Multiphysics Modeling of Selective Laser Sintering/Melting

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Title Multiphysics Modeling of Selective Laser Sintering/Melting

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Author Ganeriwala, Rishi

Publication Date 2015

Peer reviewed|Thesis/dissertation

eScholarship.org Powered by the California Digital Library University of California Multiphysics Modeling of Selective Laser Sintering/Melting

Rishi Kumar Ganeriwala

A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy

Engineering - Mechanical Engineering

and the Designated Emphasis

Energy Science and Technology

in the

Graduate Division of the University of California, Berkeley

Committee in charge: Professor Tarek I. Zohdi, Chair Professor Daniel Kammen Professor Hayden Taylor

Fall 2015 Multiphysics Modeling of Selective Laser Sintering/Melting

Abstract

Multiphysics Modeling of Selective Laser Sintering/Melting by Rishi Kumar Ganeriwala Doctor of Philosophy in Engineering - Mechanical Engineering and the Designated Emphasis in Energy Science and Technology University of California, Berkeley Professor Tarek I. Zohdi, Chair

A significant percentage of total global employment is due to the manufacturing industry. However, manufacturing also accounts for nearly 20% of total energy usage in the United States according to the EIA. In fact, manufacturing accounted for 90% of industrial energy consumption and 84% of industry carbon dioxide emissions in 2002. Clearly, advances in manufacturing technology and efficiency are necessary to curb emissions and help society as a whole. Additive manufacturing (AM) refers to a relatively recent group of manufacturing technologies whereby one can 3D print parts, which has the potential to significantly reduce waste, reconfigure the supply chain, and generally disrupt the whole manufacturing industry. Selective laser sintering/melting (SLS/SLM) is one type of AM technology with the distinct advantage of being able to 3D print metals and rapidly produce net shape parts with complicated geometries. In SLS/SLM parts are built up layer-by-layer out of powder particles, which are selectively sintered/melted via a laser. However, in order to produce defect-free parts of sufficient strength, the process parameters (laser power, scan speed, layer thickness, powder size, etc.) must be carefully optimized. Obviously, these process parameters will vary depending on material, part geometry, and desired final part characteristics. Running experiments to optimize these parameters is costly, energy intensive, and extremely material specific. Thus a computational model of this process would be highly valuable. In this work a three dimensional, reduced order, coupled discrete element - finite difference model is presented for simulating the deposition and subsequent laser heating of a layer of metal powder particles sitting on top of a substrate. Validation is provided and parameter studies are conducted showing the ability of this model to help determine appropriate process parameters and an optimal powder size distribution for a given material. Next, thermal stresses upon cooling are calculated using the finite difference method. Different case studies are performed and general trends can be seen. This work concludes by discussing future extensions of this model and the need for a multi-scale approach to achieve comprehensive part-level models of the SLS/SLM process. i

To my parents, Manju and Suri ii

Contents

List of Figures v

List of Tables ix

1 Introduction 1 1.1 Additive Manufacturing Techniques ...... 1 1.2 Applications of AM ...... 5 1.3 Outline of this Work ...... 6

2 Energy and Societal Impacts of Additive Manufacturing 9 2.1 Manufacturing Energy Use and CO2 Emissions ...... 9 2.2 Life Cycle Impacts of AM Technologies ...... 11 2.3 Societal Impacts of AM ...... 13

3 Selective Laser Sintering Process and Considerations 18 3.1 SLS Process Description ...... 19 3.2 Process Parameters ...... 22 3.2.1 Laser power ...... 22 3.2.2 Scan speed ...... 22 3.2.3 Spot size ...... 23 3.2.4 Hatch spacing ...... 23 3.2.5 Scan strategy ...... 23 3.2.6 Powder material and manufacturing ...... 24 3.2.7 Powder size distribution ...... 25 3.2.8 Layer thickness ...... 26 3.2.9 Surrounding gas atmosphere ...... 26 3.3 Material Properties of Parts Fabricated via SLS/SLM ...... 26 3.3.1 Density ...... 26 3.3.2 Microstructure ...... 27 3.3.3 Strength ...... 27 3.3.4 Hardness and surface roughness ...... 28 3.4 Other Considerations during SLS ...... 28 iii

3.5 Previous Modeling Attempts ...... 30

4 Powder Deposition and Laser Heating Model Description 33 4.1 Particle Dynamics ...... 33 4.2 Particle Thermal Eﬀects ...... 37 4.2.1 Particle-to-particle heat transfer ...... 38 4.2.2 Laser beam modeling ...... 39 4.2.3 Phase change ...... 40 4.2.4 Thermal softening and melting of particles ...... 42 4.3 Finite Diﬀerence Modeling of Substrate ...... 43 4.4 Numerical Solution Scheme ...... 45 4.4.1 Time-stepping ...... 45 4.4.2 Binning and OpenMP Parallelization ...... 47 4.5 Programming Algorithm ...... 48

5 Powder Deposition and Laser Heating Model Validation and Results 51 5.1 Material Properties and Parameter Values ...... 51 5.2 Model Validation ...... 51 5.3 Numerical Examples ...... 54

6 Residual Stress Modeling 60 6.1 Background ...... 60 6.2 Modeling Framework ...... 62 6.2.1 Mechanical eﬀects - balance of linear momentum ...... 62 6.2.2 Thermal eﬀects - balance of energy ...... 65 6.2.3 Numerical solution scheme ...... 66 6.3 Numerical Examples ...... 67 6.3.1 Cooling of a solid block ...... 67 6.3.2 Cooling of a porous block ...... 75 6.3.3 Cooling of a single laser scan ...... 81

7 Conclusions and Future Extensions 90 7.1 Summary of this Work ...... 90 7.2 Model Limitations and Future Extensions ...... 93

Bibliography 97

A Economics and Projected Growth of Additive Manufacturing 110

B Numerical Derivatives 114 B.1 Spatial Derivatives using Finite Diﬀerences ...... 114 B.2 Time Marching Schemes ...... 117 B.2.1 Euler Methods ...... 117 iv

B.2.2 Runge-Kutta Schemes ...... 118 B.2.3 Other Schemes ...... 120

C Basic Parallelization Techniques 121 C.1 Binning Algorithm ...... 121 C.2 OpenMP ...... 122 v

List of Figures

1.1 List of common industrial AM processes, adapted from [66] ...... 3 1.2 Devices produced using AM techniques. SLS produced fuel nozzle for GE LEAP jet engines (left) [5] and acetabular cup for a hip implant (right) [3] 7

2.1 Global energy related CO2 emissions by scenario. 450 scenario is considered necessary to limit global temperature rise to 2 ◦C [50] ...... 10 2.2 US greenhouse gas emissions by sector [133] ...... 10 2.3 LCAs showing largest impact sources for the SLS of PA2200 nylon (left) and SLM of stainless steel (right) [57] ...... 13 2.4 IEA projections of energy related CO2 emissions per manufacturing sector by 2050 (left) and employment per manufacturing sector by 2050 (right). Two business as usual (BAU) scenarios and the eﬃcient G2 scenario are depicted [102] ...... 16

3.1 Different stages depicting the solid state sintering of metal powders [69] . 19 3.2 Schematic of a typical SLS/ SLM set-up ...... 20 3.3 EOS P 800 SLS machine [1] ...... 20 3.4 Different scanning strategies used during SLS. Typical parallel zig-zag pattern (left) and island scanning strategy (right). Dashed lines refer to laser path...... 24 3.5 Bimodal packing distribution ...... 25 3.6 Microstructure of SLM-processed Inconel 718 [51] ...... 27 3.7 Comparison against standard bulk material properties of yield strength, tensile strength, and breaking at elongation for stainless steel parts produced via SLM. Bar color indicates direction tested [65] ...... 28 3.8 SEM images depicting balling behavior of a single laser scan at different scan speeds [74] ...... 29 3.9 Warping of parts due to uneven thermal expansion/contraction [89] . . . . 30

4.1 Discrete element representation of pre-sintered powder particles (SEM image from [151]) ...... 34 4.2 Particle-to-particle overlap ...... 35 4.3 Heat transfer to an individual particle ...... 38 vi

4.4 Contact area of two intersecting particles ...... 39 4.5 Gaussian laser beam depiction ...... 40 4.6 Laser beam illustration as a series of light rays (length of each ray corresponds to intensity). Note that this approach is not used in the current work...... 41 4.7 Illustration of how heat capacity is updated (left) and ensuing phase change diagram (right) ...... 42 4.8 Finite difference stencil and coordinate system used for an arbitrary material property A. Indices i, j, and k are used to represent the x, y, and z coordinates, respectively...... 44 4.9 Depiction of boundary conditions used for the finite difference mesh: all B.C.s (left), Neumann B.C. for top face (right) ...... 45 4.10 Simulation flow chart ...... 47 4.11 Illustration of binning algorithm. Only particles in the shaded boxes will be checked for contact with the red particle in the center box...... 48

5.1 Comparison of experimental melt pool size by Khairallah and Anderson (left) [58] and simulation melt pool size depicted in red (right). Note that the ﬁgure on the left is Figure 5(a) from [58]. The experimental melt height, width, and depth are 26 µm, 75 µm, and 30 µm, respectively. The simulation melt pool dimensions are 30 µm, 85 µm, and 20 µm...... 54 5.2 Screenshots depicting the deposition of a layer of 316L SS particles (600 particles total) ...... 55 5.3 Screenshots showing the temperature evolution of a layer of 316L SS particles and the underlying 316L SS substrate as a laser is passed over (temperature in Kelvin). Top view on upper row. Cross-sectional view from the side on bottom row...... 56 5.4 Screenshots depicting the melt pool (red) of a layer of 316L SS particles and the undelying 316L SS substrate as a laser is passed over. Top view on upper row. Cross-sectional view from the side on bottom row...... 56 5.5 Melt pool size as laser power is varied from 40 - 400 W , scan speed is constant at 2.0 m/s ...... 57 5.6 Melt pool size as scan speed is varied from 0.4 to 4.0 m/s, laser power is constant at 200 W ...... 57 5.7 Melt pool from a single laser pass over a mono-modal (left) and bimodal (right) powder distribution. Note that solid particles are blue, molten ones are green, and gaseous ones are red. Notice how some small particles are vaporized in the bimodal distribution. Laser power is 200 W and scan speed is 2.0 m/s. All parameters between the two simulations are identical other than particle size distribution...... 59

6.1 Boundary conditions for residual stress calculations ...... 69 vii

6.2 Initial temperature distribution (temperature in K) ...... 69 6.3 Average temperature of the fully dense 1 mm x 1 mm x 1 mm block of 316L SS ...... 72 6.4 Average Von Mises stress of the fully dense 1 mm x 1 mm x 1 mm block of 316L SS ...... 72 6.5 Average equivalent strain of the fully dense 1 mm x 1 mm x 1 mm block of 316L SS ...... 73 6.6 Temperature distribution (in K) of the fully dense 1 mm x 1 mm x 1 mm block of 316L SS at t = 0 (left), t = 0.5 ms (middle), and t = 1 ms (right) 73 6.7 Von Mises stress distribution (in MP a) of the fully dense 1 mm x 1 mm x 1 mm block of 316L SS at t = 0 (left), t = 0.5 ms (middle), and t = 1 ms (right) ...... 74 6.8 Equivalent strain distribution of the fully dense 1 mm x 1 mm x 1 mm block of 316L SS at t = 0 (left), t = 0.5 ms (middle), and t = 1 ms (right) 74 6.9 Phase distribution of 70% dense 1 mm x 1 mm x 1 mm block of material. Red points are phase 1 (316L SS) and blue are phase 2 (argon gas) . . . . 76 6.10 Average temperature of the 70% dense 1 mm x 1 mm x 1 mm block of 316L SS ...... 77 6.11 Average Von Mises stress of the 70% dense 1 mm x 1 mm x 1 mm block of 316L SS ...... 77 6.12 Average equivalent strain of the 70% dense 1 mm x 1 mm x 1 mm block of 316L SS ...... 78 6.13 Temperature distribution (in K) of the 70% dense 1 mm x 1 mm x 1 mm block of 316L SS at t = 0 (left), t = 0.5 ms (middle), and t = 1 ms (right) 78 6.14 Von Mises stress distribution (in MP a) of the 70% dense 1 mm x 1 mm x 1 mm block of 316L SS at t = 0 (left), t = 0.5 ms (middle), and t = 1 ms (right) ...... 79 6.15 Equivalent strain distribution of the 70% dense 1 mm x 1 mm x 1 mm block of 316L SS at t = 0 (left), t = 0.5 ms (middle), and t = 1 ms (right) 79 6.16 Finite diﬀerence mesh of a layer of powder particles ...... 80 6.17 Phase distribution of the SLS simulation with using pre-deposited particles, sampled from a 100 µm x 100 µm x 100 µm domain. Red points are phase 1 (316L SS) and blue are phase 2 (argon gas) ...... 81 6.18 Average temperature of the 100 µm x 100 µm x 100 µm domain ...... 82 6.19 Average Von Mises stress of the 100 µm x 100 µm x 100 µm domain . . . 82 6.20 Average equivalent strain of the 100 µm x 100 µm x 100 µm domain . . . 83 6.21 Temperature distribution (in K) of the 100 µm x 100 µm x 100 µm domain at t = 0 (left), t = 0.5 µs (middle), and t = 1 µs (right) ...... 83 6.22 Von Mises stress distribution (in MP a) of the 100 µm x 100 µm x 100 µm domain at t = 0 (left), t = 0.5 µs (middle), and t = 1 µs (right) ...... 84 6.23 Equivalent strain distribution of the 100 µm x 100 µm x 100 µm domain at t = 0 (left), t = 0.5 µs (middle), and t = 1 µs (right) ...... 84 viii

