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Dong-Hyun Cho

Mechanical Engineering

A Thesis Presented

By

to

The Department of

in partial fulfillment of the requirements

for the degree of

Master of Science

in the field of

Northeastern University

Boston, Massachusetts

August 2016

DETERMINING THE TEMPERATURE FIELD OF SELECTIVE LASER MELTING

PROCESS FOR DIFFERENT HEAT SOURCE PATHS

Mechanical and Industrial Engineering

ii

ABSTRACT

The temperature field of Selective Laser Melting (SLM) process for two different

patterns was investigated to find out which pattern is more effective to reduce the

temperature gradient. A three-dimensional finite difference simulation model has been

investigated with MATLAB to simulate the transient temperature field of powder bed,

Inconel 718, including phase change. The goals of this thesis are developing

mathematical formulations to generate moving heat source paths and comparing the

temperature gradients created by Hilbert and raster curves. Hilbert pattern tends to hover

over a local neighborhood longer than raster case. This phenomenon reduces local

temperature gradient, and consequently reduces thermal and residual stresses in a

solidified metal part. The obtained results show Hilbert pattern has less steep temperature

gradient than raster scan due to the relatively slow phase change pace caused by elbow-

shaped parts of the pattern.

SLM process is an additive manufacturing process to build full density metallic parts

whose properties are comparable to those of bulk materials. This technology is an

alternative method to conventional manufacturing methods such as die casting, since it

can fabricate intricate features and designs. SLM is promising technology, however,

using the intense laser causes deleterious effects such as high residual stress, plastic

deformation, warpage, and cracking. Several studies have been conducted to understand

the interaction between undesirable effects and input parameters. Comprehending

temperature distribution is one of major research focuses.

iii

ACKNOWLEDGEMENTS

I want to show my sincere gratitude to my advisor Professor Sagar Kamarthi for

providing me with the opportunity to work on this research project towards my graduate

degree and for giving me his support, motivation, patience and guidance during the whole

time of my research.

I also would like to show my thanks to Professor Gregory Kowalski for his valuable

advice in this project. I have learned more about this topic, engineering and research in

general, and myself during this experience than I ever thought possible.

At last, I want to thank my wife, my adorable daughter, and my parents for their

consistent supports and love. Most of all, I would like to thank JESUS who has called me

into fellowship in His grace and guided my life.

iv

NOMENCLATURE

Symbols Description Unit

Thermal conductivity of powder bed

Thermal conductivity of substrate

Density of solid phase

Density of liquid phase

Specific heat capacity of solid phase

Specific heat capacity of liquid phase

Velocity of the moving heat source

Incident laser power per laser spot area

Absorption coefficient

Extinction coefficient

s Path length of the incident laser

Total path length of the laser radiation

through the power bed

q Heat flux by laser radiation

Pseudo energy source term

Enthalpy change ( )

Mass fraction of the control volume of solid phase -

v

Mass fraction of the control volume of liquid phase -

Future solid mass fraction -

Present solid mass fraction -

Future liquid mass fraction -

Present liquid mass fraction -

Switch of phase change -

Switch of phase change activation -

Temperature of ambient air

Temperature of substrate

Melting point

Future temperature of i-node

Present temperature of i-node

Thickness of powder bed

Thickness of substrate

Heat transfer coefficient to the environment

Unit node volume (

Solid phase mass of i-node volume

Liquid phase mass of i-node volume

vi

TABLE OF CONTENTS

Chapter 1 Introduction………………………………………………………………..1

1.1 Additive manufacturing…………………………………………………...1

1.2 Process of Selective Laser Melting………………………………………..2

Chapter 2 Literature Review….………………………………………………………4

2.1 Balling effect………………………………………………………………4

2.2 Residual stress……………………………………………………………..4

2.3 Porosity……………………………………………………………………5

2.4 Surface roughness…………………………………………………………6

2.5 Research history of moving heat source………...………………………...7

2.6 Temperature distribution study in SLM…………………………………...9

Chapter 3 Research purpose…….…………………………………………………..12

Chapter 4 Problem setup…………………………………………………………….13

4.1 FEA model……………………………………………………………….13

4.2 Material property……………………………………………………..….14

4.3 Phase change problem …………………………………………………...15

4.4 Thermal modeling………………………………………………………..17

Chapter 5 Numerical simulation…………………………………………………….19

5.1 FDM formulation……………………………………………………..….19

vii

5.2 Geometry, initial, and boundary conditions…………………………...…21

5.2.1 Simulation modeling dimensions …………………………..……21

5.2.2 Mesh generation ………………………………………………....22

5.2.3 Initial conditions ………………………………………………...23

5.2.4 Boundary conditions …………………………………………….23

5.3 Patterns on mesh………………………………………………………....24

5.4 Simulation model and calculation procedure…………………………….25

5.4.1 Calculation procedure……………………………………………25

5.4.2 Occupied equations………………………………………………26

Chapter 6 Result and discussion…………………………………………………….28

6.1 Time to reach each step………………………………………………..…28

6.2 Temperature field of Raster scan pattern……………………………..….29

6.3 Temperature field of Hilbert pattern……………………………………..31

Chapter 7 Conclusions……………………………………………………………....33

REFERENCES…………………………………………………………………………..34

APPENDIX…………………………………………………………………………...….38

viii

LIST OF TABLES

Table 1: The methods of 3D printing……………………………………………….2

Table 2: Inconel 718 chemical composition………………………………………14

Table 3: Inconel 718 properties……………………………………………………15

Table 4: Modeling dimensions…………………………………………………….21

Table 5: Initial values……………………………………………………...………22

Table 6: Patterns of raster scan and Hilbert curves………………………………..23

Table 7: Required time to melting, fully melted, and fully solidified…………..…26

ix

LIST OF FIGURES

Figure 1: Selective laser melting process……………………………………………3

Figure 2: Residual stress mechanism………………………………………………..5

Figure 3: Formation of porosity a) high scanning speed, b) low laser power, c) hatch

