Figure 18: Predicted melt pool and thermal distribution of Inconel 718 from Ref. [58]: A) cross-

sectional view (y–z) of the temperature contour (°C) and melt pool boundary (indicated by

the black line) from the FE model; B) longitudinal view (x–z) of the temperature contour

(°C) and melt pool boundary (indicated by the black line) from the FE model. Simulated

with a laser power of 200 W and scanning velocity of 960 mm/s .............................................29

Figure 19: Temperature history of Inconel 718 with 3 tracks before and after point of interest

from Ref. [8]. (Simulated at power of 200 W and scanning velocity of 960 mm/s) ................30

Figure 20: Summarization of thermal models used in the present study based on types of

heating input, important input parameters, important outcomes, and the physical time

involved with each model. ...............................................................................................................32

Figure 21: Mechanical properties of Ti-6Al-4V at solid and liquid conditions as a function of

temperature: (A) Yield strength, (B) Elastic modulus, and (C) Thermal expansion coefficient

[59] ......................................................................................................................................................33

Figure 22: Mechanical properties for Inconel 718 at solid and liquid conditions as a function of

temperature [74] ...............................................................................................................................34

Figure 23: Multiscale modeling scheme used in the present study to obtain the thermal and

mechanical responses in the SLM process ..................................................................................35

Figure 24: A) Cross-sectional view (y-z) of the temperature contour (oC) and melt pool boundary

(indicated by black line) from Finite Element model and B) Longitudinal view (x-z) of the

temperature contour (oC) and melt pool boundary (indicated by black line) from Finite

Element model. (Simulated with laser power of 200 W, scanning velocity of 960 mm/s, and

absorptivity of 0.5) ............................................................................................................................37

xiv

Figure 25: Plan view of a Rosenthal plot of the melt pool boundary, calculated for Inconel 718

with absorbed power of 142 W and V=0.96 m/s. The point heat source is at the intersection

between horizontal and vertical axes. ...........................................................................................37

Figure 26: Melt pool width comparison between the experimental results from Ref. [9] with

prediction from Rosenthal equation and Finite Element model. Shaded area shows the

range of predictions when varying the absorptivity from 0.3 to 0.87 while the dashed lines

are for the fitted absorptivity of the two approaches. ..................................................................39

Figure 27: Illustration of the melt pools in two layers, where regions with and without remelting

are identified. .....................................................................................................................................41

Figure 28: A) temperature as a function time from various locations within the melt pool, B)

temperature gradient as a function time, and C) temperature gradient as a function of

temperature during the cooling process (Simulated by the FE model with laser power of

200 W, scanning velocity of 960 mm/s, and absorptivity of 0.5) ...............................................42

Figure 29: Comparison of temperature gradient, cooling rate, and solidification rate from Finite

Element model and Rosenthal equation. Shaded area indicates sensitivity to absorptivity in

the range 0.3-0.87. Dashed lines indicate results from the fitted absorptivities of 0.4 and 0.5

for the Rosenthal equation and Finite Element simulation, respectively. ................................44

Figure 30: Comparison of solidification map from the Finite Element model and Rosenthal

equation. Results are from an absorptivity of 0.4 (Rosenthal equation) and 0.5 (Finite

Element model). ................................................................................................................................45

Figure 31: Comparison of PDAS prediction from the Finite Element model, Rosenthal equation,

and the experimental result from Ref. [62]. Shaded area indicates result’s sensitivity of

various absorptivity from 0.3-0.87. Dashed lines indicate results from the fitted

absorptivities of 0.4 and 0.5 for Rosenthal equation and Finite Element model, respectively

.............................................................................................................................................................47

Figure 32: A) A testing schematic to investigate a single scanning vector under different laser

energy and B) A rectangular sample printed with a same printing direction under different

laser energies ....................................................................................................................................49

Figure 33: Three-dimensional illustration of a temperature contour (oC) and melt pool

configuration at power of 200 W and scanning velocity of 1357 mm/s ....................................51

Figure 34: Top views of the surface contour of the single scanning lines at the fixed scanning

velocity of 1357 mm/s and various laser power: (A) 50 W, (B) 100 W, (C) 150 W, (D) 200 W

(E) 300 W, and (F) 400 W ...............................................................................................................52

xv

Figure 35: Comparison between the experimental fused powder width and numerical melt pool

width ...................................................................................................................................................53

Figure 36: Illustration of melting depth: (Left) melt pool fully covers the powder layer, where lack

of fusion is not likely occurred, while (Right) the lack of fusion likely occurred due to

insufficient coverage of melt pool. .................................................................................................54

Figure 37: Melt pool width under various combination of laser powers and scanning velocities:

A) energy density = 36.9 J/mm3, B) energy density = 49.1 J/mm3, and C) energy density =

73.7 J/mm3 .........................................................................................................................................56

Figure 38: Cross section of a single track made under the scanning velocity of 1357 mm/s: A)

laser power = 200 W, B) laser power = 300 W, and C) laser power = 400 W ........................59

Figure 39: Processing window for the SLS process of Ti-6Al-4V ......................................................61

Figure 40: Two-dimensional model to study the residual stress development for multiple-

scanning tracks in a plane ...............................................................................................................64

Figure 41: Heating sequences from 1st – 5th strip and the temperature distribution during heating

and cooling (recorded at a middle of strip) ...................................................................................65

Figure 42: A) Residual stress (Pa), σxx, after cooling, and B) Residual stress (Pa), σzz, after

cooling ................................................................................................................................................66

Figure 43: Comparison of the average von Mises stress when the plane heating (Zhao et al.