6.24 Initial temperature distribution of the 300 µm x 500 µm x 300 µm powder- steel mixture after a single laser scan. All nodes over 1100 K are assumed to have molten and be solid 316L SS...... 86 6.25 Phase distribution of the 300 µm x 500 µm x 300 µm powder-steel mixture after a single laser scan. Red points represent solid 316L SS and blue represents unmolten powder...... 86 6.26 Average temperature of the 300 µm x 500 µm x 300 µm powder-steel mixture ...... 87 6.27 Average Von Mises stress of the 300 µm x 500 µm x 300 µm powder-steel mixture ...... 87 6.28 Average equivalent strain of the 300 µm x 500 µm x 300 µm powder-steel mixture ...... 88 6.29 Temperature distribution (in K) of the 300 µm x 500 µm x 300 µm powder- steel mixture at t = 0 (left), t = 5 µs (middle), and t = 10 µs (right) . . . 88 6.30 Von Mises stress distribution (in MP a) of the 300 µm x 500 µm x 300 µm powder-steel mixture at t = 0 (left), t = 5 µs (middle), and t = 10 µs (right) ...... 89 6.31 Equivalent strain distribution of the 300 µm x 500 µm x 300 µm powder- steel mixture at t = 0 (left), t = 5 µs (middle), and t = 10 µs (right) ...... 89

7.1 Fluid ﬂow modeled by the Lattice Boltzmann method around a Discrete Element sphere, demonstrating couple feasibility (color represents ﬂuid velocity) [28] ...... 94 7.2 DMG Mori LASERTEC 65 laser deposition welding and milling machine. Outer case (left) and close-up of a part being built (right) [6] ...... 95

A.1 Worldwide sales of industrial AM systems since 1988 [143] ...... 111 A.2 AM Revenue forecasts through 2020 [82] ...... 111 A.3 Metal AM powder demand forecasts by industry [30] ...... 112

B.1 Illustrative mesh of a domain used in the ﬁnite diﬀerence method . . . . . 115

C.1 Illustration of binning algorithm. Only particles in the shaded boxes will be checked for contact with the red particle in the center box...... 122 C.2 Linked list binning for a 2D example problem ...... 123 ix

List of Tables

5.1 Material properties for 316L stainless steel as a function of temperature and phase [48, 124] ...... 52 5.2 Remaining material properties and simulation parameters used [48, 2] . . 52 5.3 Aﬀect of powder size distribution on loose bed density, last row represents a bimodal distribution ...... 58

6.1 0.2% Yield strength, ultimate tensile strength, and instantaneous coeﬃ- cient of thermal expansion for 316L SS [124, 48] ...... 70 6.2 Additional material properties used in this simulation [101, 124] ...... 70 6.3 Argon material properties [77] ...... 75 6.4 Powder material properties [77, 116, 86, 124] ...... 85 x

Acknowledgments

I would like to thank many people for their inspiration and help in completing this manuscript. First, I would like to thank Professor Tarek Zohdi for his years of guidance and advice. I have learned more during the past 4+ years in the CMRL than at possibly any other time of my life. Thanks for all you’ve taught me and for always pushing me to keep working. I would also like thank my other commitee members, Professor Daniel Kammen and Professor Hayden Taylor. Thank you Professor Kammen for teaching me so much about energy and really fostering my love for sustainable technologies that can benefit society. Thank you Professor Taylor for your inquisitive nature and offering me useful feedback on my research. Additionally, I would like to thank all the members of the CMRL lab at Berkeley. Our countless discussions have been really useful in helping me formulate this thesis and solve problems I encountered along the way. Thanks to the rest of my family and friends too, for helping me along this process. Most importantly, I would like to thank my mom and dad, Manju and Suri Ganeriwala. You guys have always served as the best example for me and instilled in me the value of hard work and doing things the right way. Thanks for your continued love and support. This work was primarily funded by the Siemens corporation. Many thanks go to Ramesh Subramanian, Marco Brunelli, Nicolas Vortmeyer and Vinod Philip for their support over the last 4 years. Thanks to all other Siemens employees who attended my monthly research updates and offered useful feedback along the way. This work was also funded in part by the Army Research Laboratory through the Army High Performance Computing Research Center (cooperative agreement W911NF-07-2-0027). Many thanks go to Raju Namburu for his support. 1

Chapter 1

Introduction

3D printing is a buzz phrase thrown around by so many in our society today. But what is it really? And how does it work? Well, turns out that 3D printing is actually just a phrase given to a variety of diﬀerent techniques that fall under the general category of additive manufacturing (AM). AM is deﬁned as the process of building 3D objects by adding layer-upon-layer of material. This is in contrast to traditional machining which typically removes material from a larger block, also called subtractive manufacturing. AM, or 3D printing, has seen its use rapidly grow in recent years, with a recent Economist article even describing the digitization of manufacturing as the third industrial revolution due to its potential to radically change the manufacturing industry [85], a thought that has since been echoed by many others [90]. In this chapter the main commercialized AM techniques will be discussed, followed by a look at just a few of the potential applications. This chapter will conclude with a section outlining the remainder of this manuscript.

1.1 Additive Manufacturing Techniques

Within the broad category of additive manufacturing, or 3D printing, there exist many diﬀerent techniques that can be used depending on the material and desired part characteristics. However, they generally follow the same basic steps [31]:

1. A 3D model of the part to be produced is digitally produced, typically using CAD software. That file is converted into a type as specified by the AM technology (often a STL file).

2. The ﬁle is sliced into layers by software on the computer and sent to the AM machine.

3. The AM machine builds the part up layer-by-layer. The fabrication path and method is controlled by computer software.

4. The ﬁnal part is built and removed from the machine. Any support structures or adhering material is removed. Minimal post-processing operations may be necessary 2

depending on the technique used and desired part characteristics.

The most commonly used AM technologies are stereolithography (SLA or SL), fused deposition modeling (FDM), inkjet printing (IJP), laminated object manufacturing (LOM), and selective laser sintering (SLS) [66]. These processes and more are summarized in Figure 1.1. Stereolithography was patented in 1986 by Charles Hull [46] and commercialized by 3D Systems Inc. This technique uses a photo-sensitive polymer resin and a UV laser to build parts layer-by-layer. A wiper blade deposits a thin layer of resin onto the build platform. A UV laser scans over the resin in the shape of the desired object, curing the resin it contacts. The build platform is lowered and a new layer of resin is deposited on top. Once the resin has settled the UV laser scans over it again joining it to the underneath surface. The part is built up in this manner. Support structures are required to attach the part to the build platform. Upon completion the part is taken out of the resin bath and support structures are removed. Often the part may continue to be cured in a separate chamber under UV exposure. This process can offer high resolution parts but typically only for small part sizes. Additionally, the photo-polymer resins can be quite expensive, can be toxic, and are limited in material. The entire process typically takes from a few hours up to a couple days depending on part size and resin characteristics. Fused deposition modeling was developed by S. Crump in the late 1980s [15] and commercialized in 1990 by Stratasys. This technique extrudes molten material from a movable nozzle. This nozzle traverses over the build platform depositing thin layers of the molten material. After each layer, the build platform lowers and the nozzle deposits a new layer on top of the previous layers. The molten material solidifies and bonds with the previous layer almost immediately after extrusion from the nozzle. Support structures are needed to support the weight of the part. While initially only used with polymer plastics, many new materials (wax, metals with binder, and ceramics with binder) have also been incorporated [66]. FDM is generally the cheapest AM technology and is commercially available in desktop sizes for personal use. The main limitations are surface finish and accuracy due to the seam line between layers and delamination of the layers if not done correctly. Build time is typically on the order of hours to days depending on part size. Computer software can automatically make a solid model either partially or completely hollow if desired by the user (often hollow parts are sufficient for prototyping purposes). This greatly speeds up build time and decreases material use. Inkjet printing has origins tracing back to the late 19th century and was first commercialized by Siemens in 1951 [70]. It was originally developed for printing 2D images by ejecting ink droplets onto a substrate. The ink contains solutes dissolved in a solvent, which evaporates upon deposition on the substrate, leaving only the original solute. Ex- tensions to 3D printing have come about through the use of pre-patterned substrates at multiple layers of processing. Many materials have been introduced in IJP to additively produce a variety of products including solar cells, sensors, and thin-film transistors [45]. The main drawbacks are expensive ink cartridges and fragile print heads. 3

Figure 1.1: List of common industrial AM processes, adapted from [66] 4

Laminated object manufacturing was patented in 1988 by Michael Feygin [25]. In this process, layers of adhesive coated sheet materials are glued together. 3D parts are created by laminating and adhering 2D cross-sections on top of each other. Each cross- section is cut to shape often using a laser cutter. Many materials can be additively manufactured in this way including paper, polymers, composites, ceramics, and metals [66].This easily automated process is relatively cheap and safe. However, it tends to suffer from accuracy problems (especially in the build direction) and does not offer the best part quality. Often post-process machining and/or drilling are necessary to produce accurate final parts. While typically an adhesive is used to bond layers together, LOM can also include the bonding of two thin sheets of metal via ultrasonic consolidation [97]. Selective laser sintering was developed in the late 1980s at the University of Texas at Austin by Carl Deckard and his advisor Joe Beaman [18]. In this process powder particles are fused together via either full or partial melting from a laser beam. A wiper deposits a thin layer of powder particles on the fabrication bed. A laser scans over the particles in the shape of the desired object causing the particles to melt and fuse together. The fabrication bed lowers a tiny bit and a new layer of powder is swept on top with the wiper blade. The laser scans over this layer causing the powder to fuse with the layer underneath it. The final part is built up layer-by-layer this way. Upon completion the final part is removed and adhering particles are cleaned off the final part. No support structures are needed due to the presence of the powder bed. Unsintered powders can typically be recycled and reused, though this may vary with material. Materials suitable for SLS processing include polymers, ceramics, and metals. Sometimes a binder material is used as well. SLS offers the advantage of being able to rapidly produce parts of complicated geometries in one step; however, many issues can arise if not performed correctly. As developing a computational model of SLS is the main focus of this dissertation, this process will be discussed in much more detail in later chapters. Also note here that in the case of full melting of metal powders, this technique is frequently referred to as selective laser melting (SLM) though the overall process is the same for SLS and SLM. Many other AM technologies exist besides the ones explicitly described in this section. However, many of these techniques are variants of the techniques used in the described AM technologies. For example laser cladding and direct metal deposition work by shoot- ing metal particles onto a surface while those particles are simultaneously melted via a laser beam. These techniques are used to additively manufacture and to repair existing parts. Selective electron beam melting (SEBM) is identical to SLS/SLM except an electron beam provides the heat source and the process is performed in a vacuum. Three- dimensional printing deposits powder material on a substrate. This powder is selectively joined via a binder. Chemical vapor deposition (CVD) and lithography processes are also occasionally considered 3D printing techniques though they are not further addressed in this work. Since AM is such a recent field, there is still an ever-growing amount of research into these processes and ways to expand material capabilities, improve product performance, and decrease processing time. New technologies are continuously developed and commercialized. Some of these newer, not as proven, technologies are briefly discussed in 5 the Conclusions chapter.