distance…………………………………………………………………....6

Figure 4: Comparison of results of cooling time………………………………….…8

Figure 5: Variation of temperature with different beam diameter during metal laser

sintering……………………………………………………………………9

Figure 6: Proposed scanning sequence for producing large layer on powder bed…10

Figure 7: Variation of temperature at the center: (a) first layer, (b) second, (c) third,

and (d) fourth………………………………………………………….....10

Figure 8: Steady-state temperature variation with number of layers added………..10

Figure 9: Schematic of FEA system………………………………………..………13

Figure 10: Solidification of a liquid from a cold plane surface……………………...16

Figure 11: 3D and 2D view of unit calculation model……………………………....19

Figure 12: Laser beam spot on mesh……………………………………………...…21

Figure 13: Nodes numbering on mesh……………………………………………….22

x

Figure 14: Boundary conditions……………………………………………………..23

Figure 15: Temperature field of raster scan at final step (surface plot)…………......27

Figure 16: Temperature field of raster scan at final step (contour plot)……..………28

Figure 17: Temperature field of Hilbert pattern at final step (surface plot)................29

Figure 18: Temperature field of Hilbert pattern at final step (contour plot)...……....29

1

Chapter 1

Introduction

1.1 Additive manufacturing

Additive manufacturing (AM) was developed to make a prototype in the early 1980s. The

fundamental of AM technology is building an object by adding ultrathin layers of a

material layer by layer rather than subtracting a material, such as CNC milling and

turning. 3D printing technology, the industrial version of AM, is aiming at producing

functional parts that can be applicable to various fields which include aerospace,

automotive, biomedical industries and so on. AM reduces production cost because it

makes parts lighter, uses less material than subtractive technologies. Also, this process is

an effective method to produce complex shapes, whereas conventional technologies

would require welding or not possible to fabricate [1]. Comparing with die-casting or

injection molding, AM does not need a modification process such as revising molds.

Simply CAD data modification are needed. However, in worst case of the injection

molding, molds have to be changed totally for an exterior design modification. It is great

cost waste and time consuming work. AM also has many area of improvement such as

distortion, cracking, surface deterioration and so on. On the other hand, the conventional

technologies can handle the issues with relative ease such as controlling temperature of

molds or holding pressure at the end of process. The issues of SLM process will be

commented deeply, and one of the issues will be investigated in this paper. Many other

developed technologies of AM are introduced in Table 1.

2

Table 1. The methods of 3D printing [2]

Technologies Technical Features Materials

Fused deposition

modeling (FDM)

Safe, simple, and low cost. But

bad precision.

Thermoplastics, HDPE, Eutectic

metals, edible materials, RTV,

silicone, Porcelain, Metal clay

Laminated object

manufacturing (LOM)

Low cost, no chemical reaction in

the process. But narrow applicable

range.

Paper, metal foil, plastic film

Stereolithography (SLA) Optical fabrication, efficient. But

high cost.

Photopolymer

Direct metal laser

sintering (DMLS)

Efficient, low cost. But bad

precision for some little parts.

Almost any metal alloy

Electron-beam melting

(EBM)

High powder. But small

application scope in materials.

Titanium alloys

Selective laser sintering

(SLS)

No need support structures. But

pollute environment.

Thermoplastics, metal powders,

ceramic powder

Selective laser melting

(SLM)

High density, good mechanical

performance, good precision. But

balling phenomena, porosity, and

crack.

Titanium alloys, Co-Cr alloys,

Stainless Steel

1.2 Process of Selective Laser Melting

SLM is a typical powder-based AM process. SLM was developed in 1995 to

enhance the density of a metal product. SLM features fully melting of powder, thus near

full density parts can be produced. For example, the density of a formed titanium model

shows it is higher than 92 percent of a titanium block [3]. SLM process is not limited to

the model processing but for the practical application.

3

Figure. 1 shows the schematic of SLM process. The process starts by dividing the product

data into layers. The bottom layer of the divided data is the first layer of the process. The

fabrication piston moves in the vertical axis as much as one layer, usually 20-100

micrometers thick, to distribute the metal powder. The powder delivery system and roller

evenly distribute the metal powder onto a substrate plate, and then the scanner system

melts the layer selectively. In the next step, the fabrication piston goes down to fill one

layer of a metal powder, and the added layer is fused selectively by the scan laser beam.

Each layer is added as previously stated until the part is complete. Selective laser melting

process is conducted in the chamber of an inert gas to prevent oxidation.

Figure 1. Selective laser melting process

In this working process, the energy of the laser beam transfers from the top surface to the

bottom surface through various physical phenomena such as heat transfer, phase change,

fluid flow within the molten pool, and chemical reactions. These physical phenomena

cause some problems such as balling effect, porosity, surface deterioration, crack, and

residual stress. Studies in the following have been investigated about the issues.

4

Chapter 2

Literature Review

2.1 Balling effect

Balling phenomenon is a result of powder particles regrouping under the conditions of

inhomogeneous heating. During laser processing, the particles absorb the radiation at the

outer surface of the powder bed. Laser intensity penetrated through pores of powder

decreases, and it results irregular distribution of the heat on the powder layer. The

irregular sintering or melting of the particles locates on different levels. As a result, the

upper layer is formed denser than lower layers. The upper part of the melt fraction

increases in the process and leads to a spherical droplet [4]. Also, a molten pool with

length to diameter ratio greater than π will cause a balling effect. Higher scanning speed

leads to an increasing length to diameter ratio in a molten pool [5]. Balling phenomena

can be controlled by using compact powders and optimizing scanning speed because the

way reduces steep temperature gradient and the molten pool size [6-7].