[59]) and the multiple heating strips (region thermal model) are applied. ...............................67

Figure 44: Residual Stress evolution after 22 layers added. Base plates extended from y=0 to

y=1.97mm. .........................................................................................................................................69

Figure 45: A) Full part geometry, B) Aluminum stand and substrate, and C) Fabrication of the

first layer .............................................................................................................................................72

Figure 46: Laser Scanning Pattern of One Layer at Different Magnifications .................................72

Figure 47: Schematic showing details of the embedded thermocouple within the build plate. ....74

Figure 48: Schematic of the build showing positions of embedded thermocouples as dashed

circles, and points of interest within the sample design. ............................................................74

Figure 49: A) Three-dimensional temperature contour maps showing the thermal evolution

obtained from the numerical model; B and C) Temperatures at the points of interest. The

temperature is shown just before the next powder layer was deposited. ................................76

Figure 50: Points of interest with numbered labels. .............................................................................77

Figure 51: Numerical prediction of the thermal history at point 7 and 14: Left) Time reset to zero

at every new layer and Right) Time continuously increasing. ...................................................79

xvi

Figure 52: A) Transient temperature history comparison between numerical results and

measurement and B) Peak temperature history comparison between numerical results and

measurement including data uncertainty ......................................................................................80

Figure 53: A) Optical micrograph showing melt pools, B) Melt pool width comparison at points 7

and 8, C) Melt pool depth comparison at point 7 and 8. ............................................................81

Figure 54: A) adjusted thermal history consisting of 10 layers for points 8 and 12, B) adjusted

thermal history consisting of 10 layers for points 10 and 14, and C) thermal history of point

14 plotted on a CCT diagram for Inconel 718 from ref. [91]. .....................................................83

Figure 55: Temperature gradient (G) and solidification rate (R) at various depths in the melt pool

for (A) points 8 and 10 and B) points 12 and 14. C) and D) show optical micrographs at

lower and higher magnification, in the vicinity of point 7 ............................................................85

Figure 56: Illustration of melt pools in two layers, where regions with and without remelting are

identified. ............................................................................................................................................86

Figure 57: Calculated temperature gradient (A), solidification rate (B) and PDAS (C) at various

depths within the melt pool for points 8, 10, 12, and 14 .............................................................87

Figure 58: PDAS measurements at points 7 and 8 .............................................................................87

Figure 59: Comparison of measured and calculated dendrite spacing, showing slight over-

prediction. ..........................................................................................................................................88

Figure 60: A) three-dimensional model and mesh used for thermal and mechanical simulation;

B) the numerical prediction of Tsub during the build .....................................................................91

Figure 61: A) temperature profile of 20 scanning tracks with L of 5 mm; B) temperature profile of

20 scanning tracks with L of 10 mm. Simulated with laser powder of 200W and scanning

velocity of 1357 mm/s. A black solid line indicates Televated ........................................................92

Figure 62: A) development of σxx and σzz in Ti-6Al-4V block plotted at 0.5, 2.5, 10, 30, and 40

mm; B) three-dimensional von Mises stress of as-built condition; C) comparison of σxx between as-built condition as well as after detached from the substrate and cut in half; D)

three-dimensional von Mises stress after detached from the substrate and cut in half ........94

Figure 63: Comparison of the experimental and numerical results of the residual stress

transverse to the built direction at 7 different points along the height ......................................96

Figure 64: (A) Ti-6Al-4V blocks to be cut and subjected to the residual stress measurement, (B)

Ti-6Al-4V bridge structures to be cut and measured bending angle, and (C) Schematic of

unidirectional scanning pattern having different scanning lengths as used in the present

study ...................................................................................................................................................97

xvii

Figure 65: Processing map showing the influence of tscanning and E on (A) Surface temperature

elevation and (B) Average residual stress in a deposited-layer ................................................99

Figure 66: (A) Bending angle after a substrate detachment of samples B1 and B5, and (B)

Numerical prediction of sample B1 showing von Mises stress and deformation with 2x

magnification ...................................................................................................................................101

Figure 67: Bending curvature comparison between measurement and numerical prediction. ...102

Figure 68: Residual stress measurement comparison between measurement and numerical

prediction. ........................................................................................................................................103

xviii

List of Symbols

Symbol Description of symbol

Rosenthal equation

Q Power absorbed by the part (W)

V Beam travel velocity (mm/s)

T Local temperature

T0 Plate/preheating temperature

ξ Distance from the beam position along the travel direction

r Radial distance from the beam position

ρ Density (kg/m3)

Cp Specific heat capacity (J/kg/K)

k Thermal conductivity (W/m/K)

α Thermal diffusivity, given by k/ρ/Cp W Melt pool width (µm)

Gaussian power distribution

P Power input (W)

q Heat flux distribution (W/m2)

r0 Laser radius (µm)

x0 Center of laser on x-axis

y0 Center of laser on y-axis

λ Absorptivity

Thermophysical properties of materials

AH Surface area of voids

Cp, sensible Sensible heat (kJ/kg/K)

Cp, modified Modified Cp for solid-liquid transition (kJ/kg/K)

hrad Radiative heat transfer coefficient (W/m2/K)

ks Thermal conductivity of solid (W/m/K)

keff Thermal conductivity of powder (W/m/K)

Lheat Latent heat of fusion (kJ/kg)

xix

Tm Melting temperature (K)

∆Tm Melting range (K)

ε Emissivity

εeff Emissivity of powder

εH Emissivity of voids

εl Emissivity of liquid

εs Emissivity of solid

ρeff Density of powder (kg/m3)