1.2 Applications of AM

Additive manufacturing offers the ability to rapidly produce parts of complicated geometries in ways previously unfeasible with traditional techniques. As such, AM is frequently used in rapid prototyping (RP) and rapid manufacturing (RM) of parts. While many people may think of 3D printing as a hobbyists endeavor; its uses are actually far greater. While only 25% of the AM market was involved the direct manufacture of end- use products as of 2011, it is the industry’s fastest growing segment with a 60% annual growth rate [13]. SLS in particular has applications in a vast number of industries including (but certainly not limited to) tooling, biomedical, aerospace, automotive, energy, and consumer products. SLS and other AM techniques have vast potential for rapid tooling (RT) due to their ability to quickly fabricate (1) sacrificial patters using polymers; (2) insert, tool, core, and molds using composites; (3) metallic molds with cooling channels; and (4) ceramic based molds [67]. According to the National Center for Manufacturing Science, AM can reduce die production time by 40% [87]. SLS/SLM has also shown the capability of producing hard metal parts such as tungsten carbide-cobalt (WC-Co) and titanium carbide-nickel (TiC-Ni) used in machining tools and abrasion resistant coatings [83, 27]. In the biomedical industry many researchers have demonstrated the ability for SLS to produce tissue engineering scaffolds, valves, stents, and many types of implants [127, 140, 111]. AM allows for easy customization of bio-products to fit each individual. Custom fit personal hearing aids and dental products (implants, crowns, bridges, etc.) can also be made faster and in higher quality using SLS [67]. Using UV light to cure polymers, LUXeXceL has shown the ability to 3D print functional lenses [107]. AlpZhi is similarly 3D printing lenses. AM techniques have also been shown to significantly decrease production time of surgical tools [87]. One of the biggest adopters of AM (and SLS/SLM in particular) has been the aerospace industry. In addition to using AM for rapid prototyping of design changes, entire parts are now fabricated using SLS, which have the benefits of being lighter, cheaper, and quicker to manufacture. GE has made headlines for their new fuel nozzle in the LEAP engines which are entirely made using SLM (see Figure 1.2) [4]. Many other casings, panels, and duct work is fabricated using SLS, again saving time, money, and weight (a huge factor in airplane design). Molds and templates for cast parts can be 3D printed in a matter of hours, not days or weeks as typically required using conventional machining [42]. Similarly, the automotive industry has demonstrated the potential for AM to reduce part build time, costs, and weight. Many clips, brackets, panels, and HVAC systems can quickly be customized for each individual and car model and produced using SLS [67]. Other companies, such as Local Motors and Divergent Microfactories, have taken this a step further and are 3D printing entire cars. Divergent Microfactories has been able to 6 significantly reduce car weight and environmental impacts without sacrificing safety or performance using modular, 3D printed components [16]. Other sectors of the energy industry (in addition to aerospace and automotive) also see significant benefits through the use AM. Companies such as Siemens and GE are testing SLM produced components in their gas turbine blades. AM allows them the opportunity to reconfigure turbine blade design to enhance cooling and improve efficiency, while simultaneously reducing build time [137]. Thermal barrier coatings also show the potential to be 3D printed onto turbine blades. However, 3D printed turbine blades still suffer from performance issues and thus have not yet been introduced into deliverable products; this is very much an active area of research. A technique known as laser-driven non-contact transfer printing has shown the ability to produce flexible inorganic solar cells by depositing prefabricated material from a stamp onto a substrate [73]. Roll-to- roll printing is additionally being tested for the fabrication of solar cells. 3D printing of optical materials allows for customized transparency and varying indices of refraction. With improvement such technology could also have applications for solar cells [107]. SLS has been able to improve the performance of fuel cells by being able to make more efficient flow channels in a carbon plate, increasing chemical reaction between flowing gas and the catalyst [67]. In the consumer industry, AM techniques have many applications in producing en- closures, fixtures, seals, gaskets, hinges, etc. Clothing, jewelry, and art have also taken to the design freedom enabled using AM. Additionally, shoes and athletic apparel have adopted AM as a way to quickly fabricate customized products [67, 45]. Fully functional, 3D printed electronics is another growing industry [139]. However, it is important to note here that while AM has clearly seen its use sky- rocket in a number of industries, there are still significant limitations preventing increased growth. Namely, AM techniques are currently limited by size of parts produced, material imperfections, cost, and the number of available materials. Currently only hundreds of materials are able to be 3D printed, compared to thousands available for conventional manufacturing processes [131]. As continued research into these limitations proceed and new techniques become commercialized, the applications of AM will only continue to grow.

1.3 Outline of this Work

This chapter was meant to introduce the reader to the basics regarding additive manufacturing: techniques and some applications. Chapter 2 will delve further into the energy and societal impacts of AM. Chapter 3 will then begin to focus speciﬁcally on the SLS process. In Chapter 3 a more in depth discussion is provided about how the SLS process works, the diﬀerent process parameters, properties of parts fabricated using SLS, other considerations and limitations, and previous modeling attempts. In Chapter 4, the deposition and laser heating model of the SLS process is described. A coupled discrete 7

Figure 1.2: Devices produced using AM techniques. SLS produced fuel nozzle for GE LEAP jet engines (left) [5] and acetabular cup for a hip implant (right) [3] 8 element - finite difference model is used to simulate the deposition and laser heating of a layer of powder particles. The mechanical model and thermal model are described, along the the numerical solution scheme and programming algorithm. Chapter 5 provides validation and results of the deposition and laser heating model described in Chapter 4. Different parameter studies are performed and numerical examples are shown. Chapter 6 presents the modeling of sintered parts as they cool down and thermal stresses develop. A background of residual stress and previous modeling attempts is first provided. Next, the finite difference scheme used to simulate the thermal stresses upon cooling is described. Results and numerical examples are provided for different cases and initial conditions. Chapter 7 concludes this work by discussing the main takeaways and observations from the modeling results. Future improvements and work to be done regarding the complete modeling of SLS is discussed along with extensions of the work to other AM techniques. 9

Chapter 2

Energy and Societal Impacts of Additive Manufacturing

The need to improve sustainability in all sectors of our society is perhaps more pressing than ever before due to global climate change and temperature rise as a result of anthropogenic greenhouse gas emissions. There is no one magic technology that can help curb our CO2 equivalent emissions down to the required rates to limit temperature rise ◦ to the target of 2 C over pre-industrial levels. Figure 2.1 depicts global CO2 emissions under different scenarios. The 450 scenario is that necessary to limit total atmospheric CO2-equivalent to 450 ppm, which is considered necessary to limit global temperature rise to 2 ◦C [50]. Clearly, significant energy efficiency and conservation measures are needed in virtually every sector of our society to decrease emissions and meet this target. This section analyzes manufacturing’s impact on greenhouse gas (GHG) emissions and the role additive manufacturing may play in these emissions and on our society as a whole.

2.1 Manufacturing Energy Use and CO2 Emissions Manufacturing is an indispensable part of our society. It is necessary to produce goods and services which we all use on a daily basis. The industrial sector, including manufacturing, mining, and construction, accounts for nearly 25% of all global employment [21]. Indeed manufacturing has also been listed as the most important cause of economic growth according to the Roosevelt Institute [106]. While clearly a strong driver for the global economy, manufacturing also is a major culprit in the climate change problem. Figure 2.2 shows the US GHG emissions by sector, in which it is clear that industrial emissions constitute a very signiﬁcant portion of all US emissions [133]. According to the US Energy Information Administration (EIA), manufacturing accounts for nearly 20% of total energy usage in the United States [132]. In fact, manufacturing accounted for 90% of US industrial energy consumption and 84% of industry carbon dioxide emissions in 2002 [109]. 10

Figure 2.1: Global energy related CO2 emissions by scenario. 450 scenario is considered necessary to limit global temperature rise to 2 ◦C [50]

Figure 2.2: US greenhouse gas emissions by sector [133] 11

Just as a move away from fossil fuels to clean sources of energy is considered vital to reducing GHG emissions, a move towards more efficient and sustainable manufacturing processes is also necessary, especially as the amount of raw materials processed has steadily increased over the past decades. For example, the annual global production of aluminum more than doubled between 1980 and 2005 [21]. While the energy intensity (energy per $1 of goods) of US industry has been reduced by 50% over the past three decades, an alarming trend has been observed. As new manufacturing processes, which can work at finer scales and smaller dimensions, are commercialized, the specific electricity requirements (J/kg) for these processes is increasing and the process rate (kg/hr) is decreasing. Additionally, high exergy materials are increasingly used in inefficient ways [36]. Thus it is important to take a closer look at the impacts of AM before deciding upon its potential sustainability impacts.

2.2 Life Cycle Impacts of AM Technologies

When evaluating new technologies, such as additive manufacturing, it is important to analyze the total environmental impacts in an objective and quantifiable manner. A commonly used technique for such type of analysis is called the life cycle analysis (LCA). LCAs analyze the environmental impacts of a process as it relates to numerous different factors, including GHG emissions, deforestation, toxicity, ozone depletion, eutrophication, and others. Each of these factors is then weighted appropriately to determine a final environmental impact score. LCAs typically include complete cradle-to-grave analyses of processes/machines including the material extraction, manufacturing, transportation, use, and end-of-life phases [138]. As additive manufacturing technologies are still relatively recent, not too many comprehensive LCA studies have been performed comparing the effects of AM vs. conventional manufacturing. However, the general consensus is that AM processes have better environmental characteristics as compared to traditional machining [81, 123, 45]. The main reasons lie in reduced material consumption, less tooling required, and the elimination of often toxic cutting fluids used in machining. One study by Serres et al. [114] found a 70% environmental impact reduction for CLAD (a laser based AM process) as compared to machining. Other studies for laser based AM found similar impact reductions [91, 88] stemming from the ability to remanufacture (fix) broken parts using direct metal deposition and the reduction in time necessary to produce certain parts of complicated geometries. Additionally, dies for injection molding can be created in much shorter time frames using laser based AM [87]. However, other studies have found conflicting reports regarding the environmental benefits of AM. One study found that casting is actually significantly better than AM in terms of energy usage, although AM beats casting in terms of other environmental impacts such as material use [45]. Another found that that milling performed comparably with inkjet printing and SLA in terms of total environmental impact [23]. This study actually 12 found that how the machine is used was more important than the fabrication technique, in that a lot of power is consumed when machines are simply idling. Thus having the machines constantly in use significantly reduced the impacts per amount of material processed. However, a follow up study found that AM did possess environmental benefits when producing parts of complicated geometries [24]. The main reason for this lies in the fact that the geometry of the part typically does not matter in the amount of time required for fabrication using AM techniques. However, when trying to produce a part with complicated geometry (such as a hollow part with small features) using conventional milling, many turns and cuts are needed which significantly increases the time required for fabrication. Baumers et al. [8] measured the energy use during the SLS process. In this study the authors mentioned the need to include utilization percentage when performing LCAs of these machines as it will have a significant affect on the environmental impacts. They measured that warm-up and cool down time can sometimes consume upwards of 20% of the total energy required to build parts in an SLS machine. Kellens et al. [57] performed a more detailed analysis of the lifecycle impacts of SLS/SLM machines. Figure 2.3 shows the main impact sources for the production of nylon 12 based PA2200 parts using an SLS machine and the production of stainless steel parts using an SLM machine. In the LCA for the production of nylon parts in an SLS machine, waste powder production is actually the leading source of environmental impacts, followed by energy use. This demonstrates the need for more efficient recycling of powder during SLS operations. In the LCA for the production of stainless steel using an SLM machine, energy use is the leading source of environmental impacts, followed closely by the production and use of the nitrogen gas atmosphere in the chamber. A better sealed chamber could decrease the impacts of nitrogen (or other inert gas) by reducing the amount needed to continually fill the process chamber. Powder waste is only a very small percentage of total impacts during this SLM process suggesting that metal powder is produced and recycled more efficiently than polymer powders in SLS. However, these results could vary depending on specific material, machine, and batch size [57]. In any case, it is worth further investigating powder and compressed air production practices as improvements in these areas will improve the sustainability of the SLS process as a whole. Overall, firm conclusions regarding energy use in AM vs. conventional machining are still unclear due to a lack of literature on the subject and the variability in energy measurement methodologies used by each researcher. While there have been attempts to quantify the direct lifecycle impacts of AM technologies, there are many indirect effects which could significantly alter global manufacturing energy consumption. The ability to fabricate net-shape parts of virtually any geometry allows for the redesign of currently used parts. For example, it is easy to imagine formerly solid parts that could now be made hollow due to ease of manufacturing. This would not only save material but would reduce costs and energy use associated with manufacturing. Lighter parts in an airplane or car would also lead to large fuel savings. Parts could be designed with fewer components reducing total manufacturing time and material usage. In fact, AM has been able to reduce the number of steps required to produce surgical 13

Figure 2.3: LCAs showing largest impact sources for the SLS of PA2200 nylon (left) and SLM of stainless steel (right) [57] tools from 62 to 7 [88]. GE has been able to produce lighter fuel nozzles for their airplane gas turbine engines using SLM. They were able to reduce 18 separate components, which required multiple machining and welding steps, into one additively manufactured nozzle. This nozzle is 25% lighter and can be made in much less time [4]. In total, GE has been able to realize a weight reduction of over 500 pounds per engine through 3D printing of various components, which can amount to huge fuel savings and CO2 reductions over the life of the engine [42]. Divergent Microfactories has been able to reduce the weight of a car chassis from around 1000 pounds down to 100 pounds through the use of their node based assembly system made possible by AM and the use of carbon fiber. Not only have they dramatically cut down on material use in the manufacturing of their vehicle (a significant portion of a car’s LCA), but they claim their 3D printed car would have one-third the emissions of a comparable electric vehicle [16]. This is an impressive feat which illustrates the potential environmental benefits of AM, even if some of their claims may be partially exaggerated. AM could affect the supply chain since parts would no longer need to be made in large quantities as the price per part does not depend on scale when using AM. This would allow less overstocking of parts and decrease the need for huge manufacturing plants. More manufacturing could be performed locally and on-demand which would decrease energy used for transportation, packaging, and storage of components. All these indirect impacts are currently not reflected in LCAs of specific technologies and should be the subject of continued investigation.