2.2 Residual stress

Residual stress remains in the results of SLM process by the effect of a temperature

gradient. A temperature gradient mechanism results from large thermal gradients that

occur around the laser spot. The steep temperature gradient has developed owing to the

rapid heating of the upper surface by the laser beam and the rather slow heat conduction.

5

Simultaneously, the material strength is reduced because the expansion of the heated top

layer is restricted by the underlying material which has elastic compressive strains. When

the material’s yield strength is reached, the top layer will be plastically compressed. In

absence of mechanical constraints, a counter bending away from the laser beam would be

perceived. During cooling process, the plastically compressed upper layers start shrinking

and a bending angle towards the laser beam develops. Thus tensile stress is introduced in

the added top layer and compressive stress is in the below as shown in Figure 2 [8]. This

mechanism is a reason of distortion or cracking in parts.

Figure 2. Residual stress mechanism

One of solutions of this problem is preheating of the base plate and the wall uniformly.

400 ° C preheating can reduce the residual stress approximately 40% (from the yield

stress). Since increased molten pool size by preheating can be eliminated by controlling

of laser power or velocity, and preheating does not increase the molten pool length

considerably [9].

2.3 Porosity

Less than 80 W laser power produced pores which resulted into an approximate 50%

drop in material strength from the stainless steel 316L bulk values of uniaxial loading

6

[10]. Pore’s shape or formation is influenced by parameters such as hatch distance,

scanning speed, and laser power. If the hatch distance is too large, the pores will appear

waffle shape. High scanning speed and low laser power show irregular pores. Whereas

pores can be reduced with low scanning speed, high laser power, and narrow pattern [24].

Figure 3. Formation of porosity a) high scanning speed, b) low laser power, c) hatch distance

2.4 Surface roughness

The surface roughness has a possibility to initiate cracks. It is critical to commercial

product so some SLM machines require post processing such as surface machining,

polishing, and shot peening. Rippling effect that occurs due to surface tension forces

affects a melt pool at the top surface. It is because a surface temperature difference

between the laser beam and the solidifying zone. Mumtaz and Hopkinson investigated

that high peak laser power can reduce top and side surface roughness, since recoil

pressure flattens out the melt pool and increases wettability which is good to reduce

balling formation [11].

7

2.5 Research history of moving heat source

Above issue study shows how important the temperature distribution is in SLM process.

For this, many researchers have conducted the temperature distribution analysis. The

beginning was the research about moving heat source and its application to metal

treatments by Rosenthal. He developed analytical solutions which were estimating the

time and cooling rate in welding problems, and showed these solutions were in good

agreement with the experimental results. Rosenthal’s research has become the foundation

for the subsequent studies such as the analysis of weld pools in welding processes, or

SLS and SLM process [12]. Eagar and Tsai, the pioneer of two-dimensional heat source,

found the geometry of the melt pool produced by a Gaussian heat source. They focused

on the functional relationship between parameters of process and material, since the

theory could calculate the changes of the weld geometry [13]. The significance of this

research is both material and process parameters are important in the resulting

temperature field. Goldak et al. introduced a double ellipsoidal moving heat source model

to handle for both shallow and deep penetration welding process. The model was the first

three-dimensional heat source. They calculated the cooling time (800 to 500°C) for

sallow and deep penetration welds using the finite element method double ellipsoid

model, conventional Rosenthal analytical solution, and experiment. As shown in Figure 4,

the finite element method double ellipsoid cooling time values were much closer to the

experimental values than Rosenthal’s [14].

8

Figure 4. Comparison of results of cooling time [14]

Kolossov et al. developed a three-dimensional heat transfer model of the SLS process. It

was the non-linear model by adopting the parameters of the thermal conductivity and the

specific heat as a temperature dependent variable for the phase change. They simulated

the three-dimensional finite analysis based on the continuous media theory. The result of

the model indicated that the evolution of thermal conductivity determines the behavior

and development of the temperature field, and it was confirmed by means of

experimental test on a titanium powder bed [15]. Patil and Yadava developed a thermal

model to calculate the temperature distribution within a single titanium layer during metal

laser sintering. The results computed by the model showed the effect of process

parameters such as beam diameter, laser on-time, laser off-time, and hatch spacing. The

temperature increased with increased in beam diameter, but the temperature decreased

after 0.15mm as shown in Figure 5 [16].

9

Figure 5. Variation of temperature with different beam diameter during metal laser sintering [16]

Dong et al. developed a transient three-dimensional finite element model for taking into

account the thermal and sintering phenomena in selective laser sintering process. Their

results demonstrated about the relationships of maximum temperature versus laser speed,

maximum temperature versus laser power, and temperature versus powder bed depth for

different preheating and laser power [17].

2.6 Temperature distribution study in SLM

Following papers present the studies of SLM temperature distribution. Matsumoto et al.

calculated the temperature distribution and stress of SLM process within a single layer.

They obtained a solution to prevent the distortion of the solid layer by designing shorten

scanning track as shown in Figure 6 [18]. However, they did not suggest the effective

range of scanning track length, and consider the phase change.

10

Figure 6. Proposed scanning sequence for producing large layer on powder bed [18]

Roberts et al. used an innovative three-dimensional finite element method simulation

technique known as the element birth and death for predicting the transient temperature

distribution for multiple layers. The technique had been employed by Gan et al [23].