ρs Density of solid (kg/m3)

σs Stefan–Boltzmann constant (W/m2/K4)

φ Void fraction

Thermal model

q’’avg Average heat flux (W/m2)

H Hatch spacing (µm)

l Layer thickness (µm)

E Energy density (J/mm)

Tactual Actual substrate temperature

Tsub Solid substrate temperature

Televated Surface elevated temperature

tlocal Laser exposure time on a local spot (s)

tscanning Time spending on scanning a single track (s)

τ Heat diffusion time with a distance of a layer thickness (s)

Mechanical model

D Stiffness matrix (Pa)

σ Stress tensor (Pa)

εtotal Total strain

εp Plastic strain

εth Thermal strain

εe Elastic strain

Tref Reference temperature for a stress-strain free condition

xx

Solidification model

δ Primary dendritic arm spacing (µm)

DT0 Solidification interval (K)

D Liquid diffusivity (m2/s)

k0 Partition coefficient

G Gibbs–Thomson coefficient (m K)

1

1. Introduction

1.1 Selective laser melting

The additive manufacturing (AM) process refers to a technique, which forms a three-

dimensional object layer-by-layer. AM process has several advantages over the conventional

manufacturing. (i) It is able to fabricate a complex geometry, which is sometimes not possible to

be produced by the subtractive manufacturing process. (ii) This technology involves fewer

processing steps such that the product can be directly printed from a three-dimensional model.

(iii) The AM is cost-effective, especially when the only or small production volume is needed. (iv)

The multi-component parts, which previously need to be manufactured separately, are able to

be produced as an integrated product [1]. A very wide-range of materials and manufacturing

techniques are available nowadays. Bourell et al. summarized currently available materials with

the associated AM process as shown in Table 1 [2]. The present study only interests in the

powder bed fusion process, which is interchangeably knwon as the selective laser melting

(SLM) or direct metal laser sintering (DMLS) process.

The schematic of the SLM process is shown in Figure 1. Powders are melted by a laser spot,

travelling over a powder layer according to pre-defined scanning paths, where molten powders

are solidified and bonded. Additionally, after finishing a scanning of a current layer, the building

platform is moving down while the re-coating blade deposits powder on a previously built layer.

These steps repeat until a complete component is successfully constructed. The use of the AM

has experienced dramatic growth, especially in aerospace industry, medical implants and

devices, as well as art industries because these industries often require the customized product

rather than the mass production. The example of AM product could be seen in Figure 2. To

date, many metals are available in the market including aluminum, nickel alloy, stainless steel,

and titanium alloy.

The AM process allows users to have a control over many processing parameters, where

altering each parameter could affect thermal behavior, solidification behavior, microstructure,

mechanical properties, surface roughness, as-built density, fatigue life, residual stress, and

distortion [3-6]. Primary parameters in the AM are laser power, scanning velocity, hatch spacing,

scanning stripe width, layer thickness, laser diameter, spot size, and recoating time between

2

layers. Typical parameters used to build Inconel 718 product in the EOS M280 machine is

shown in Table 2.

In successive sections, the previous research on influence of processing parameters on the

defect generation, and residual stress formation as well as the utilization of numerical models to

reveal the physical characteristics in the AM process are reviewed.

Table 1: Current commercial materials directly processed by AM, by AM process category from

Ref. [2]

3

Figure 1: A typical layout of the SLM system

Figure 2: Exemplary metal parts fabricated by SLM/EBM: (a) aircraft bracket (approx. 150 mm

width), by Airbus; (b) acetibular cup for human hip implant (approx. 75 mm diameter), by Arcam;

(c) injection mold tool with internal cooling passages (approx. 200 mm width), by Sodick. The

parts are post-processed, including support removal, machining, and polishing, in addition to the

SLM/EBM process as the construction method. Pictures are taken from Ref. [7]

4

Table 2: Typical processing parameters used for Inconel 718 product in EOS M280 from Ref. [8]

Parameter Values Parameter Values

Laser power 285 W Laser diameter 75 μm

Laser speed 960 mm/s Laser type Ytterbium fiber

Hatch spacing 0.11 mm Wavelength 1070 nm

Scanning stripe width 10 mm Spot Size 70 to 80 μm

Layer thickness 40 μm Focal Length 410 mm

Recoater material High speed steel Time for one layer 40-60 s

Time between layers 13 s

1.2 Melt pool modeling in the SLM process

As the quality of final products strongly depends on the choice of processing parameters,

researchers have concentrated on understanding the relationships among part quality,

microstructure, mechanical properties and processing conditions. Sadowski et al. determined

the melt pool shape and size as functions of several processing parameters. This allowed

proper operating conditions to be identified that reliably gave fully fused scanning tracks [9].

Wang et al. experimentally studied Inconel 718 parts made by AM and found that

microstructures and mechanical properties seemed to be independent of part height [10].

Moreover, columnar microstructures were observed throughout the part while the quasi-static

mechanical properties of AM parts are often comparable or superior to those made by

conventional methods.

Even though experimental studies are necessary to characterize AM parts, they are time-

consuming and limited to the chosen technique used. Mathematical methods are a

supplemental route to explore and understand the fundamental behavior during the fabrication

process. Two major approaches for obtaining results are analytical and numerical solutions. The

analytical model of Rosenthal can give a quick estimation (in minutes) of the thermal

characteristics of powder-bed fusion [11], where Figure 3 displays the melt pool formation

obtained from the Rosenthal’s equation. However, the derivation of Rosenthal’s equation relies

on several simplifying assumptions, which raises concerns about its accuracy and generality.