2.3 Societal Impacts of AM

An article from The Economist described the digitization of manufacturing (made possible primarily by AM) as the third industrial revolution, a sentiment that has been echoed by many since then [85]. By being able to create a part on a computer and subsequently 3D print it, anyone can become a designer and manufacturer. Since using a 14

3D printer is often as simple as reading a manual, and a few trial and error prints, the need for highly skilled machinists will decrease as more people and companies purchase AM machines. Overall the way our society approaches manufacturing in the future could be significantly different from the conventional methods used in the past as AM technologies continue to develop. In a literature review of the societal impacts of AM technologies, Huang et al. [45] found that the main benefits of AM use can be summarized by: • Customized additive manufacturing of healthcare products • Reduced material use and energy consumption • Supply chain reconfiguration by on-demand and on-site manufacturing Many healthcare products are already being designed and built more efficiently using AM techniques [67, 140]. AM allows for mass customization of parts and quicker build times for things such as hearing aids, implants, tissue scaffolding, stents, and dental parts. Additionally personal protective equipment such as helmets, protective garments, and even athletic equipment can be custom fit to individuals providing safety benefits. In the previous section material use and energy benefits were discussed by noting how AM drastically reduces cutting waste and can greatly shorten the fabrication time required for parts with complicated geometries. Parts can be redesigned to have fewer components and more complicated structures. Design changes can be implemented quickly and without the need for significant process restructuring. Mass customization of parts becomes much more feasible as you would no longer need to overhaul an entire factory if the shape of one part is changed. The amount of time required for manufacturing of these parts can also be decreased. One of the biggest impacts of AM can be in how the supply chain is structured. AM can improve the efficiency of supply chains through just-in-time manufacture and waste reduction. Parts can be made at the shop floor instead of requiring delivery by vendors. Build-to-order strategies can be implemented ensuring that overstocking does not occur [130]. A concept known as distributed manufacture is gaining attention as AM allows for more small, localized manufacturing facilities [45]. This will reduce the need for large warehouses, packaging, and transportation of goods. Holmstrom et al. [44] suggested that integrating AM into the spare parts supply chain can be approached in one of two ways: 1. Using centralized AM to replace inventory holding 2. Distributed AM to replace inventory holding and conventional distribution The centralized approach is desirable when the parts requiring AM are limited in quantity and response time is not as critical. The distributed approach is more appropriate when there is a significantly high demand of AM producible replacement parts to justify the necessary investment in AM machinery [39]. 15

The growth of distributed manufacturing could also have significant health benefits if this causes a move away from large, centralized manufacturing facilities in remote locations. Many companies are predicting that AM could lead to more manufacturing re- turning to US as the need for cheap labor in China and other countries decreases [13, 85]. By moving manufacturing closer to company headquarters and end-use, companies can save on transportation costs and be able to more quickly change designs in products. Less large factories could improve health as many low-wage workers in developing countries do not have appropriate personal protective equipment and adequate ventilation. Less conventional machining would also decrease the exposure to toxic cutting fluids for these workers. Additionally, noise reduction could improve hearing as AM machines are typically much quieter than milling counterparts [45]. However, AM is also not without its drawbacks. While toxic cutting fluids are not used, many resins used in SLA processes are toxic and pose health risks if touched or inhaled. For many materials, the biological effects are still not fully understood [45]. Additionally, inhalation of fumes released during FDM could be dangerous if proper ventilation is not present as ultrafine particle emissions released during FDM have been shown to be toxic to rats and mice [126]. Since many FDM machines are bought for use in private residences, there are no safety regulations in place to ensure they are used in areas with adequate ventilation. An additional concern could lie in the fact that the automization of manufacturing could cause massive job loss, an even more significant problem considering that the population will only continue to rise as the 21st century progresses. However, IEA projections actually show that many new jobs can be created with the use of more sustainable manufacturing techniques. Figure 2.4 (from a 2009 IEA report) shows projections of the CO2 emissions and employment per manufacturing sector in 2050 for two different business as usual scenarios and a G2 scenario representing increased efficiency. While the CO2 emissions are understandably much lower in the efficient G2 scenario, this forecast also predicts the creation of more jobs under the G2 scenario [102]. Also worth noticing in this figure are the large CO2 emissions impacts and also employment for the manufacturing of metals and plastic, materials currently used in AM techniques. While plastics may still be a source of health concerns due to the large impacts and toxicity of the resins used in SLA and even some of the FDM plastics, the emissions due to metals could certainly be reduced using techniques such as SLS/SLM by drastically reducing material waste and overstocking of parts. Also note that these are only impacts associated with manufacturing of these materials. However, transportation emissions can also be reduced due to AM since many parts can potentially be made locally and thus not require long-distance shipping. Finally, it is important to note here that AM will never completely replace conventional manufacturing techniques. In terms of cost and energy use, casting still significantly outperforms AM when it is required to produce large quantities of the same part [45]. Additionally, limitations still exist in AM regarding size, imperfections, performance, materials, and cost. Until all these issues can be addressed, the widespread adoption of 16

Figure 2.4: IEA projections of energy related CO2 emissions per manufacturing sector by 2050 (left) and employment per manufacturing sector by 2050 (right). Two business as usual (BAU) scenarios and the eﬃcient G2 scenario are depicted [102] 17

AM will remain limited. However, the amount of research currently going into improving existing AM technologies and creating new ones is steadily increasing, showing signiﬁcant promise for its growth [13]. While it clearly has the potential to signiﬁcantly improve upon the sustainability of the manufacturing industry, more research is needed in this area to better quantify the impacts. Some of the potential economic impacts of AM are discussed in Appendix A. 18

Chapter 3

Selective Laser Sintering Process and Considerations

Sintering traditionally refers to the heating of a compacted powder material to a temperature below the melting point, but high enough to allow bonding of the individual particles. This typically involves heating the pre-compacted powder to 70 – 90% of the melting temperature of the particles. The primary sintering mechanisms, which allow bonding to occur, are diffusion and vapor phase transport, where metal atoms release to the vapor and then re-solidify at convergent geometries (often the interface of two particles). Alternatively, in a mixture of powder particles, it is common to melt the powder with a lower melting temperature, which will then surround the solid particle due to surface tension (liquid-phase sintering). The final density of sintered parts can reach over 99% of the theoretical density of the resultant alloy. However, varying amounts of porosity may remain due to trapped gases remaining in voids after compaction, which may be desirable depending upon the application [54]. Conventional sintering involved placing the powdered material into a die where it is compressed and placed in a furnace. Solid-state sintering, where the powders do not fully melt, requires many hours at temperatures close to the melting temperature to allow the powders to bond together. Figure 3.1 depicts the traditional sintering of metal powders in a furnace. Another common sintering technique is electric current assisted, or spark plasma, sintering. In this process, powders are placed into a die connected to electrodes. The die is mechanically compressed while an electric current is run through the electrodes. The powders heat up due to Joule heating arising from their internal resistance. This process has the advantage of producing net shape parts in a much shorter time, and can be more energy efficient than traditional sintering [95]. Finally, there is selective laser sintering (SLS), which was first invented in 1986 at the University of Texas, Austin [125]. Since it’s original invention, it’s use has continued to grow rapidly with potential applications is virtually every industry where goods/products are made. The remainder of this chapter will describe the SLS process in much more detail. 19

Figure 3.1: Diﬀerent stages depicting the solid state sintering of metal powders [69]

3.1 SLS Process Description

Selective laser sintering differs from traditional sintering in that the powders are not consolidated prior to heating and often full melting occurs. Note that in the case of full- melting for metals, the process is frequently referred to as selective laser melting (SLM). SLS typically refers to partial melting for metals, or the cases where the process is applied to ceramics and/or polymers [68, 67]. However, the distinction between SLS and SLM is often vague. Note that many other names may be used to describe this process, including direct metal laser sintering (DMLS) or direct metal laser melting (DMLM), among others [90]. However, all of these terms refer to the same basic process, with possible slight differences in some of the process parameters. For the remainder of this work, the process shall be referred to as selective laser sintering (SLS) or occasionally selective laser melting (SLM) for simplicity. Figure 3.2 depicts a schematic of a typical SLS/SLM machine. An actual EOS SLS machine is shown in Figure 3.3. First a thin layer of powder particles, with sizes ranging from approximately 10 to 100 µm, is deposited onto a substrate via a roller. Each layer is typically on the order of 20 to 100 µm thick. Many different powder materials can be used including metals, polymers, ceramics, and composites. A laser is then directed via a scanner system over the powders in the pattern of the desired shape, either partially melting (SLS) or fully melting (SLM) the powders. As the laser selectively melts the particles, they fuse and bond together. Next, the fabrication bed lowers, a new layer of powder is placed on top by the roller, and the laser selectively scans over the surface again. This layer-by-layer, additive manufacturing process is repeated until the final part has been fabricated. The laser beam typically has a nominal diameter on the order of 0.5 mm or less. A 20

Scanner system Object being Laser Roller fabricated Unsintered Powder supply powder

Powder Fabrication delivery piston piston

Figure 3.2: Schematic of a typical SLS/ SLM set-up

Figure 3.3: EOS P 800 SLS machine [1] 21

pulsed or continuous wave CO2 laser is typically used for polymer materials, while an Nd:YAG or fiber laser is more typically used for metals. The reason for this lies in the fact that polymers and ceramics (oxides) absorb the 10.6 µm wavelength emitted by the CO2 laser better, while metals and ceramics (carbides) better absorb the 1.06 or 1.08 µm wavelengths emitted by Nd:YAG or fiber lasers, respectively [67, 41]. Typically, the laser power can vary anywhere from approximately 30 - 400 W depending on the material and desired part characteristics (although up to 1 kW or even higher power lasers have also been used) [90]. Typically higher powers are used for full melting in SLM while lower laser powers may be used for the partial melting which occurs in SLS. Additionally, scan speed may be varied from 2 to 700 cm/s or more [125, 33]. In SLS, a binder material with a lower melting temperature may be applied to the particles. This binder material is melted, bonding together particles with higher melting temperatures. The binder also helps ensure even distribution of each layer of powder particles [145]. To aid in the process, frequently the sintering chamber is heated to minimize required laser energy and expedite the process. Preheating the chamber has the added benefit of reducing thermal gradients and thus reducing the residual stresses present in the final part. Additionally, an inert gas, usually nitrogen or argon, is present to avoid oxidation or burning of the powders. Upon completion of the SLS process, the final part is removed and often bead blasted to remove unsintered, adhering particles [125]. In the case of SLS, the final parts frequently still have a significant amount of porosity. In these cases additional operations, such as post-sintering, hot isostatic pressing, or infiltration, may be required to increase the part density [64]. Post-sintering works by holding the post-SLS part at high temperatures (approximately 70% of the melting temperature) for hours or even days to allow solid- state sintering and densification to occur [67]. Hot isostatic pressing involves holding the post-sintered part at an elevated temperature while an inert gas, typically argon, is pumped in at a high pressure causing consolidation of the part by plastic deformation. Infiltration involves injecting epoxy or some other flowable material with lower melting point into the porous part, essentially plugging up the holes [78, 134]. Due to the high heating and cooling rates present in SLS (up to 106 oC/s) large residual stresses may build up in the final part [135]. This high cooling rate also causes anisotropy of the final part as long thin grains form in the z-direction (or the vertical build direction), and inhibits the formation of certain microstructures. Typically martensitic microstructures are formed in steels [67]. Post SLS heat treating, such as annealing, can be performed to relieve residual stresses and change the microstructure [113]. SLS has the advantage of being able to rapidly produce parts of relatively complex geometries, in a timely and cost effective manner. It is especially useful in its ability to make metallic parts, which would otherwise be impossible to form using other rapid prototyping, rapid tooling, and rapid manufacturing techniques as shown in the works by Simchi and Pohl [120, 121]. Maeda and Childs and Fischer et al. [83, 26] have demonstrated the ability for SLM to produce parts made from hard metal powders, such as WC-Co and TiC-Ni used in machining tools and abrasion resistant coatings. SLS has also seen uses in RP and RM for polymeric and ceramic materials. Tan et al., Williams et al., 22 and Schmidt et al. [127, 140, 111] have successfully laser sintered polymeric biomaterials which can be used to produce tissue engineering scaffolds and certain types of implants. SLS is also currently employed to produce parts in the automotive, aerospace, biomedical, and energy industries, among others.

3.2 Process Parameters

While there are clearly a vast, and growing, number of applications for the SLS process, if not applied correctly, the resulting parts can have many defects, especially high residual stresses, microcracking, delamination of layers, and/or high porosity [64]. An optimal set of process parameters need to be applied to ensure high quality of the fin- ished product. Such process parameters may include (but certainly aren’t limited to) laser power, scan speed, spot size, hatch spacing (distance between successive parallel passes of the laser), and scan strategy. Additionally powder material, size distribution, layer thickness, and conditions in the surrounding atmosphere can significantly affect final part quality. Clearly, the optimal set of parameters will vary depending on material and desired final part characteristics. This section discusses each of these process parameters in more detail.