Briefly, all the elements are activated visually, however, they do not add to the overall

stiffness of the matrix. They obtained the result of temperature variation of layers and

steady-state temperature variation of layers as shown in Figure7 and 8 [19].

Figure 7. Variation of temperature at the center: (a) first layer, (b) second, (c) third, and (d) fourth [19]

Figure 8. Steady-state temperature variation with number of layers added [19]

11

Song et al. calculated the temperature distribution of the melted Ti6Al4V powder for

considering the density, porosity, and hardness in SLM process. It was controlled by the

processing parameters without a post-process. They simulated the temperature

distribution based on a three-dimensional finite element model and then conducted

experiments. The result showed the condition of laser power of 110W and scan rate of

0.2 m/s fabricated the Ti6Al4V parts with lower porosity and higher density. With the

increase in scan rate, the structure exhibited more pores and less density [20]. Hussein et

al. simulated the temperature and stress fields in single 316L stainless steel layers without

support in SLM process. They developed a three-dimensional transient finite element

model for predicting the temperature and stress fields. The simulation results provided

that support structures can help withstand residual stress forces and dissipate heat during

part building because the cooling rates of the layer on the solid substrate were higher than

the layer on the powder bed [21]. Criales and Ozel analyzed the effect of process

parameters such as average power and powder material’s density for the prediction of

temperature profile in SLM. They also calculated the temperature profile along the z-

direction, depth, for understanding of transient behavior of the temperature rise. The

purpose of their research was to improve the efficiency of SLM process by selecting best

process parameters [22].

The above research conducted computer simulations, mainly finite element analysis, to

understand the behavior of SLM or SLS process because it is effective to control various

parameters such as properties of powder, laser power, speed and paths, and initial and

boundary conditions. Also, the simulation has a benefit to reduce the experiment cost.

12

Chapter 3

Research purpose

SLM is an innovative technology with lots of benefits such as short lead time, mass

customization, reduced parts count, complex shapes, and less material waste. However

there are several technological issues mentioned above due to the local heating and

melting, rapid cooling and re-heating during process. Several efforts have been made to

overcome and understand these issues, but technical challenges still remain.

The aim of this paper is investigating the reduction of the temperature gradient with

comparing two different patterns, Hilbert space and raster scan. Hilbert pattern is a fractal

space filling curve which preserves locality and has an effect of significant reduction in

temporal thermal gradient due to a localized revisiting of heated points [25]. Raster scan

strategy is reported to significantly decrease macroscopic warpage and global residual

stresses, and several research facilities have adopted this strategy [26]. Comparing these

two patterns, the more effective pattern will be determined to reduce the temperature

gradient so eventually thermal stress or residual stress will be decreased.

13

Chapter 4

Problem setup

4.1 FEA model

Figure 9. Schematic of FEA system

This model, Figure 9, is one layer of metal powder which is consisted of Inconel 718 on a

steel substrate. It is assumed that the laser spot is stationary and the powered metal

platform is moving. The laser is applied from the top of the nodal volume, and the laser

radiation absorbed by the powder material is an internal heat generation source term. The

energy balance of this model on an incremental control volume is used to determine the

temperature, temperature gradient, melting, solidification, and the heat affected zone.

Interparticle radiation or convection are not considered, as previous studies have shown

them to be negligible. Convection occurs between upper the X-Y plane of the powder bed

and ambient air, and conduction occurs between lower the X-Y plane of the powder bed

14

and the substrate. The X-Z plane at this system’s boundary is considered to be adiabatic,

but a node volume with a face on the Y-Z plane on the system’s boundary has convection.

Arrays will be populated at the start of the simulation and updated at each time step.

4.2 Material property

Inconel 718 is chosen as the material of SLM process. It was developed by Wiggin

Alloys in England [25]. Inconel 718 is highly utilized Nickel-base superalloys [26].

Inconel alloys are well suited for extreme environments subjected to high pressure and

kinetic energy because it has characteristics such as high-strength, corrosion-, oxidation-,

and fatigue-resistance, as well as the wide temperature range of use at -253°C to 704°C

[27]. Also, the alloys have excellent weldability when compared to the nickel-base

superalloys hardened by aluminum and titanium. For this reason, this austenitic

superalloy has been used for jet engine and high-speed airframe parts such as wheels,

buckets, spacers, and high temperature bolts and fasteners. Typical compositions weight

percent are shown in Table 2.

Table 2. Inconel 718 chemical composition [27]

Composition Weight percent

Nickel (Ni) 50.00-55.00

Chromium (Cr) 17.00-21.00

Iron (Fe) Balance

Niobium (plus Tantalum) (Nb) 4.75-5.50

Molybdenum (Mo) 2.80-3.30

Titanium (Ti) 0.65-1.15

Aluminum (Al) 0.20-0.80

Cobalt (Co) 1.00 max.

15

Carbon (C) 0.08 max.

Manganese (Mn) 0.35 max.

Silicon (Si) 0.35 max.

Phosphorus (P) 0.015 max.

Sulfur (S) 0.015 max.

Boron (B) 0.006 max.

Copper (Cu) 0.30 max.

For FEA simulation, properties of Inconel 718 have been used as shown in Table 3.