5

Figure 3: Plan view of a Rosenthal plot of the melt pool boundary, calculated for Inconel 718

with absorbed power of 142 W and V=0.96 m/s. The point heat source is at the intersection

between horizontal and vertical axes

On the other hand, numerical models often have fewer assumptions than those of the analytical

solution, which make them more realistic. However, the required calculation time is often

drastically longer than that of the former approach. For example, models developed in our

research group usually take 4-6 hours with Intel Xenon Processor E5 2.8 GHz with 4 cores to

complete a thermal calculation over a 2 mm scanning length (equivalent to approximately 1 ms

in a physical time) with mesh numbers around 80,000. Nevertheless, despite these differences,

both calculation approaches have been broadly used. Tang et al. applied the Rosenthal

equation along with their proposed criterion to identify conditions that would cause lack of fusion

condition in several alloy systems [12]. Their analysis accurately estimates the lack-of fusion

porosity of as-built parts under various fabricating conditions.

In addition, with the analytical solution and existing correlations, Liang et al. developed a

process map that can predict the primary dendritic arm spacing in Inconel 718. Results were

validated with experiments [13]. Romano et al. used the Finite Element method to calculate

melting and solidification in the SLM process [14]. Simulation results from Romano et al. tended

to over-estimate the melt pool width, especially at high power input. Therefore, a correction

factor for the effective absorptivity has been proposed to be incorporated with numerical results

such that the estimation is in better agreement with measurements. It should be noted that while

previous FE studies often consider the powder as continuum material [14-17], recent studies by

Yan et al. have considered the presence of powder particles in the computational domain. The

multi-scale modeling techniques are introduced to enable the comprehensive examination from

the powder-scale to the layer-scale [18, 19]. Consequently, Yan’s models represent realistic

6

simulation conditions and can capture complicated physical behaviors such as the interaction

between powder particles, as well as the balling effects and the non-uniformity of a scanning

track induced by the surface tension as seen in Figure 4. However, such models have high

computational cost (reported 140 hours for 4 ms physical time with an Intel Core i7-2600 CPU).

Previous studies have shown that both analytical and numerical techniques can be used to

describe the AM process and give satisfactory results. Yet, these two techniques are often

studied separately, and a comprehensive comparison between the two approaches is still

lacking.

Figure 4: Simulation results of formation process of nonuniform single track (cross section view).

The black dashed curve represents the bottom boundary of melted region in the substrate. The

arrows are velocity vectors. Images are taken from Ref. [19].

1.3 Defects in SLM products and processing window

Despite the potential of AM technology to revolutionize the AM industry, additive manufacturing

has encountered with the problem of part defects due to the improper operating conditions in

production. They can lead to poor mechanical properties, poor surface finish or even cracking in

the products. Thus, many researchers have attempted to understand the influence of operating

parameters on the generation of defects.

7

As many factors can influence the outcome of SLM process, Nelson has introduced the use of

the energy density or the Andrew number, which considers the deposited laser energy per unit

area [20]. The proposed parameter was designated as a single indicator to describe the

processing outcome. Thijs et al. carried out a study performing experiments of various energy

densities and scanning strategies to understand the microstructure of the Ti-6Al-4V products

[21]. They found that parts with high porosity are resulted from the insufficient energy density.

Song et al. further emphasized the importance of processing parameters by showing that they

strongly influence the microstructure, roughness, densification, and hardness of products [22].

To provide the comprehensive understanding of the interplay between processing parameters,

the processing window has been established. Beuth and Klingbeil proposed the process map,

which can be used to predict the melt pool size, thermal gradient, and maximum residual stress

based on various processing parameters [23]. Gong et al. experimentally studied the defect

generation in Ti-6Al-4V powder bed processes with samples made under various energy

densities [24]. The study then introduced the processing window, in which proper operating

condition will result in parts with high solidity. In addition to Ti-6Al4V, the processing windows for

other common materials have been developed by various researcher. Dewidar et al. [25] and

Yadroitsev et al. [26] established the processing map for the stainless steel while Olakanmi

created several processing windows accounted for the aluminum alloy with various

compositions [27]. Figure 5 shows the example of the processing window taken from Yadroitsev

et al. [26]. Axes represent laser power and scanning speed. The process map indicates the

desirable operating area, where the fully continuous scanning tracks could be achieved while

the undesirable area will show the opposite.

8

Figure 5: Single track process map for the first layer of SS grade 316L (~25 µm) powder.

Powder layer thickness is 50 µm from Ref. [26]

However, it is also found that the notion of the energy density is not strongly emphasized in the

interpretation and the creation of the processing windows even though it can simplify the

consideration of multiple parameters. Furthermore, the applicability of the process map is limited

to specific setting such as the layer thickness and the laser diameter. If one of the two

parameters changes, the interpretation of the processing map could be less reliable. Also, since

most of the operational guidelines are empirically determined from experiments, the

construction of the guideline can be time-consuming, labor-intensive, and financially demanding.