3.2.1 Laser power Laser power typically ranges from 30 - 400 W depending on the material and desired part characteristics. SLS machines are typically rated up to 50 W as this is adequate to melt most polymers and for partial melting of metals. Metal SLM machines have laser powers on the order of hundreds of watts to ensure full melting at reasonable scan speeds [67]. Laser cladding machines and other variants of SLM machines designed to melt much thicker layers at a time can employ lasers on the order of kilowatts [115]. The laser energy density is a key consideration in ensuring adequate melting of powder particles and the underneath layer, while minimizing ablation of the particles, which can lead to increased porosity and uneven surface ﬁnishes. The laser energy density is typically deﬁned as [67] Laser power Laser energy density = (3.1) Spot size ∗ Scan speed

3.2.2 Scan speed Similar to laser power, scan speed is another parameter which can be adjusted to ensure proper melting. A lower scan speed will increase the laser energy density and thus lead to a larger melt pool. Faster scan speeds will decrease run time. However, faster scan speeds have also been shown to decrease melt pool width, which can lead to balling (due to Rayleigh instabilities, discussed further in Section 3.4) and uneven surface finishes [74, 144]. Additionally very high scan speeds may not allow sufficient time for the heat 23 to diffuse across the powder bed. Typical scan speeds range from from 2 to 700 cm/s, though this can be varied.

3.2.3 Spot size Most SLS machines use a Gaussian laser where the power decreases exponentially away from the center of the laser. The spot size of a Gaussian laser is defined as the point where the beam intensity falls to 1/e2 of the peak intensity at the center. A larger spot size will allow for faster scanning of an individual layer by increasing the melt pool width after a single pass. However, larger spot sizes will also decrease resolution of the final part. Spot size can be changed on a laser with a technique known as laser defocus, which changes the focal point of the laser to a plane below the build platform. However, this will also change the spatial distribution of laser intensity, which will affect melt pool geometry [10, 122].

3.2.4 Hatch spacing The hatch spacing is deﬁned as the distance between parallel laser passes (see Figure 3.4). The ideal hatch spacing will be dependent on the melt pool size and the laser spot size. It should be large enough to decrease run time but still small enough such there is some remelting between parallel laser passes. This is necessary to ensure proper bonding and to decrease the chance of balling [84].

3.2.5 Scan strategy The scan strategy can have a very large effect on the characteristics of your final part. A good scanning strategy can significantly decrease distortion, anisotropy, and porosity in the final part [117]. Typically laser scanning is done in parallel lines in a zig-zag pattern, either along the length or width of the part. Alternating the scanning strategy between adjacent layers can also help decrease anisotropy and increase final part density [63]. Generally it has been shown that shorter scan lengths are desired to decrease the chance of balling. Additionally shorter scan lengths helps ensure that cooling and solidification does not occur before the laser comes back for an adjacent pass. Solidification is undesired between adjacent laser passes as this can cause improper bonding between layers and an uneven surface. To ensure shorter scanning lengths an island strategy may be employed where the build layer is divided into small blocks, or “islands”, each of which is scanned independently in an alternating pattern. This strategy has been shown to reduce balling and additionally distortion due to uneven thermal contraction when cooling [67]. The typical parallel zig-zag and island scanning strategies are shown in Figure 3.4. Note that many other scanning strategies exit, such as spiral, diagonal, fractal path, and others [146]. It is also generally preferred to build objects in a manner such that the z-direction 24

Hatch spacing

Figure 3.4: Diﬀerent scanning strategies used during SLS. Typical parallel zig-zag pattern (left) and island scanning strategy (right). Dashed lines refer to laser path.

(vertical build direction) corresponds to the smallest dimension of the part, if possible. This decreases fabrication time and increases part accuracy [67].

3.2.6 Powder material and manufacturing Clearly the powder material will have a big impact on the final part characteristics. Generally, powders which are suited for welding applications perform well when additively manufactured via SLS. More ductile materials are better able to withstand the thermal stresses that arise when cooling, whereas as brittle materials have a greater tendency to form microcracks when cooling. This is especially present in the SLM of Inconel (a superalloy frequently used in gas turbines among other applications) [55, 122]. More ductile materials are better able to withstand residual stresses by plastic deformation and thus resist forming microcracks. The manufacturing technique used to create the powders can also be important. Metal powders are typically manufactured via atomization, a process where molten metal is forced through a nozzle at moderate to high pressures. A gas is introduced into the metal stream just before leaving the nozzle to induce turbulence. The collection chamber is also filled with a fluid promoting turbulence of the molten metal. This process is capable of forming fine micron sized powders with spherical shapes. Gas atomization has been shown to produce more spherical powders for steel when compared against water atomization. These more spherical powders have better packing and wetting characteristics upon melting, leading to a more dense final part. Additionally less oxidation of the powders is present using gas atomization [75]. 25

Figure 3.5: Bimodal packing distribution

Note that powder particles may be formed by other techniques such as pulverization or chemical reactions. Sometimes these can produce particles of highly irregular shapes as compared with the spherical particles formed by the atomization technique (for metal powders) or co-extrusion processes (for polymer powders) [110]. These irregular shapes inhibit the flowability of such powders, leading to increased porosity. Additionally, polymer powders of less than 50 µm in size attract to each other due to inter-particle forces, further decreasing flowability. To combat this frequently a lubricant or some other small powder is added to the mixture, which can increase flowability and loose bed density [32].

3.2.7 Powder size distribution The powder size distribution has a very significant effect on the densification and certain material properties of the final part. Simulations and experiments by Korner et al. [60] on selective electron beam melting (SEBM), similar to SLM except that an electron beam acts as the heat source and the process occurs in a vacuum, found that loose powder bed density played the biggest role in densification of the final part. As a result there has been investigations into using bimodal, or even trimodal, packing distributions where smaller particles fill the interstitial spaces between larger particles to increase the loose bed density (as depicted in Figure 3.5). The problem with such powder distributions, however, lies in the fact that the smaller particles may evaporate before the bigger particles fully melt. These evaporated particles may then get trapped as gas bubbles in the final part, increasing the final porosity [56]. Using finer powders can decrease the surface roughness of the part. However, using too fine of powders decreases the packing density of the loose bed. Hence performing SLS with sub-micron sized powders is limited in application and use due to difficulties in producing dense parts when using such small powders [67]. 26

3.2.8 Layer thickness The thickness of each layer is an additional consideration when determining SLS process parameters. Obviously, the part can be built faster if thicker layers are used; however, this will also decrease the smoothness and resolution of the part. Meanwhile, thinner layers have been shown to be optimal for allowing any trapped gas bubbles to escape before solidiﬁcation occurs [63]. The thickness of each layer is additionally limited by the melt depth of the laser, as some remelting of the underneath layer is required for proper bonding to occur. The optimal layer thickness will depend on material, desired part resolution, powder size distribution, and input laser energy density.

3.2.9 Surrounding gas atmosphere The atmosphere of the surrounding environment is critical in producing usable parts via SLS. If oxygen is present in the atmosphere, metals will very quickly form oxides which will inhibit melting and consolidation of the powders. Oxides have also been shown to increase the prevalence of balling in the melt pool. Thus the SLS chamber is ﬁlled with an inert gas, typically argon or nitrogen, and the partial pressure of oxygen is closely monitored. Creating a vacuum in the SLS chamber is also avoided as this can lead to increased ablation of particles [67]. Increasing the pressure of the chamber can decrease ablation.

3.3 Material Properties of Parts Fabricated via SLS/SLM

If done correctly, many parts created via SLS/SLM can be used in industrial applications. However, it is important to note that parts created via SLS tend to have slightly diﬀerent material properties as compared to conventionally forged or cast parts.

3.3.1 Density Obviously the density of parts produced via SLS is a critical consideration. If optimal process parameters are used, often times up to 99.9% of the theoretical bulk density can be achieved for steels and titanium allows [147, 63]. Other metals, including superalloys, can also achieve nearly full density under proper processing conditions. The main sources of lingering porosity in a material are due to only partial melting of the powder and entrapped gas bubbles. Remelting of each layer, by essentially scanning the laser twice over each layer, has been shown as an effective technique for increasing final part density [64]. In the case of porous parts, post-processing techniques can be used to increase the density including post-sintering, hot isostatic pressing, and infiltration. 27

Figure 3.6: Microstructure of SLM-processed Inconel 718 [51]

3.3.2 Microstructure Another feature characteristic of parts produced via SLS/SLM is anisotropy. Due to the extremely high cooling rates present (up to 106 oC/s) in the build (or vertical) direction, metals have a tendency to form long, very fine columnar grains. These grains solidify before dendrites can form, causing an anisotropic microstructure (typically martensitic) [10]. Due to this anisotropy, build direction has a significant effect on the final part strength, as this will vary in different directions [118]. Figure 3.6 depicts the microstructure of SLM-processed Inconel 718 [51]. Another consideration is residual stresses and microcracking which may arise in more brittle materials due to the high cooling rates and uneven thermal gradients. Preheating the chamber and the base plate has been shown to reduce residual stresses by decreasing the thermal gradients [119]. Additionally post-SLS heat treating, such as annealing, can be performed to relieve residual stresses and change the microstructure [113].

3.3.3 Strength The yield strength of parts produced via SLS/SLM can often be higher than that of materials produced with conventional methods. This can be explained by the martensitic microstructure and the ﬁne columnar grains formed due to the high cooling rates. Additionally, the tensile strength is often comparable to, or higher than that of conventionally produced materials [122]. However, the ductility (measured by the strain at breaking), is almost always less for parts produced via SLM. The spatially distributed melt pool boundaries restrict the ductility, explaining why SLM parts are typically more brittle [118]. Figure 3.7 shows a comparison against standard bulk material properties of the yield strength, tensile strength, and breaking at elongation for stainless steel parts 28

Figure 3.7: Comparison against standard bulk material properties of yield strength, tensile strength, and breaking at elongation for stainless steel parts produced via SLM. Bar color indicates direction tested [65] produced via SLM [65].

3.3.4 Hardness and surface roughness The hardness of a material is a measure of how resistant it is to shape change. While this will clearly depend on the porosity of the sample, for fully dense steel parts the hardness has been shown to be comparable with conventionally wrought products of the same alloy [128]. The surface roughness of SLS parts is strongly dependent on the particle size, as partially molten particles will typically adhere to the edges of any sintered part. Studies have shown that remelting of top layers can be used to reduce the surface roughness of the top surface at the expense of increased processing time [147].

3.4 Other Considerations during SLS

During the SLS/SLM process, care must be taken to ensure that balling does not occur. Balling is a phenomena attributed to Plateau-Rayleigh instabilities where the melt pool will tend to ball up rather than forming one continuous track if the length to width ratio of the track is too high. This can cause an uneven surface ﬁnish and uneven layer height, leading to unusable parts. In studies performed on SLM of stainless steel and pure nickel powder, Li et al. [74] found two main types of balling that occur. Type 1 is ellipsoidal balls with dimension of approximately 500 µm which occur due to poor wetting ability of the material. The other type are small spherical balls of approximately 10 µm size, which 29

Figure 3.8: SEM images depicting balling behavior of a single laser scan at different scan speeds [74] do not have a significant effect on final part quality. One of the primary factors which leads to balling is the length to width ratio of the track. If the ratio of track length to track width is greater than approximately π, balling will occur due to Plateau-Rayleigh instabilities [67]. The length to width ratio can be decreased by increasing laser power or spot size or decreasing the scan speed or scan length. Preheating the chamber can also reduce this ratio. Figure 3.8 depicts the balling phenomena as a function of scan speed for a given laser power and spot size [74]. Clearly the balling behavior becomes more prevalent at higher speeds. The oxygen content is another factor which has a significant effect on the amount of balling that occurs. More oxygen in the chamber leads to the increased formation of oxides which have poor wetting characteristics. Decreasing the oxygen percentage in the chamber has been shown to lead to decreased balling under the same processing parameters [74]. Other studies have found that increased melt pool temperature decreases balling, as higher temperature molten metals have better wetting characteristics [64]. Note that wetting is dependent on interfacial energies between the surfaces of the liquid, solid, and air. Another factor which has a strong influence on the melt pool shape is Marangoni convection, which is surface tension driven convection that occurs in the melt pool. For many materials surface tension is a strong function of temperature. Thus convection currents form in the center of the melt pool (where temperature is highest) either to or from the outer edges of the melt pool, depending on whether the material’s surface tension increases or decreases with temperature. The inclusion of oxides or any other impurities also has a very strong influence on surface tension, and hence Marangoni convection [76]. Depending on the nature of these convective currents, this can lead to the formation of a shallow and wide melt pool to a narrow and deep melt pool, or somewhere in between. A couple of final considerations to take into account during SLS are the delamination of layers or part warping if process parameters are not applied correctly. Delamination 30

Figure 3.9: Warping of parts due to uneven thermal expansion/contraction [89] of layers occurs due to insufficient bonding between successive layers. This can either be due to poor wetting characteristics in the material, most commonly due to the presence of oxides, or due to insufficient melting. By adjusting the laser energy density such that some of the previously melted layer gets re-melted during each pass, increased bonding between layers will occur and delamination can be avoided. Warping occurs due to thermal gradients in the part and thermal expansion mismatch. When heated the material will want to expand, but may be constrained from doing so due to surrounding material. Conversely, hot material will want to contract when cooling, but may again be constrained from doing so by surrounding material. Both of these scenarios can cause warping in the final part, as depicted in Figure 3.9 [89]. Preheating the chamber and re-melting can relieve these thermal stresses and reduce or even eliminate warping in the final part. Post-process machining or heat treatment may also be used.