Table 3. Inconel 718 properties

Properties Physical constants Correction factor

(powder)

Density (Solid powder) 8,192 0.9

Density (Liquid phase) 7,585 -

Specific heat capacity (Solid powder) 435 J ) 0.9

Specific heat capacity (Liquid phase) 778 J ) -

Thermal conductivity 11.4 W ) 0.9

Enthalpy of melting 227 kJ -

Melting point 1609 K -

4.3 Phase change problem

SLM process repeats phase change such as melting or solidification of a material in a

short time. Therefore, understanding phase change problem is an essential part in

simulation of temperature field. Phase change problem is also referred to as moving

boundary problem. Such problem is inherently transient, since the location of the

interface between the two phases changes with time as latent heat is absorbed or released

at the interface. Heat conduction problems involving the phase change are mostly

16

associated with Josep Stefan who appears to have been the first to make an extensive

study of them in relation to the melting of the polar ice cap [28].

Consider a one-dimensional case in which the phase change takes place at the melting or

solidification temperature, and thus the two phases are separated by a sharp interface as

shown in Figure 10. This case is assumed that heat transfer takes place only by

conduction in both phases.

Figure 10. Solidification of a liquid from a cold plane surface [29]

The solidification model is depicted in figure 10. Initially the liquid temperature, , is

higher than the fusion temperature, . If the temperature of the liquid surface at is

lowered to and maintained, the liquid starts to solidify at and the interface

moves gradually in the positive x direction with velocity,

Since densities of the solid and the liquid are assumed to be constants, and the solid phase

is bounded the surface at , the velocity of the solid phase relative to the interface

17

must be equal to the interface velocity . Therefore, the law of conservation of mass

requires that

An energy balance on the control volume per unit area gives

where and are the specific enthalpies of the liquid and solid phases. Using the

previous relationship, the energy balance can be written as

where =

For the case of melting, the energy balance at the interface would be given by

This principle is applied to the thermal modeling in the following.

4.4 Thermal modeling

A mathematical description of figure 9 is presented here. The three-dimensional energy

balance for this system is:

18

The thermal conductivity and the specific heat capacity are constants, but the density and

the specific heat changes as a phase status, solid or liquid.

Internal heat generation source term is:

A standard transient energy balance with no phase change would only consider the time

rate of change of a single phase internal energy. However, for the phase change, , a

pseudo energy source term that is switched on during the phase change in nodal volume

must be included.

To keep track of the phase direction (melting or solidification), is a switch, activated

in a binary sense depending on the recent temperature history. Initially, is set to zero

assuming one is starting from the solid phase. When the phase change temperature is

reached from a liquid state, and are set to unity, and the internal energy term

for the element is set to zero. This action turns off the internal energy term and turns on

the term.

19

Chapter 5

Numerical simulation

5.1 FDM formulation

Figure 11. 3D and 2D view of unit calculation model

The analytical solution can be converted into the finite deference formulation using the

unit calculation model. Unit calculation model is depicted above. The lengths of and

are identical. will be calculated by the relationship of adjacent nodes. Each node

volume is identical, but and are not considered as nodes. Arrays will be filled with

the unit model formulation focused on i-node by finite difference method.

20

Analytical solution:

Using the finite difference method, future temperature of i-node can be obtained from the

analytical solution.

Simplified letters are described in the following.

21

5.2 Geometry, initial, and boundary conditions

5.2.1 Simulation modeling dimensions

Table 4. Modeling dimensions

Dimension Value Unit

(powder thickness) 30

(substrate thickness) 20 mm

(x-dir. node distance) 150

(x-dir. node distance) 150

(x-dir. total length of powder bed) 2.85 mm

(y-dir. total length of powder bed) 2.85 mm

D (laser spot diameter) 150

Figure 12. Laser beam spot on mesh

22

Due to the thin one layer powder bed, the temperature gradient along the z-direction is

negligible. The unit node area is generated as a rectangle. The area of laser spot is fully

occupied in the unit node area.

5.2.2 Mesh generation

Figure 13. Nodes numbering on mesh

23

5.2.3 Initial conditions

Table 5. Initial values

Variables Value Unit

(All node temperature) 300 K

(Substrate temperature) 300 K

(Ambient temperature) 300 K

(Velocity of heat source) 0 m/s

(Switch of phase change) 0 -

(Switch of phase change activation) 0 -

(Solid mass fraction) 1 -

(Liquid mass fraction) 0 -

At t=0, temperature of powder bed is set as the ambient temperature, , and solid phase.

5.2.4 Boundary conditions

Figure 14. Boundary conditions

The X-Z plane at this system’s boundary: Adiabatic

The Y-Z plane at this system’s boundary: Convection

24

5.3 Patterns on mesh

Table 6. Patterns of raster scan and Hilbert curves

Patterns on numbered mesh

Raster scan

Hilbert curves

Starting point and steps

Starting node number : 126

Ending node number : 267

64 steps

Starting node number : 126

Ending node number : 267

64 steps

The two patterns located exactly in the center of powder bed. The mesh consists of four

hundreds nodes as shown in Figure 13, and the node numbers are assigned from the left

side to the right side and from the bottom to the top sequentially. Both of patterns start

the identical position as shown in the picture above and also finish the identical node. The

total length of the two pattern is sixty four node length, and the area of these patterns is

1.44 . All of conditions that the starting and ending position, the area, and total

length of patterns have set identically for the simulation results estimation.

25

5.4 Simulation model and calculation procedure

5.4.1 Calculation procedure

Checking conditions for the center node volume

Eq2 Eq2 Eq2

Eq2 q Eq2

Eq2 Eq2 Eq2

if > : Reached melting point

Eq2 Eq2 Eq2 Begins phase change with constant temperature,

Eq2 Eq2

Eq2 Eq2 Eq2

if < 0 : Completely melted

Eq3 Eq3 Eq3 Powder bed moves for one time step

Eq3 Eq5 q

Eq3 Eq3 Eq3

Stop after moving for one time step

Eq2 Eq2 Eq2

Eq2 Eq4 q

Eq2 Eq2 Eq2

Solidification starts with constant temperature,

Eq2 Eq2 Eq2

Eq2 ?