On the other hand, various numerical modeling has been performed to investigate the thermal

history as well as the melting and solidification characteristics of the AM process. Zhao et al.

developed the three-dimensional finite element model to predict the thermal history and melt

pool configuration by comparing results between uniform and spherical laser heating

distributions [15]. The model is able to show the temperature evolution during the melting and

solidification. However, the numerical validation with an experimental result is not presented in

the study. Fu and Guo carried out numerical analysis to study the thermal characteristics of the

selective laser melting in multiple layers, in which the numerical simulation gives the reasonable

9

melt pool size compared with experiments [28]. Dong et al. developed a model to study the

effects of the scanning velocity, laser power, pre-heating temperature, and a laser diameter on

the maximum temperature and average density of a sintered product [29]. Even though the

model can predict the average density reasonably well, it did not describe the likelihood of

having incomplete and excessive melting from poor operating conditions. Loh et al. developed

the finite element model, which is used to examine the volume shrinkage and the material

evaporation [30]. The parametric study is performed to show the influence of the processing

parameters on the melt penetration, melt pool width, and the rate of the temperature change in

the material. Li and Gu also perform the similar study but further emphasize the parametric

effects on the cooling rate and the temperature gradient [31]. Despite capability of

aforementioned research in depicting the influence of processing parameters on a thermal

development and melt pool dimension in SLM process, previously developed models have not

made a strong association of how they could be used to determine suitable processing

parameters in reality.

1.4 Residual stress in SLM products

The presence of large thermal gradients during the metal deposition process results in

undesirable residual stress and product distortion. Improving a build plan could reduce distortion

and residual stresses; however, it is usually not clear. It is also difficult to record the fast-

changing temperature history under the laser point via experimental method. As of today, it

usually takes an average of three trials to make a good reliable product. To optimize the build

plan without the expensive trial-and-error iterations, and to better understand the heat transfer in

SLM processes, thorough experimental studies and accurate predictive models are needed.

Experimental studies were conducted to gain the better understanding of the residual stress

mechanism in AM products. Casavola et al. fabricated samples with various thicknesses from

the AISI 18 Maraging steel using the SLM process [32]. The hole drilling method with a strain

gauge was used to obtain the residual stress values along the depth. The high tensile stress

was observed near the top free surface, and the stress magnitude reduced toward inner layers,

where the stress reduction was found to be more pronounced for thicker specimens. Sochalski-

Kolbus et al. used the neutron diffraction to compare the residual stress from a simple block

made by the electron beaming melting (EBM) and direct metal sintering (DMLS) process [33].

10

The part produced by EBM showed much lower residual stress than one made by DMLS,

mainly owing to the preheating step in EBM. Mishurova et al. experimentally identified the use of

higher laser energy on lowering the residual stress [34]. They made the bridge-like structures

from Ti-6Al-4V with the energy density varied from 53 – 291 J/mm3. While the laser power was

fixed, the scanning velocity was adjusted to achieve various energy densities. The study found

that the obvious increase in bending angle after the cut was noticed when the energy density

reduced from 116.7 to 53 J/mm3. The primary reason was that the slower scanning speed

induced the larger volume affected by the irradiated energy, thus, the thermal gradients in the

specimen decreased. While experiments enhanced the understanding of the residual stress

formation in the AM part, they are subjected to several shortcomings: i) most of the residual

stress measurements are destructive methods, ii) only the final residual stress state can be

measured, and measuring the residual stress development during the build poses a great

challenge, iii) the detachment of the base plate is usually required for the measurement, thus,

the residual stress relaxation is unavoidable, and iv) the residual stress measurement is

laboriously intensive and resource-consuming.

Therefore, the thermo-mechanical modeling is an alternative route to investigate the residual

stress development in AM products. Researchers have developed various analytical models

[35, 36] to describe the residual stress and distortion, while many other researchers have used

finite element analysis (FEA) to model the direct metal laser sintering process and study its

effects on the part [37-41]. For the application of titanium alloy, a number of early efforts have

been undertaken to develop its laser additive manufacturing processes [42-47]. A numerical

study of a residual stress formation in a single track was carried out by Vastola et al. [48].

Various laser powers and beam sizes were parametrically studied, where numerical results

revealed that the maximum von Mises stress in a single track was not truly affected by a choice

of power and beam size. Figure 6 displays the as-built residual stresses which are parallel and

transverse to a scanning vector as well as a residual stress formation throughput the depth.

Most of the previous research focused on the material property measurement, laser-powder

interaction and metallurgical research investigations. However, in-depth thermal history and

thermo-mechanical coupling and the physics effects of multi-layer deposition has still been

lacking.

11

Figure 6: Panels (a) and (b) show the color map of the s11 and s22 stress components,

respectively, at a vertical midplane cutting through the simulation domain. Panels (c) and (d)

show the line plot of the same stress components as previously mentioned, along the vertical

line shown in panel (a). Images taken from Ref. [48]

1.5 Product-scale simulation

Despite its merit, these numerical models described in Section 1.4 were particularly designed to

address a residual stress formation in a single layer, or at most multiple layers. Using these

models to calculate the entire product behavior can be computationally excessive or even

impossible as the time and length scales are too distinctive (for example, one minute cycle in a

single layer vs. tens of hours in the actual production). Consequently, the product-scale part is

needed to cope with this computational difficulty.

Pan computing, which was later acquired by Autodesk, has published numbers of impressive

studies on the product-scale simulation of the laser-based directed energy deposition (DED)

12

with the residual stress and distortion experimentally validated [49, 50]. However, the model

developed for the DED process cannot be applied to the powder-bed based process, which has

much smaller powder layer and laser diameter as well as the need to consider the influence of

scanning strategies [51]. Even though recent studies from Pan computing have addressed the

thermal and mechanical behaviors in the laser powder-bed process, the analysis was limited to

simple geometries with the maximum height of approximately 10 mm [51, 52]. To improve the

computational capability, Li et al. proposed the multiscale modeling approach by calculating the

local residual stress based on different scanning strategies. The pre-calculated stress is then

mapped on to 5x5 mm domain to obtain the macroscale part distortion [53]. Using the

“temperature-thread” was also another multiscale technique found in the study by Li et al. [54].