3.5 Previous Modeling Attempts

Many researchers have already proposed diﬀerent ways to simulate this process (see Zeng el al. [150] and Schoinochoritis et al. [112]for a very detailed review of many diﬀerent approaches). These approaches can generally be lumped into three main categories:

1. Empirical models based oﬀ experimental results

2. Continuum based models (finite elements, finite differences, finite volumes, etc.)

3. Particle-scale, discrete models

Empirical models are useful in optimizing process parameters for a speciﬁc material and powder size; however, their applications are limited when trying new powders and/or materials. Continuum based models can oﬀer decent part-level simulations but fail to capture powder bed inhomogeneities and the stochastic nature of the process. Particle- scale models can capture more of the physics at the size of individual particles, but can be very computationally expensive. 31

Nelson et al. [94] used empirical data to create a 1D heat transfer model of SLS that can predict sintering depths. Simchi and Simchi and Pohl [120, 121] used empirical results to determine a relationship between energy input and densification during SLS. Kolossov et al. and Dong et al. [59, 19] have created 3D finite element (FE) models for the temperature evolution during laser sintering. The latter model also predicts densification. Matsumoto et al. [86] proposed a FE method for calculating the temperature and stress distribution in a single layer of sintered material. Antony et al. [7] used FEM and experimental analysis to look at SLM of 316L stainless steel powders. Investigation was made into the effects of wetting angle and balling. Illin et al. combined the finite element method with empirical correction factors to determine the melt size and temperature distribution during laser melting [47]. Dai and Gu [17] used a commercial finite volume software to simulate SLM for copper alloys. Effects such as Marangoni convection and entrapped gas bubbles are included in this simulation, which is compared with experiments in predicting resultant part density. Gusarov and Kruth [35] provide an analytical equation for the penetration of a laser into a powder bed as a function of powder bed density and particle size. This work was followed with a finite difference (FD) simulation of heat transfer during SLM [34]. Korner et al. [60] used a modified Lattice Boltzmann approach to produce a 2D model of selective electron beam melting and followed up on this by successively assembling multiple layers using the same approach in a later work [61]. This work showed that loose powder bed density had the biggest effect on final part density and captures the stochastic nature of the powder bed. Khairallah and Anderson [58] use a multiphysics Arbitrary La- grangian Eulerian (ALE) code to fully simulate the SLM process. This approach appears novel in that individual particles are modeled and meshed up in determining the melt pool size. Additionally, the effect of surface tension on melt pool geometry is demonstrated. The main drawback of this approach is the large computational effort required to simulate relatively short time scales. Kovaleva et al. [62] use a discrete element (DE) approach to model individual particles and determine the melt pool size during SLM by summing the total volume of all melted spheres, offering a qualitative depiction of the melt pool. In many of the previously described works, the material was treated as a continuum medium and effective material properties were used. Additionally, due to the difficulty of accounting for localized phase change in FE models, this effect was usually neglected. Empirical models have the drawback of being process and material specific. Meanwhile, particle scale models, such as the one by Khairallah and Anderson [58], are typically extremely computationally expensive. In the following chapter, a reduced-order discrete element model is presented to model the deposition and subsequent laser sintering/melting of the powdered particles. The interaction of a single layer of these DE particles with an underneath substrate is also modeled. The solid substrate is modeled via the FD method. An algorithm for dealing with the change in material properties due to phase changes is presented. This approach has the advantage of eliminating the need of using homogenized effective properties for the powder bed, can capture particle dynamics and the stochastic nature of the powder bed, and yet is reduced order so that computation time does not 32 exceed a few hours (on a laptop computer) for any of the simulations run. This allows for quick optimization of process parameters for different materials and/or powder size distributions. 33

Chapter 4

Powder Deposition and Laser Heating Model Description

A multiphysical modeling approach has been employed to simulate the SLS process for a single layer of particles. A discrete element approach was used to model particle-to- particle and particle-to-wall mechanical and thermal interactions. In this discrete element model, individual powder particles are modeled as discrete, thermally and mechanically interacting spheres as shown in Figure 4.1. Particle to underneath substrate interactions are modeled using the finite difference method for the solid substrate. The temperature of each particle is assumed to be uniform, due to the low Biot numbers and in an effort to reduce computation time. Additionally we assume that the particles are small enough so that the effect of their rotations with respect to their center of mass is negligible to their overall motion. The modeling approach can be characterized in two parts: (1) simulation of the deposition of the powder particles; (2) simulation of the temperature evolution of the particles and underneath substrate after a single pass of a laser beam. The model builds upon a previous work by the author [29] and adapts approaches previously developed in other works by Zohdi [156, 154] and Campello and Zohdi [11].

4.1 Particle Dynamics

A simple model of the deposition of the particles is described in this section. In an eﬀort to simplify the layer deposition process, the authors assume that the particles are being dropped into the domain from a height of approximately 0.3 mm or less. These particles are allowed to settle themselves into a layer by gravity. Starting from Newton’s second law, we can determine an equation for the motion of the i-th particle (starting from a sample of N non-intersecting particles): total con fric env grav mix¨i = Fi = Fi + Fi + Fi + Fi , (4.1) where m represents mass, x represents position, and F represents force. Contact forces con fric env (Fi ), friction forces (Fi ), environmental forces (Fi ), and the force due to gravity 34

Figure 4.1: Discrete element representation of pre-sintered powder particles (SEM image from [151])

grav (Fi ) are considered. con The contact force, Fi , is modeled via a standard Hertzian contact model for intersecting spheres [52]. This theory assumes that the contact area between the particles is small with respect to the dimensions of each particle and with respect to the relative radii of curvature of the surfaces. Additionally, the strains are considered small and within the elastic limit, and the surfaces are considered frictionless. These assumptions can certainly be justiﬁed for the case of elastically loaded, smooth metallic spheres, as is the case in the present work. Note that other contact models exist and may be used depending on the situation, such as the “hard” particle momentum-based model as described by Luding [80]. In the case of Hertzian contact, it follows that if the distance between two particles (i and j) is less than the combined radii of the particles than a particle-to-particle contact force exists: If kxj − xik ≤ Ri + Rj, then 4√ Fcon = − R∗E∗δ3/2n − dδ˙ n , (4.2) ij 3 ij ij ij ij

Nc con X con Fi = Fij , (4.3) j=1 ∗ ∗ where Nc is the number of particles in contact with particle i, R and E are the eﬀective radius and Young’s modulus of the interacting particles given by 35

i j nij

tij

Figure 4.2: Particle-to-particle overlap

R R R∗ = i j , (4.4) Ri + Rj

∗ EiEj E = 2 2 . (4.5) Ej(1 − νi ) + Ei(1 − νj )

Here Ri, Ei, and νi represent the radius, Young’s modulus, and Poisson’s ratio, respectively, of particle i. In Equation 4.2 δij refers to the overlap between particles i and j (see Figure 4.2) given by

δij = |kxj − xik − (Ri + Rj)| . (4.6) ˙ The rate of change of overlap δij is given by

˙ δij = (vj − vi) · nij, (4.7) where nij is the unit normal vector between particles i and j, calculated as

xj − xi nij = . (4.8) kxj − xik Finally the damping parameter d in Equation 4.2 is calculated according to the method of Wellmann and Wriggers [136], where q √ ∗ ∗ ∗ 1/4 d = 2ζ 2E m R δij . (4.9) In this equation, m∗ = mimj is the eﬀective mass and ζ is a damping parameter you mi+mj must set. ζ = 1 is critically damped, ζ > 1 is overdamped, and ζ < 1 is less than critically damped. Note that if ζ = 0, there is no damping force, as in an elastic collision. In the case of particle contact with a wall, the same contact model is used except that we now replace the property values of particle j with those of the wall. In the case of the ∗ eﬀective radius, we take the limit as Rj → ∞, which yields R = Ri. The overlap is given by

δiw = |kxw − xik − Ri|, (4.10) 36

where xw is the shortest distance between the particle center and the plane the wall is located on. The rate of change of overlap is

˙ δiw = −vi · niw, (4.11) assuming the walls are stationary. The normal vector niw is the unit vector orthogonal to the plane of the wall, facing from the particle to the wall. To model friction forces between particle-to-particle or particle-to-wall collisions, we assume a continuous sliding model for simplicity. The friction force is given by

fric con Fij = µd k Fij k tij, (4.12)

Nc fric X fric Fi = Fij . (4.13) j=1

In Equation 4.12, µd is the dynamic coeﬃcient of friction and tij is the unit tangential vector between the particles, given by

vtj − vti tij = , (4.14) k vtj − vti k where vt represents the tangential velocity, vti = vi − (vi · nij)nij. Again the frictional force between the particles and the wall is modeled in the same manner except that the velocity of the wall (equal to 0 in this simulation) replaces the velocity of particle j. Note that a stick-slip friction model could be introduced without too much additional eﬀort, but it was deemed unimportant to the overall simulation as the particles are initially dropped and then settle into an equilibrium position, after which there is no other driving force to cause motion. Stick-slip friction would only apply if there was a driving force to cause motion once the particles have settled into their equilibrium positions. env The environmental force, Fi , comes into play as a force opposing the motion of the particle due to the surrounding environment (i.e. drag force as the particle moves through the ambient environment). This environmental force could also be used to model the eﬀects of lubricant binders which may be placed in the powdered material. Assuming that no binders are present, we approximate this environmental force with Stoke’s drag [93]:

env Fi = −6πµRivi, (4.15) where µ is the dynamic viscosity of the air in the sintering chamber (typically argon or nitrogen gas). Note that Stoke’s flow is a valid approximation only in situations where the Reynold’s number is well below unity, i.e. Re 1. This condition is satisfied in the present situation since the radius of the particles and the velocities experienced will both be small. Recall the Reynold’s number is defined as 37

ρvL Re = , (4.16) µ where ρ is density of the fluid, v is the relative speed of the particle compared to the fluid, and L is the characteristic length, or diameter of the particle. If we assume an argon kg 3 −5 environment, which has a density of ρAr = 1.78 /m and viscosity of µAr = 2.1∗10 P a∗s at standard temperature and pressure [77], and that L = 50 µm (the max diameter of particles in this simulation), we obtain Re ≤ 0.33 which is in the range appropriate for Stoke’s flow. Note that 0.33 is a worst case scenario as the speed, v = 0.078 m/s, was calculated as if an object was falling 0.3 mm with no drag force (i.e. greater than the highest possible speed in this simulation). In reality the Reynold’s number will be smaller since most particles will be smaller than 50 µm and will have lower speeds. grav Finally there exists the gravitational force, Fi , which is the driving force causing the particles to settle in a layer. The gravitational force acts in a downwards, or −z, direction and is given by

grav Fi = −migzˆ, (4.17) where zˆ is the unit vector in the z direction and g = 9.8 m/s2 is the acceleration due to gravity.