Eq2 Eq2 Eq2

if < 0 : Completely solidified

Eq2 Eq2 Eq2

Eq2 Eq2 ?

Eq2 Eq2 Eq2

: Laser beam position

26

Initially, arrays are occupied the initial temperature, or . At first calculation, the heat

source, q, is assigned at the center node of the picture above. Until the temperature of

center node reaches the melting point, , the program updates all the temperature field.

When the temperature of center node reaches melting point, the phase change calculation,

, starts. At that time, the center node temperature is set as the melting point and the

heat source does not move to the next position. When the center node is fully melted, the

heat source moves to the next position. For this, all the arrays reflect the velocity of the

powder bed in the temperature calculation. After this step, the center node of liquid status

is experiencing phase change again. Solidification continues with the constant

temperature of . When the liquid mass fraction, , of the center node has a negative

value, the temperature of the center node is calculated by the equation 2 instead of the

constant temperature. The most important part in this simulation is reflecting the

independency of phase change calculation. It is unpredictable how many nodes are in the

phase change status because it could be that the speed of reaching melting point is much

higher than the melting and solidification speed. The calculation independency is

programmed in the modeling.

5.4.2 Occupied equations

q

:

27

Eq. 2

:

Eq. 3

:

Eq. 4

:

Eq. 5

:

? : It is occupied one of q, Tm or Eq. 2. It depends on status of j-node.

28

Chapter 6

Result and discussion

6.1 Time to reach each step

Table 7. Required time to melting, fully melted, and fully solidified

Melting point Time to reach (s)

0.00019

Fully melted Time required (s)

0.00063

Fully solidified after melted Time required (s)

0.00152 - 0.00063

= 0.00089

29

As shown in table 7, the time to reach melting point was measured at 0.00019 seconds. It

represents that the powder of the laser spot melts immediately once the incident laser

radiates. This means extremely high speed of temperature rise occurs during the process.

Comparing the time required between fully melted and fully solidified, it is observed that

fully solidified time requires more time than the former. This fact suggests that if Hilbert

pattern is hovering in local area so that the liquid phase can be maintained longer time,

Hilbert pattern would be more effective to reduce temperature gradient than raster scan.

6.2 Temperature field of Raster scan pattern

Figure 15. Temperature field of raster scan at final step (surface plot)

30

Figure 16. Temperature field of raster scan at final step (contour plot)

The raster scan pattern took the process completion with a time of 0.0655 seconds for the

area of 1.44 . As shown in the figure 16, the temperature gradient near the heat

source made steep contour lines except the right side. Also, the results show how fast the

temperature decreased. When the process finished, there was no temperature gradient at

the first line of pattern. This means Inconel 718 powder experienced tremendously rapid

heating and cooling process because it took just 0.0655 seconds for the whole process.

31

6.3 Temperature field of Hilbert pattern

Figure 17. Temperature field of Hilbert pattern at final step (surface plot)

Figure 18. Temperature field of Hilbert pattern at final step (contour plot)

32

Hilbert pattern took the process completion with a time of 0.06775 seconds for the area of

1.44 . It took a bit longer than raster scan case. The determining factor of completion

time is the phase change duration time because this simulation model designed that the

laser heat source moves when i-node completely melted. Laser heat source position is the

checking point to figure out whether the process finished or not. As shown in the figure

18, the temperature gradient near the heat source made relatively less steep contour lines

than raster scan case. The spaces between contour lines of Hilbert pattern are forming

evenly and nearly a circle shape. By the contour plot and the equation of solid mass

fraction,

the reason turns out that the value of

of Hilbert case must be smaller than

raster scan case. The raster scan’s value of is -0.2905 and is -0.2743, but

Hilbert’s value of is -0.2470 and is -0.2323 at last step of the process. Both of

the values of Hilbert case are smaller than raster case. It has been observed when the laser

passes the elbow-shaped parts of patterns, required fully melting time was needed one

more time step than straight line section. It represents the process of Hilbert pattern

experiences melting and solidification more slowly than raster case. As a result, Hilbert

pattern is more effective to preserve the locality than raster scan. Detailed temperature

field data of both patterns are attached in the appendix.

33

Chapter 7

Conclusions

The temperature field of SLM process was investigated using three-dimensional finite

difference method to compare the effect of two different patterns. The temperature profile

was numerically solved using a finite difference method. Several assumptions to simplify

the model were made. Required time to complete the process showed Hilbert pattern case

needs longer time than raster scan case for the identical area and path length. Phase

change pace of melting or solidification affected the laser travel speed. Hilbert case had

slower phase change pace than raster scan case due to the elbow-shaped parts of the

pattern. It provides us the cooling or heating rates of Hilbert pattern case have an

advantage to reduce temperature gradient. It can be analyzed Hilbert pattern is more

effective to preserve locality than raster scan pattern. By the results of contour plot, it is

observed that the overall temperature gradient of raster case is much steeper than Hilbert

case.

To continue this study and improve the use in future work it is necessary to relax

assumptions for more accurate simulation. For example, nonlinear thermal conductivity,

nonlinear specific heat capacity, and multiple layers. Also, it is need to conduct

experiments for studying the effect of Hilbert versus a raster pattern on thermal and

residual stresses with developing the optimized modeling by regulating process

parameters and part building strategies.