The general idea was that the microscale laser-heating model thoroughly calculated a complete

thermal cycle, in which the temperature history from the microscale was extended to the

mesoscale hatch model and the macroscale part model for the fast-thermal prediction.

Figure 7: Summarization for the efficient predictive model of part distortion and residual stress in

selective laser melting .Images taken from Ref. [54]

While previous numerical studies have laid a foundation for the product-scale prediction, the

present study has identified a supplementary route, which can further improve the

computational efficiency. Even though the scanning strategy has an influence on the residual

13

stress in a solidified layer, the solid substrate temperature (Tsub), which is the temperature of the

top solid layer under a powder layer before the laser moves to a point, is found to have more

direct impact on the residual stress as it directly affects the local temperature gradient and

cooling behavior. Therefore, if the solid substrate temperature under the powder layer can be

determined, the ‘residual stress database’ based on different solid substrate temperature can be

calculated and applied to predict the residual stress on the produced top layer and facilitate the

entire part’s stress prediction. The merit of this approach is to eliminate the need of handling

every individual scanning strategies. Additionally, the residual stress determination can be

performed with the stationary solver, which can significantly minimize the required

computational resources.

Consequently, this study aims to present the multiscale modeling approach enabling the

residual stress and distortion estimation of the entire product. The interactive use of the

numerical models at the various time and length scales is utilized for the efficient and accurate

prediction. The thermal calculation is firstly performed to determine the solid substrate

temperature under the powder layer. The ‘residual stress database’, which is separately built, is

used to indicate the residual stress in a fabricated layer based on the solid substrate

temperature. Accordingly, the residual stress and distortion in a product-scale are predicted and

validated by experimental results.

1.6 Controlling of residual stress from scanning patterns

Previous numerical and experiment studies attempted to control the residual stress magnitude

by using scanning patterns. Cheng et al. numerically studied the influence of eight different

scanning patterns on the residual stress formation in Inconel 718 [55], where all scanning

pattern considered was displayed in Figure 8. While different scanning patterns exhibit different

residual stress magnitude, the residual stress fluctuation from all the scanning strategies was

within 5% compared to the average value. Residual stress results from various scanning

patterns are shown in Figure 9.

14

Figure 8: Consideration of different scanning strategies. Images taken from Ref. [55]

15

Figure 9: Max S11 and S22 stress comparison for all cases. Image taken from Ref. [55]

Instead of considering scanning patterns, Parry et al. numerically studied the residual stress in

Ti-6Al-4V parts and identified approximately 25% residual stress reduction when the scanning

length reduced from 3 to 1 mm [56]. Similar experimental results were reported by Kruth et al.

where the bridge-structures were built, and bending-curvature after a base-plate’s detachment

yielded 13% reduction when the scanning length reduced from 10 to 2 mm [57]. According to

previous studies, even though the scanning pattern can influence the residual stress, the energy

input and scanning length seem to be more dominant, which is mainly because the energy input

and scanning length strongly affect the surface temperature elevation during the build. The

higher surface temperature leads to lower yield strength and smaller thermal strain, thus,

minimizes the residual stress. However, to date, the simultaneous and comprehensive

examination on the influence of the energy input and scanning length on the residual stress

magnitude is still lacking. Also, when addressing the influence of scanning strategy, most of the

previous published reports were merely either experimental or numerical studies.

Thereby, the purpose of this study was to numerically and experimentally explore a role of the

energy input and scanning length in the residual stress reduction. The numerical modeling

performed the thermal calculation to determine the surface temperature elevation associated

with choices of processing parameters. The residual stress of a deposited layer was predicted

according to a degree of surface temperature elevation. The processing guideline showing the

correlation between parameters of interest and residual stress magnitude is introduced. Several

16

samples are made under various choices of scanning length to validate the numerical prediction

through the comparison with distortion and residual stress measurement.

1.7 Objectives

The ultimate objective of my study is to understand the influence of processing parameters upon

critical aspects in the SLS product from the micro- to product-scale, including defect generation,

residual stress, distortion, and microstructure prediction. The investigation is largely based on

FE models, in which the present study emphasizes on the interaction among models with

different length and time scale to achieve the complete prediction’s capability as shown in

Figure 10. While the study is computational, sufficient experimental investigations are made to

validate simulation results. Lists of objectives are as the following:

1. Investigate advantages and disadvantages of using analytical and numerical approaches

for the melt pool formation, thermal history, microstructural prediction in SLM products.

(Published in Engineering [58])

2. Develop a numerical-based processing window to understand the effects of processing

parameters on defect generations including lack of fusion, excessive melting, and balling

effects. (Published in Journal of Material Processing Technology [16])

3. Study the residual stress initiation in SLM products both in single and multiple layers.

The influence of parameters such as pre-heating temperature and energy density is also

determined. (Published in Additive Manufacturing Journal [59])

4. Establish the multi-scale simulation to study the thermal, microstructural, and residual

stress development in a product during the build as well as predict the deformation of an

as-built product. (Published in Progress in Additive Manufacturing Journal [8])

5. Control the residual stress magnitude in the SLM products by optimizing scanning

lengths and energy inputs as well as build the complete process map regarding the

influence of scanning length and energy input on the surface temperature elevation and

as-built residual stress within a layer

17

Figure 10: Overview of the computational scheme

18

2. Experimental approach

All samples in the present study was made by either EOS M280 or EOS M290. The

specification of the machine is reported in Table 3. The fabrication was performed under the

room temperature of approximately 25 °C, where there was no heat-treatment performed after

the operation. The powder distribution and morphology of Inconel 718 are shown in Figure 12A

and Figure 12B. The average powder size was 32.2 μm. Prior to part building, the building

chamber was evacuated and subsequently back-filled with argon to ensure that the oxygen

concentration was less than 1000 ppm.