4.2 Particle Thermal Eﬀects

Assuming a lumped capacitance model where the temperature ﬁeld is uniform throughout each particle (consistent with the low Biot numbers resulting from the small particle sizes), the thermal governing equation in integral form is [49]:

Q · ndA + HdV = ρcT,˙ (4.18) ˆ∂Ω ˆΩ ˆΩ where Q represents heat transfer, H is a source term to account for laser heat input, ρ is density, c is the constant pressure specific heat capacity of the material, T is temperature, and Ω represents the domain of interest. Again recall the Biot number is defined as h L Bi = conv , (4.19) K where hconv is the convective heat transfer coefficient, L is a characteristic length (the diameter of the particle), and K is the thermal conductivity of the particle. Assuming W 2 that hconv = 40 /m K (an appropriate value for this situation [49]), L = 50 µm, and K = 32.4 W/m∗K (the maximum thermal conductivity for steel at the melting point [124]), we obtain a Biot number of Bi ≤ 6.2 ∗ 10−5 which is well below the required value of Bi = 0.1 that is typically used when determining the validity of a lumped capacitance thermal model. 38

Laser irradiance

i j Q

Figure 4.3: Heat transfer to an individual particle

4.2.1 Particle-to-particle heat transfer Discretizing Equation 4.18 over each particle (see Figure 4.3) we obtain

˙ Qi + Hi = miciTi. (4.20) If we then break down the heat transfer term in Equation 4.20 into it’s conductive, convective, and radiative parts we get

cond conv rad Qi = Qi + Qi + Qi , (4.21)

Nc X Tj − Ti 4 4 Qi = K Aij + hconv(Ti − Tenv)Ai + σSB(Ti − Tenv)Ai. (4.22) j=1 kxj − xik

In Equation 4.22 Nc represents the number of particles in contact with particle i, K is the thermal conductivity of the material (assuming it is isotropic), Aij is the contact area between particles i and j, hconv is the convective heat transfer coeﬃcient, is the −8 W 2 4 material emissivity, σSB is the Stefan-Boltzmann constant (σSB = 5.67 ∗ 10 /m K ), 2 and Ai = πRi is the area of the particle facing the surrounding environment. Note that convective and radiative heat transfer is assumed to only occur for exposed particles on the top surface. Thus these particles will exchange heat with the environment, which is at a temperature of Tenv. Convective and radiative heat transfer is assumed to cancel out for individual particles within the powder bed and is thus not considered for particles not on the top, exposed, layer. Additionally, conduction is only assumed to occur between contacting particles. Particle-to-gas conduction in the gaps between particles is not considered due to the much lower thermal conductivity of the gas environment (argon in this simulation). Thermal expansion of the particles is not currently considered. For two contacting particles the contact area (see Figure 4.4) is given by

2 2 2 !!!2 2 Rj − Ri − d Aij = πh = π Risin arccos , (4.23) −2dRi where d = kxj − xik. The law of cosines is invoked in this derivation. 39

Ri R h j i j d

Figure 4.4: Contact area of two intersecting particles

4.2.2 Laser beam modeling

Assuming a Gaussian laser is used, the laser heating term, Hi in Equation 4.20, is given as

Hi = αI(r, z)Ai, (4.24) where α is the absorptivity of the material at the wavelength of the laser (0 < α < 1), I(r, z) is the laser intensity [W/m2] as a function of radial distance (r) and depth (z) from 2 the beam center, and Ai = πRi is the area of the particle receiving direct radiation from the laser. The laser beam is assumed to be Gaussian (as is typically the case in the SLS/SLM process) so that the intensity decreases exponentially as you increase the radial distance from the beam center (see Figure 4.5). A Beer-Lambert type model for the laser penetration is used, where the intensity is assumed to decrease exponentially as a function of penetration depth. Thus the intensity can be expressed as

−βz (−2r2/w2) I(r, z) = I0e e , (4.25) 2∗P ower where I0 = πw2 is the peak intensity, w is the beam spot size measured to where the intensity falls to 1/e2 of the peak intensity, and β is the optical extinction coefficient [41]. In continuous media, the optical extinction coefficient is directly related to the imag- inary portion of that material’s index of refraction, k, and the wavelength of incident radiation, λ: 4πk β = . (4.26) λ Note that k is a function of wavelength. In the case of metals, the penetration depth is on the order of nm [108, 104], which is smaller than any length-scale present in this work. However, the laser energy will penetrate much deeper in a powder bed than a solid material due to the multiple reflections that take place off the powder particles. Multiple approaches exist for modeling laser penetration into a powder bed. One option is to model the physical action of the laser beam hitting the surrounding particles using a ray tracing 40

Figure 4.5: Gaussian laser beam depiction or geometrical optics approach. Geometrical optics is applicable when the characteristic interaction length between the light waves is at least an order of magnitude larger than the wavelength of the light waves[148]. This approach represents the light waves as a series of rays (see Figure 4.6). These rays then refract, reflect, transmit, and get absorbed as heat depending upon material values and the wavelength of the light [40]. However, care should be used in this situation since carbon dioxide lasers emit on vibrational transitions of the CO2 molecule in the infrared region between 9 and 11 µm. Nd:YAG or fiber lasers emit at approximately 1.06 and 1.08 µm [41]. Since the interstitial gaps between the micron- sized particles used in SLS will clearly be less than an order of magnitude larger than the wavelength of these three most frequently used lasers, a ray-tracing or geometric optics approach will not be entirely valid. Full-scale modeling of the laser can be performed via solving Maxwell’s equations since lightwaves are truly just oscillating electromagnetic waves [40]. However, this approach is very computationally expensive and thus currently not deemed necessary for this application. An alternative approach for estimating the optical extinction coefficient in a metal powder bed is proposed by the theory of Gusarov et al. [34, 35]. In this theory the extinction coefficient for a homogeneous absorbing powder is given as a function of the powder bed porosity, γ, and the particle diameter, D. This theory has been shown to agree with ray tracing simulations in the limit of a deep powder bed [35] and is given as

3(1 − γ) β = . (4.27) 2γD

4.2.3 Phase change There are a couple diﬀerent methods to numerically account for the energy needed for phase change to occur [9, 92, 154]. One is the latent heat accumulation method wherein the temperature of a particle is ﬁxed once it reaches the melting (or vaporization) temperature. The amount of energy added to the particle over subsequent time steps is stored. 41

Figure 4.6: Laser beam illustration as a series of light rays (length of each ray corresponds to intensity). Note that this approach is not used in the current work.

Once enough energy is added to overcome the latent heat of melting (or vaporization), the particle is considered to have completed its phase change and the temperature is allowed to continue increasing. An alternative method is referred to as the apparent heat capacity method. In this technique, the energy needed for phase change to happen is taken into account by computationally raising the speciﬁc heat of the material in a small region around the melting (or vaporization) temperature. In this work, the apparent heat capacity method is used, similar to the approach used by Muhieddine et al [92] and illustrated clearly in the work by Rey [104]. The algorithm is outlined as follows and can be graphically seen in Figure 4.7:

c , T < T − ∆T  solid m 2  c +c L  solid liquid + m/s ,T − ∆T ≤ T ≤ T + ∆T  2 ∆T m 2 m 2  ∆T ∆T c = cliquid,Tm + < T < Tv − (4.28)  2 2  cliquid+cgas Lv/c ∆T ∆T  2 + ∆T ,Tv − 2 ≤ T ≤ Tv + 2   ∆T cgas, T > Tv + 2 where Tm is the melting temperature, Tv is the vaporization (or boiling) temperature, Lm/s is the latent heat of melting or solidiﬁcation, and Lv/c is the latent heat of vaporization or condensation. Note that other material properties (such as thermal conductivity, density, Young’s modulus, etc.) are also updated as a function of temperature and phase. 42 )

) Theoretical curve csolid+ Lm/T c T Implementation

A B C E - gas D - liquid/ gas c

liquid ( Temperature C - liquid c solid T B - solid/ liquid

Specific Heat Capacity ( T A - solid T Temperature (T) m Heat Input

Figure 4.7: Illustration of how heat capacity is updated (left) and ensuing phase change diagram (right)

4.2.4 Thermal softening and melting of particles As the particles heat up they become softer, which is taken into account by updating the Young’s modulus as a function of temperature according to the specific material property data. Decreasing the Young’s modulus will serve to increase the particle-to-particle overlap and increase the contact area and hence heat conduction between particles, as would be expected. However, the problem arises as to how to treat molten particles, which would be much softer and flow through pores between the solid particles. A few different techniques could be employed in this situation [155]. Namely,

1. Break up molten particles into many individual “ﬂuid” particles, which would be governed by diﬀerent constitutive laws.

2. Keep particles as they are but assign molten particles a much decreased stiﬀness to simulate the much “softer” ﬂuid.

3. Treat the molten particles as a continuum and simulate via a hybrid discrete element- continuum approach.

Option 1 may offer qualitatively realistic simulations but would be extremely computationally expensive. Option 2 is more of an ad-hoc approach wherein a decreased stiffness would allow for accurate heat transfer modeling but not realistic simulations of fluid flow upon melting. Option 3 may also be realistic but would also require a significant increase in computational effort, and to the author’s knowledge, has not been done in the case of phase change. Galindo-Torres [28] and Wellman and Wriggers [136] have demonstrated the feasibility of coupling discrete element particles with a continuum model for modeling fluid flow around particles and large scale granular materials, respectively. However, nei- 43 ther of the mentioned works incorporated the immediate switch from a discrete element particle to a continuum method as would need to be the case for melting. In this work, the author has decided to employ approach 2 as this requires the least additional computational expense and serves the purpose of modeling heat transfer, if not fluid flow. Thus, once particles melt (are above the melting temperature), the Young’s modulus of the particle is significantly reduced so as to increase the contact area between the molten particle and any neighboring ones. As this is a more ad hoc approach, it was necessary to figure out exactly how “soft” the molten particles should become to model heat transfer effectively. To this end a few observations must be made:

• Heat transfer between two particles is proportional to the contact area of those particles (Equation 4.22).

• Contact area is a function of particle-to-particle overlap (Equation 4.23).

con grav • From Equation 4.1 we can state that in equilibrium 0 = F + Fi or √ i PNc 4 ∗ ∗ 3/2 ˙ j=1 − 3 R E δij nij − dδijnij = migˆz. Understanding that E∗ must be decreased to increase the contact area, knowing that the ﬁnal cooling rate should be on the order of experimentally observed value of 106oC/s [67], and only considering conductive heat transfer (Equations 4.20 and 4.22), one can calculate the necessary Young’s modulus for liquid particles. Note that other researchers have similarly used an artiﬁcially decreased Young’s modulus to simulate porous media [86].

4.3 Finite Diﬀerence Modeling of Substrate

The ﬁnite diﬀerence method is employed to model heating of the solid substrate underneath the powder particles. To represent conduction within the substrate we can recast Equation 4.18 as

∂ ∂T ! ∂ ∂T ! ∂ ∂T ! ρcT˙ = ∇ · (K∇T ) + H = K + K + K + H, (4.29) ∂x ∂x ∂y ∂y ∂z ∂z

Using a centered diﬀerence approximation for the partial derivatives in space we obtain

∂ ∂T ! 1 K(x + ∆x, y, z, t) + K(x, y, z, t)! T (x + ∆x, y, z, t) − T (x, y, z, t)! K ≈ ∂x ∂x ∆x 2 ∆x 1 K(x, y, z, t) + K(x − ∆x, y, z, t)! T (x, y, z, t) − T (x − ∆x, y, z, t)! − . (4.30) ∆x 2 ∆x 44

A(i,j,k+1) A(i,j+1,k) z y A(i+1,j,k) x A(i-1,j,k) A(i,j,k)

A(i,j-1,k) A(i,j,k-1)

Figure 4.8: Finite diﬀerence stencil and coordinate system used for an arbitrary material property A. Indices i, j, and k are used to represent the x, y, and z coordinates, respectively.

∂ ∂T ∂ ∂T We can discretize ∂y K ∂y and ∂z K ∂z in an analogous manner. An illustration of the general finite difference stencil is shown in Figure 4.8. To solve Equation 4.29 using the FD scheme, it is necessary to prescribe the appropriate boundary conditions on the faces. Since the domain is much larger than the spot size of the laser beam, fixed Dirichlet boundary conditions (i.e. T (0, y, z, t) = T (Lx, y, z, t) = T (x, 0, z, t) = T (x, Ly, z, t) = T (x, y, 0, t) = T0) were prescribed on all side faces and the bottom face of the substrate mesh as shown in Figure 4.9. A Neumann, flux-type B.C. was prescribed on the top face by calculating heat transfer from the particles to the substrate. This heat transfer amount (in units of W ) was divided by the area of one grid square on the top layer of the FD mesh to calculate the flux (W/m2) experienced by those mesh points. Each nodal point is assumed to receive 1/4 of the total heat flux experienced by the square enclosed by those points (see Figure 4.9). Thus the boundary condition for the

∂T top face can be expressed as K ∂z = qi,j where qi,j is the portion of flux hitting nodal Lz point (i, j) on the top face of the substrate. The derivative term is approximated using the second order accurate, 3 point finite difference stencil of the form

∂T 3T (x, y, L , t) − 4T (x, y, L − ∆z, t) + T (x, y, L − 2∆z, t) z z z . (4.31) ∂z ≈ 2∆z Lz The same system of updating the speciﬁc heat to account for phase changes is used in the FD substrate as well (see Equation 4.28). However, in this instance the density is also updated when the material changes phase from a solid to a liquid (and potentially even to a gas). Note that the thermal conductivity is also updated to the liquid value after phase change. Also note that heat capacity, thermal conductivity, Young’s modulus, and density are updated as a function of temperature within the solid phase. 45

Q [W]

Prescribed flux from DEM particles on top face

Laser (i,j+1) (i+1,j+1) q = Q/(∆x*∆y) ∆y 2 L Ly [W/m ] z y z (i,j) ∆x (i+1,j) x Lx

qi,j=qi+1,j=qi,j+1=qi+1,j+1=1/4*q T=T0 (on all faces except top)

Figure 4.9: Depiction of boundary conditions used for the ﬁnite diﬀerence mesh: all B.C.s (left), Neumann B.C. for top face (right)