34

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38

APPENDIX

Appendix A: MATLAB code of Main calculation

clear; clc; close;

Inputs;

[ X,Y,XY ] = Mesh( Lx,Ly,Me,Ne );

[Ti] = T_initial(Ta,Me,Ne);

[Node_num] = Node_table( Me,Ne );

dt=dx/move; %time/step

for s=1:50

if s==1

Tv=Ti; % Initial temperature

else

Tv=Tf;

end

39

% m=Node_num(Hilbert(loc),1); % x-node number of

the laser

% n=Node_num(Hilbert(loc),2); % y-node number of

the laser

m=Node_num(Raster(loc),1); % x-node number of the

laser

n=Node_num(Raster(loc),2); % y-node number of the

laser

[Tp] = T_p(Tv,dt,c1,c2,vel,rho1,rho2,m,n);

[q] = qG(dt,m,n);

Tf=Tp+q;

% % Melting point check

if Tf(m,n) > Tm

Tf(m,n)=Tm;

vel=stop;

c2=c;

rho2=rho;

DXps=( Cil*(Tf(m-1,n)-Tf(m,n)) +Cij*(Tf(m+1,n)-

Tf(m,n)) +Cim*(Tf(m,n-1)-Tf(m,n)) +Cip*(Tf(m,n+1)-Tf(m,n))

40

+Cia*(Ta-Tf(m,n)) +Cis*(Ts-Tf(m,n)) -Cv*c*stop*(Tf(m-1,n)-

Tf(m+1,n)) )/(dh*Ms);

Xps_p = Xps_p + DXps*dt; % Future solid mass

fraction

% % Phase change check (Solid to Fluid)

if Xps_p < 0 % Completely melted

vel=move; % Powder bed movement

c2=cf; % Liquid phase

rho2=rhof; % Liquid phase

loc=loc+1; % Next step of Laser

Xps_p=1; % Reset initial value

mm = [mm; m];

nn = [nn; n];

Xpf_pp =[Xpf_pp; Xpf_p];

end

% % Solidification

for iSol = 1:length(mm)

Tf(mm(iSol),nn(iSol)) = Tm;

DXpf=( Cil*(Tf(mm(iSol)-1,nn(iSol))-

Tf(mm(iSol),nn(iSol))) +Cij*(Tf(mm(iSol)+1,nn(iSol))-

Tf(mm(iSol),nn(iSol))) +Cim*(Tf(mm(iSol),nn(iSol)-1)-

41

Tf(mm(iSol),nn(iSol))) +Cip*(Tf(mm(iSol),nn(iSol)+1)-

Tf(mm(iSol),nn(iSol))) +Cia*(Ta-Tf(mm(iSol),nn(iSol)))

+Cis*(Ts-Tf(mm(iSol),nn(iSol))) -Cv*cf*stop*(Tf(mm(iSol)-

1,nn(iSol))-Tf(mm(iSol)+1,nn(iSol))) )/(dh*Mf);

Xpf_pp(iSol) = Xpf_pp(iSol) + DXpf*dt; %

Future liquid mass fraction

end

% % Phase change check (Fluid to solid)

ind=find(Xpf_pp

42

Appendix B: MATLAB code of Inputs

dx=150e-6;

dy=150e-6;

D=150e-6;

LP=200;

Ilas=LP/(D^2*pi/4); %Laser power W/m2

Me=19; %Horizontal element number

Ne=19; %Vertical element number

Lz=30e-6; %Powder thickness(m)

Ls=0.02; %Substrate thickness (m)

Lx=dx*Me; %m

Ly=dy*Ne; %m

%%%%%%%%%% Inconel 718 Properties %%%%%%%%%%

rho=8192*0.9; %Density of soild powder(90%) kg/m3

rhof=7585; %Density of the liquid phase(1600K) kg/m3

c=435*0.9; %Specific heat of soild powder(90%) J/(kg K)

cf=778; %Specific heat of the liquid phase(1600K) J/(kg K)

K=11.4*0.9; %Thermal conductivity of a powder(90%) W/(m K)

Ks=50; %Thermal conductivity of a substrate W/(m K)

assumed a steel

43

h=3.81e3; %Heat transfer coefficient to the environment

W/(m2 K)

dh=227e3; %Enthalpy change of melting J/kg

Ta=300; %Ambient Temperature K

Ts=300; %Substrate temperature K

Tm=1609; %Melting point K

a=2.67e4; %Absorption coefficient 1/m

b=3.2e4; %Extinction coefficient 1/m

stop=0; %Velocity of stop m/s

move=0.6; %Velocity of moving m/s

qg=Ilas*(a/(b*Lz))*(1-exp(-b*Lz)); %Heat generation from

the laser radiation W/m3

V=dx*dy*Lz; %Unit volume m3

Ms=rho*V; %Mass of soild powder kg

Mf=rhof*V; %Mass of the liquid phase kg

Cij=1/(dx/(2*K*dy*Lz)+dx/(2*K*dy*Lz)); %Ci,j W/K

Cia=1/(Lz/(2*K*dy*dx)+1/(h*dx*dy)); %Ci,infinite W/K

Cis=1/(Lz/(2*K*dy*dx)+Ls/(2*Ks*dy*dx)); %Ci,sub W/K

Cim=1/(dy/(2*K*Lz*dx)+dy/(2*K*Lz*dx)); %Ci,i-1 W/K

Cv=rho*dy*Lz; %Cv,i except c*vel W/K

Cip=1/(dy/(2*K*Lz*dx)+dy/(2*K*Lz*dx)); %Ci,i+1 W/K

Cil=1/(dx/(2*K*dy*Lz)+dx/(2*K*dy*Lz)); %Ci,l W/K

44

Cie=1/(dy/(2*K*dx*Lz)+1/(h*dx*Lz)); % yz plane BC W/K

loc=2; % Initial position of laser

Xps_p=1; % Initial mass fraction of solid powder

Xpf_p=1; % Initial mass fraction of liquid powder

vel=stop; % Initial velocity

c1=c; % Initial specific heat

c2=c; % Initial specific heat

rho1=rho; % Initial density

rho2=rho; % Initial density

mm = [];

nn = [];