Figure 11: DMLS Machine EOS M290. (Source: EOS)

19

Table 3: Technical data of EOS M290

Building volume 250 mm x 250 mm x 325 mm (9.85 x 9.85 x 12.8 in)

Laser type Yb fibre laser; up to 400 W

Precision optics F-theta lens; high-speed scanner

Scanning speed up to 7.0 m/s (23 ft./sec)

Focus diameter 100 μm (0.004 in)

Power supply 32 A / 400 V

Power consumption max. 8,5 kW / average 2,4 kW / with platform heating up to 3,2

kW

Inert gas supply 7,000 hPa; 20 m³/h (102 psi; 706 ft³/h)

Figure 12: A) Size distribution of EOS IN718 powder and B) Morphology of powder particles

from SEM

20

The process sequence is summarized in Figure 13. Firstly, the 3D model is constructed in the

CAD software while the 3D model is sent to the Materialise Magics software for generating

support structures and creating sliced files. The sliced CAD models are then imported to the

EOSPRINT software to specify printing parameters. Afterward, sliced files with printing

parameters are uploaded to the EOS machine for the fabrication. A fabricating time usually

takes about 20 – 30 hours depending on sizes and numbers of products. The post processing

such as heat treatment and surface polishing could be performed to improve the parts’ quality.

Figure 13: Process sequence of SLM process, which consists of preparing the CAD model,

assigning printing parameters and creating the sliced file, and uploading prepared file to the AM

machine for fabrication

21

3. Analytical and numerical modeling

3.1 Analytical solution for thermal result

Originally, Rosenthal developed the analytical method to predict the temperature history in

fusion welding [11]. Because of similarities between fusion welding and selective laser melting,

Rosenthal’s equation is extended to predict the thermal characteristic in laser melting. The

Rosenthal equation owes its merit to its simplicity and wide applicability. Moreover, this single

formula can predict temperature history as a function of time, temperature gradient, cooling rate,

and solidification rate. The Rosenthal equation was derived based on the following

assumptions.

i) The thermophysical properties including thermal conductivity, density, and specific

heat, are temperature-independent. The latent heat due to phase change is not

included.

ii) The scanning speed and power input are constant, leading to a quasi-stationary

condition temperature distribution around the melt pool.

iii) The heat source is a point source.

iv) Heat losses from surface convection and radiation are not considered; convection in

the liquid pool is neglected. Therefore, the heat transfer is governed purely by

conduction.

To use the Rosenthal equation in a powder-bed process, an additional assumption is that the

deposition of powder has an insignificant influence on melt pool size. Montgomery et al. [60] and

Gong et al. [61] confirmed the validity of this assumption through their experiments under similar

processing conditions used in the present study. Single scanning tracks were made on a solid

substrate with and without the presence of powder; the melt pool areas were similar and the

melt pool widths in the no-powder cases were only slightly larger than those with powder, which

means that the difference in melt pool width is insignificant in the context of this work.

The resulting analytical solution, i.e., the Rosenthal equation, is as follows:

(1) T = T0 +

Q2π kr

exp −V r + ξ( )2α

⎡

⎣⎢

⎤

⎦⎥

22

where T is the temperature, Q is the absorbed power, T0 is the temperature at locations far

away from the top surface, k is the thermal conductivity, V is the scanning velocity, and a is the

thermal diffusivity. It should be noted that in Eq. (1), the laser moves along the x-axis, in which

the moving coordinate of x-Vt is replaced by x; r is the distance from the heat source, defined as

(x2 +y2+ z2)0.5. Since the Rosenthal equation does not account for any temperature-dependent

material properties, the properties at room temperature are used as shown in Table 4.

Table 4: Room temperature thermal properties of Inconel 718 used in the Rosenthal equation.

Properties Value

Thermal conductivity 11.4 (W/m K) [14]

Density 8220 (kg/m3) [14]

Heat capacity 435 (J/kg K) [14]

Absorptivity 0.3-0.87 [14, 60, 62, 63]

3.2 Numerical solution for thermal result

To model the complete thermal evolution in the AM, the multiscale modeling is utilized for the

analysis at both micro- and macro-scale and to improve the computational efficiency. The

overall computational scheme consisted of three models, i) point thermal model, ii) region

thermal model and iii) product thermal model, accounting for three different time and length

scales. The multi-scale thermal modeling involved substantial details of logistics and

approximation, the essence of thermal models is outlined as the following:

3.2.1 Governing equation and boundary condition

To evaluate the thermal behavior, the mass, momentum, and energy equations with the initial

and boundary conditions are supposed to be solved simultaneously. However, in the present

study, the natural convection in a melt pool and the Marangoni effect are assumed to be

insignificant because of the very small dimensions and the very short solidification time of melt

pool. Consequently, the heat transfer in the melt pool is dominated by the heat conduction and

phase change [64]. Therefore, the governing equations are reduced to only the energy equation

of conduction, which is:

23

( ) ( ) qTkTcdtd

p +Ñ×Ñ=r (2)

The parameters, ρ, cp, k, T, and q are the density, specific heat, thermal conductivity,

temperature, and volumetric heat source, respectively. Moreover, the power distribution of the

laser in the DMLS is assumed to be Gaussian distribution model [28, 29], which can be defined

as:

(3)

Where P is the laser power, r0 is the laser radius, and λ is the laser absorptivity of materials.