Xpf_pp =[];

Hilbert = [126 127 128 148 147 167 187 188 168 169 189 190

170 150 149 129 130 131 151 152 132 133 134 154 153 173 174

194 193 192 172 171 191 211 231 232 212 213 214 234 233 253

254 274 273 272 252 251 271 270 269 249 250 230 210 209 229

228 208 207 227 247 248 268 267]; % Node number sequence

Raster = [126 127 128 129 130 131 132 133 134 154 153 152

151 150 149 148 147 167 168 169 170 171 172 173 174 194 193

192 191 190 189 188 187 207 208 209 210 211 212 213 214 234

233 232 231 230 229 228 227 247 248 249 250 251 252 253 254

274 273 272 271 270 269 268 267]; % Node number sequence

45

Appendix C: MATLAB code of Mesh generation

function [ X,Y,XY ] = Mesh( Lx,Ly,Me,Ne )

lx=Lx/Me;

ly=Ly/Ne;

no_node=(Me+1)*(Ne+1);

XY=zeros(no_node,2);

X=zeros(Me+1,Ne+1);

Y=zeros(Me+1,Ne+1);

for i=1:(Ne+1)

for j=1:(Me+1)

XY((i-1)*(Me+1)+j,1)=(j-1)*lx;

XY((i-1)*(Me+1)+j,2)=(i-1)*ly;

end

end

for j=1:(Ne+1)

for i=1:(Me+1)

X(i,j)=XY(i+(j-1)*(Me+1),1);

Y(i,j)=XY(i+(j-1)*(Me+1),2);

end

end

end

46

Appendix D: MATLAB code of Node table

function [Node_num] = Node_table( Me,Ne )

Node_num=zeros((Me+1)*(Ne+1),2);

for i=1:(Ne+1)

for j=1:(Me+1)

Node_num((i-1)*(Me+1)+j,1)=j;

Node_num((i-1)*(Me+1)+j,2)=i;

end

end

end

47

Appendix E: MATLAB code of Moving heat source

function [q] = qG(dt,m,n)

Inputs;

q=zeros(Me+1,Ne+1);

for i=1:Me+1

for j=1:Ne+1

if i==m && j==n

q(i,j)=dt/(rho*c)*qg;

else

q(i,j)=0;

end

end

end

end

48

Appendix F: MATLAB code of Initial temperature

function [Ti] = T_initial(Ta,Me,Ne)

Ti=zeros(Me+1,Ne+1);

for i=1:Me+1

for j=1:Ne+1

Ti(i,j)=Ta;

end

end

end

49

Appendix G: MATLAB code of Present temperature

function [Tp] = T_p(Tv,dt,c1,c2,vel,rho1,rho2,v,w)

Inputs;

Tp=zeros(Me+1,Ne+1);

for i=2:Me

for j=2:Ne

if i==v && j==w

Tp(i,j)=Tv(i,j) + dt/(rho2*c2*V)*( Cil*(Tv(i-

1,j)-Tv(i,j)) +Cij*(Tv(i+1,j)-Tv(i,j)) +Cim*(Tv(i,j-1)-

Tv(i,j)) +Cip*(Tv(i,j+1)-Tv(i,j)) +Cia*(Ta-Tv(i,j))

+Cis*(Ts-Tv(i,j)) -Cv*c2*vel*(Tv(i-1,j)-Tv(i+1,j)) );

else

Tp(i,j)=Tv(i,j) + dt/(rho1*c1*V)*( Cil*(Tv(i-

1,j)-Tv(i,j)) +Cij*(Tv(i+1,j)-Tv(i,j)) +Cim*(Tv(i,j-1)-

Tv(i,j)) +Cip*(Tv(i,j+1)-Tv(i,j)) +Cia*(Ta-Tv(i,j))

+Cis*(Ts-Tv(i,j)) -Cv*c1*vel*(Tv(i-1,j)-Tv(i+1,j)) );

end

end

end

50

%%%%% Boundary conditions %%%%%

for i=1:Me+1

Tp(i,1)=Ta; %%% Bottom side

Tp(i,Ne+1)=Ta; %%% Top side

end

%%% Left side

for j=2:Ne

for i=1

Tp(i,j)=Tv(i,j) + dt/(rho1*c1*V)*( Cie*(Ta-Tv(i,j))

+Cij*(Tv(i+1,j)-Tv(i,j)) +Cim*(Tv(i,j-1)-Tv(i,j))

+Cip*(Tv(i,j+1)-Tv(i,j)) +Cia*(Ta-Tv(i,j)) +Cis*(Ts-Tv(i,j))

-Cv*c1*vel*(Ta-Tv(i+1,j)) );

end

end

%%% Right side

for j=2:Ne

for i=Me+1

Tp(i,j)=Tv(i,j) + dt/(rho1*c1*V)*( Cil*(Tv(i-1,j)-

Tv(i,j)) +Cie*(Ta-Tv(i,j)) +Cim*(Tv(i,j-1)-Tv(i,j))

+Cip*(Tv(i,j+1)-Tv(i,j)) +Cia*(Ta-Tv(i,j)) +Cis*(Ts-Tv(i,j))

-Cv*c1*vel*(Tv(i-1,j)-Ta) );

end

end

end

51

Appendix H: Temperature field of Hilbert pattern’s last step

: Heat source position

52

Appendix I: Temperature field of Raster scan’s last step

: Heat source position