The entire computational domain is assumed to have an initial temperature of T0. The boundary

condition of the bottom surface is set to have a constant temperature of T0, which comes from

the observation that the substrate bottom temperature is mostly constant during the operation.

However, the top surface is subjected to heat losses from the convection and radiation. These

two boundary conditions could be combined by considering the radiative heat transfer

coefficient as Eq. (4).

(4)

Figure 14A summarizes the boundary conditions used for numerical modeling in the present

study while Figure 14B illustrates the three-dimensional computation domain.

q(x, y) = 2λPπr0

2

−2 x − x0( )2 + y − y0( )2( )r02

⎛

⎝⎜⎜

⎞

⎠⎟⎟

hrad = εσ s T +T02( ) T +T0( )

24

Figure 14: A) Thermal boundary conditions: i. Laser heat input on the top surface ii. Heat losses

due to convection and radiation iii. Insulated walls iv. Constant temperature at bottom, B) Bulk

3D geometry considered in the FE calculation.

3.2.2 Material thermophysical properties

The present study primarily considered Ti-6Al-4V and Inconel 718 as they are widely used in the

SLS process. The temperature-dependent thermal properties of two materials are reported

below.

• Ti-6Al-4V thermal properties

The AM process involves the rapid changes in temperature. The thermal conductivity, density,

and heat capacity at various temperatures are shown in Figure 15 [65]. Even though the

thermophysical properties of solid apparently change with the temperature, the thermophysical

properties of liquid are much less dependent upon temperature [17]. Moreover, in a powder

form, the thermal conductivity and density are functions of void fraction, where the correlation

proposed by Koh and Fortini [66] is used as shown in Eq. (5) and (6). The material properties of

powder are also shown in Figure 15. The jumps of data points in the figure are mainly due to

the change of material structures around 1000 °C. However, to improve the calculation

efficiency, the temperature dependent properties are simplified by the linear approximation.

25

Figure 15: Thermophysical properties of Ti-6Al-4V at solid, liquid, and powder conditions: A)

Thermal conductivity, B) Density, and C) Specific heat

21111

jj

+-

=S

eff

kk

(5)

jrr

-=1S

eff (6)

Additionally, a modified specific heat is used to define the presence of latent heat (Lheat) of 290

kJ/kg during the phase changing [67]. The modified specific heat is calculated over the melting

temperature range ∆Tm, which is about 100 K [68]. Therefore, the specific heat will exhibit a

rectangular block shape over the melting region, which can be described as:

(7)

The absorptivity and emissivity of materials depends on the temperature and the phase of the

material. Kwon et al. experimentally investigated the absorptivity of a solid titanium at a

wavelength of 1060 nm [69]. However, their work focused mainly on the solid phase. To cover a

broader temperature range, Boivineau et al. further addressed the absorptivity over the

temperature range from the liquidus temperature and higher [67]. In addition, a powder

absorptivity and emissivity can be obtained based on void fraction and solid properties. Sih and

mm

mmmm

mm

TTTforTTTTTfor

TTTfor

D+>D+

26

Barlow proposed a simple model to estimate the absorptivity of a powder’s surface, which

agrees closely with experimental results [70]. The effective emissivity is calculated based on the

combination of the hole and solid emissivity as shown in Eq. (8), while the overall area of holes

and the hole’s emissivity are obtained from Eq. (9) and (10), respectively. The void fraction of

0.52 is chosen in the present study [71]. Based on Eq. (8), (9), and (10), Figure 16 displays the

absorptivity value in three different conditions (solid, liquid, and powder), as a function of the

temperature.

( ) SHHHeff AA eee -+= 1 (8)

12908.1908.02

2

+-=

jjj

HA (9)

11082.31

1082.32

2

2

+úúû

ù

êêë

é÷÷ø

öççè

æ -+

úúû

ù

êêë

é÷÷ø

öççè

æ -+

=

jje

jje

e

S

S

H (10)

27

Figure 16: Absorptivity of solid, powder, and liquid of Ti-6Al-4V as a function of temperature,

where αs denotes the solid’s absorptivity, αeff the effective absorptivity of powder, αl the liquid

absorptivity, and Tm

• Inconel 718 thermal properties

The typical temperature-dependent thermal properties of Inconel 718 are shown in Figure 17. In

addition, the absorptivity of 0.5 is used as this value is reported for a laser with a wavelength of

1.06 μm, comparable with the machine of EOS GmbH, Germany used in the present study [60].

Temperature (°C)0 500 1000 1500 2000 2500

Abso

rptiv

ity

0

0.2

0.4

0.6

0.8

1,s,eff,lTm

28

Figure 17: Thermophysical properties for Inconel 718 as functions of temperature from Ref. [14]:

A) density; B) specific heat; C) thermal conductivity; D) emissivity.

3.3 Point thermal model

A point thermal model serves as the most realistic thermal model among the three models in the

present study, where it simulates the moving heat source with a carefully described heat input

distribution. Therefore, this model captures the accurate thermal history in a millisecond time

scale and is suitable to be used as the thermal input for the microstructural prediction. Our

previous studies have shown that the point thermal model can accurately predict the melt pool

dimension and microstructure according to calculated thermal behaviors such as temperature

gradient, cooling rate, and solidification rate [16, 58]. Figure 18 shows an example of numerical

results from the point thermal model of Inconel 718 [58]. The temperature and melt pool conto