4.4 Numerical Solution Scheme

4.4.1 Time-stepping To solve the coupled mechanical-thermal system, purely explicit time stepping schemes were utilized as they provided enough accuracy and efficiency for the current problem. Note that implicit schemes could easily be employed as well, with the advantage of increased stability regions allowing one to take larger time steps. However, implicit schemes require either iterations within each time step or large matrix inversions (for 3D FD schemes), which also increase computation time; thus creating a trade-off. Equation 4.1 was solved for the new position of each particle by decomposing this 2nd order differential ( ) x˙ i equation into a series of first order equations. If we define Xi = , then we can xi recast Equation 4.1 as

( 1 total ) m Fi X˙ i= i = f(Xi, t), (4.32) x˙ i where x˙ i and xi represent the velocity and position components of the i-th particle, respectively. This ﬁrst order, vector system of equations, was marched forward in time using a fourth order explicit Runge-Kutta scheme [72] as follows

Y1 = Xi(t), 46

1 Y = X (t) + ∆tf(Y , t), 2 i 2 1 1 ∆t Y = X (t) + ∆tf(Y , t + ), 3 i 2 2 2 ∆t Y = X (t) + ∆tf(Y , t + ), 4 i 3 2 ∆t ∆t ∆t X (t+∆t) = X (t)+ [f(Y , t)+2f(Y , t+ )+2f(Y , t+ )+f(Y , t+∆t)]. (4.33) i i 6 1 2 2 3 2 4 During the deposition process, only the mechanical equations required solving as the temperatures of each particle were considered ﬁxed. Temperature rise due to collisions and small deformations was assumed to be negligible and thus not considered (indeed during the sintering/melting phase, energy input from the laser dominates all other heat transfer mechanisms). After deposition of a layer of particles, their positions were assumed to be ﬁxed as a laser passed over the domain. At this time the thermal equations (Equations 4.20 and 4.29) were solved for each particle and FD node. A standard Forward Euler time-marching scheme was used for both the particles and FD nodes where ˙ Ti(t + ∆t) ≈ Ti(t) + ∆t · Ti(t), (4.34) for each particle and

T (x, y, z, t + ∆t) ≈ T (x, y, z, t) + ∆t · T˙ (x, y, z, t), (4.35) for each nodal point. The overall solution algorithm is depicted in Figure 4.10. Note that the RK-4 method was used for position since the standard Forward Euler scheme required significantly smaller time steps for stability. Thus RK-4 was actually quicker than Forward Euler even though it requires four evaluations each time step, as compared to one. This has to do with the stability region of the problem. More details regarding these numerical differentiation schemes can be found in Appendix B. Finally, it should be noted that the FD formulation presented is second order accurate in space and first order accurate in time. However, due to the stability conditions for such explicit schemes [72], the time step is proportional to the square of the grid spacing, or ∆t ∝ (∆x)2. Thus the time step must be divided by four every time the grid increment is halved. Since there are three spatial dimensions, if the grid increment in each direction is halved, this equates to a 32 fold increase in computation time. Thus, clearly an optimal trade-off exists between using small enough grid increments to accurately capture the physics of the problem, while simultaneously not requiring an inordinate amount of computation time. The convergence of this scheme was checked by calculating the same problem with grid increments continuously halved until no significant difference in results was obtained. 47

Model"deposi(on"and""hea(ng/" cooling"of"par(cles"

Calculate" Solve"for"new" Drop"par(cles" forces"on"each" Determine" par(cle" into"domain" par(cle"" laser"posi(on" posi(on"" (""""""""""""")" F = mx

Calculate"heat" Solve"for"new" Calculate" Update" transfer"to"each" par(cle"and" heat"transfer" material" par(cle"(energy" nodal" to"substrate" proper(es" balance)" temperatures""

Figure 4.10: Simulation ﬂow chart

4.4.2 Binning and OpenMP Parallelization 1" To decrease computational time, a binning algorithm is implemented. A basic check for interactions between N particles that could potentially be interacting with any other particle would require O(N 2) operations (check each of N particles to see if they are interacting with N other particles). Binning enables one to reduce the number of computations necessary from O(N 2) to O(N). This works by decomposing the domain into grid boxes, or bins (see Figure 4.11). Particle interactions only have to be checked within the same or neighboring bins. In three dimensions this equates to checking interactions within 27 bins (including the particle’s own bin). While this may seem like a lot, it is considerably less than having to check every particle. Clearly as the bin size decreases the number of checks for each particle also decreases. However, care must be taken to not let the bins get too small, otherwise some interactions may be missed. One problem that arises when decomposing the domain into bins is the issue of particles moving from one bin to another as system time evolves. Thus it is necessary to recompute each particle’s location and corresponding bin at the beginning of each time step. However, if large enough bins are used then it may only be necessary to “re-bin” the domain every few time steps, rather than every time step. A balance between using larger bins to eliminate the need for “re-binning” every time step and using smaller bins to reduce the number of particle checks must be reached. This often is a function of the dynamics of the problem at hand. In the case of this simulation, the bin length was chosen to be twice the diameter of the largest particle. Note that cube shaped bins were used in this simulation, though this is not a requirement. More details regarding the implementation of such an 48

Figure 4.11: Illustration of binning algorithm. Only particles in the shaded boxes will be checked for contact with the red particle in the center box. algorithm and other variations can be found in works by Perkins and Queiruga [100, 103]. To further decrease computational time, parallelization across multiple cores can be utilized. In the majority of this work, simulations could be run within a matter of minutes to hours on a laptop computer since the program was written in a high performance language (Fortran 90 in the present case). However, in a few instances of larger simulations (those described in Chapter 6), OpenMP parallelization was employed to utilize all available cores on the laptop computer or available computing cluster. OpenMP parallelization is easily implemented for discrete element and finite difference schemes using explicit time-stepping, as done in this work. More details regarding OpenMP and other parallelization techniques can be found in [37] and are briefly discussed in Appendix C.

4.5 Programming Algorithm

All programming was performed in Fortran 90 and for the most part simulations were carried out on a laptop computer. A pseudocode for the particle deposition process is presented in Algorithm 4.1. The laser heating pseudocode is given in Algorithm 4.2. 49

Algorithm 4.1 Particle deposition pseudocode 1. Assign number of particles and mean, standard deviation, min, and max particle diameters

2. Assign material values for each particle

3. Randomly generate position coordinates for the given particle size distribution 0.2 to 0.3 mm above the bottom surface of the domain

4. Check to ensure no particles overlap (if there is overlap, regenerate that particle’s starting coordinates)

5. Break domains into bins and assign a bin number to each particle based on position

6. Set initial velocity of each particle equal to 0

7. Begin time loop (for t < tend) - here tend is based upon time that will be required for particles to settle

(a) Determine external forces on each particle (i.e. forces due to gravity and environmental forces) (b) Check for particle-wall contact i. Only check if particle bin is neighboring a wall ii. If contact, determine contact and friction forces appropriately and store this information (c) Check for particle-particle contact i. Only check for contact between particles in the same or neighboring bins ii. If contact, determine contact and friction forces appropriately and store this information (d) Sum all force components acting on each particle (e) Calculate acceleration felt by each particle (f) Update velocity and position of each particle using RK-4 method (requires 4 iterations within each time step) (g) Update particle bin based on new position

8. Output position and particle size data to plotting ﬁles 50

Algorithm 4.2 Laser heating pseudocode 1. Read in particle position and size data from particle deposition output file 2. Assign initial temperature and material values to each particle and nodal point in finite difference mesh of substrate 3. Set laser position to be initially just outside of domain, specify laser power and speed

4. Begin time loop (for t < tend) - here tend is based upon time it will take for laser to make one pass through the domain (a) Update laser position (b) Update material properties for each particle and nodal point as a function of temperature and phase (c) Calculate forces on each particle as stiffness changes due to thermal softening (same steps as described in particle deposition pseudocode) (d) Calculate particle heat transfer i. Check for contact between particles in the same or neighboring bins ii. Calculate conduction between particles iii. Calculate laser energy absorbed by each particle depending on position and height (z-coordinate of each particle) iv. Calculate heat transfer due to convection and radiation for particles on top surface (e) Calculate heat transfer through substrate i. Loop through interior nodes ii. Calculate conductive heat flux iii. Determine laser energy input to top layer of FD mesh depending on layer height of particles on top of substrate (all laser energy which passes through the particles is absorbed as heat in the top layer of the FD mesh) (f) Calculate particle-to-substrate heat transfer and apply as boundary condition i. Calculate conductive heat transfer from particle to substrate and vice versa ii. Add this to the conduction terms for particles touching substrate and for all FD nodes contacting a particle (g) Sum all heat transfer mechanisms to determine rate of temperature change (T˙ ) for all particles and nodes (h) Update particle position and velocity using RK-4 time stepping, nodal and particle temperatures using Forward Euler 5. Output temperature, phase, and position data to plotting files 51

Chapter 5

Powder Deposition and Laser Heating Model Validation and Results

5.1 Material Properties and Parameter Values

Simulations were carried out assuming powder particles and a substrate made of 316L stainless steel (316L SS), a commonly sintered metal. The initial preheat temperature of the powder bed and substrate is set to be 90 ◦C. The specific heat, thermal conductivity, and density values of 316L SS as a function of temperature are given in Table 5.1. Note that these values are linearly interpolated at temperature values in between the specified points until 1400 K, after which the property values are fixed until phase change at 1700 K. Other material properties and simulation parameters used are provided in Table 5.2. Note that a time step size of ∆t = 2 ∗ 10−8 seconds is used when solving the mechanical equations during the deposition process and a time step of ∆t = 5 ∗ 10−9 seconds is used when solving the thermal and mechanical problem as the laser is scanned over the powder bed. These time steps were chosen such that the simulation remained stable and the results did not change when the time step was further reduced. The thermal time step of ∆t = 5∗10−9 is limited by the finite difference mesh, whereas the time step of ∆t = 2∗10−8 is adequate for the DEM-only portion of the simulation during the deposition process. The grid spacing for the finite difference mesh of the substrate is ∆x = ∆y = ∆z = 5 µm, which was chosen such that the results did not significantly change when further refined.

5.2 Model Validation

To validate the model described in the previous chapter, numerical simulations were carried out to recreate experiments conducted by Khairallah and Anderson [58]. Namely, the melt pool size predicted via this model was compared against the actual melt pool size 52

Temperature, Speciﬁc Heat, Thermal Density, Young’s T (K) c (J/kg∗K) Conductivity, ρ (kg/m3) Modulus, K (W/m∗K) E (P a) 293 452 13.3 7952 198*109 366 485 14.3 7919 194*109 478 527 15.9 7877 185*109 589 548 17.5 7831 177*109 700 565 19.0 7786 167*109 811 573 19.8 7739 157*109 922 586 21.9 7692 148*109 1033 615 23.2 7640 137*109 1144 649 24.6 7587 129*109 1255 690 26.2 7537 120*109 1700 (liquid) 815 32.4 7300 1700

Table 5.1: Material properties for 316L stainless steel as a function of temperature and phase [48, 124]

Parameter Symbol Value Poisson’s ratio ν 0.26 Damping parameter ζ 0.1 Dynamic friction coeﬃcient µd 0.1 W 2 Heat transfer coeﬃcient hconv 40 /m K Melting temperature Tm 1700 K Boiling temperature Tv 3130 K 5 J Latent heat of melting Lm 2.99 ∗ 10 /kg 6 J Latent heat of vaporization Lv 6.09 ∗ 10 /kg Material emissivity 0.33 Material absorptivity α 0.33 Powder bed porosity γ 0.55 Laser power P ower 200 W Laser scan speed vlaser 2.0 m/s Laser spot size w 54 µm

Table 5.2: Remaining material properties and simulation parameters used [48, 2] 53 during a single pass of a Gaussian laser over a single layer of powder particles sitting on a substrate. A Gaussian particle distribution is used with average diameter of 27 µm and a half-max-width of 10 µm. The particles are dropped into 1 mm x 0.5 mm cross-sectional domain from a height of less than 0.3 mm and allowed to settle via gravity to simulate the deposition process. 600 total particles are used in this simulation, yielding a layer height of approximately 30 µm as specified in [58]. The particle packing density is calculated around 45%. These particles sit on top of a substrate with dimensions of 1 mm x 0.5 mm in cross-section and a depth of 0.3 mm. A grid spacing of ∆x = ∆y = ∆z = 5 µm yields 201 x 101 x 61 grid nodes for the finite difference mesh. Figure 5.1 shows a depiction of the simulation melt pool size vs. the actual melt pool size in the experiments by Khairallah and Anderson [58]. The experimental melt height, width, and depth are 26 µm, 75 µm, and 30 µm, respectively. The simulation melt pool dimensions are 30 µm, 85 µm, and 20 µm. These results show pretty good agreement with the experiments and are also close to the simulations carried out by Khairallah and Anderson [58] of this same experiment where they predicted a melt pool height, width, and depth of 26 µm, 72 µm, and 20 µm. The main source of error is believed to lie in the fact that the current simulation does not take into account consolidation mechanisms upon melting. Namely, one would expect the melt pool to be slightly lower in height than the original powder layer as the melted liquid coalesces and densifies, which would explain why the experimental melt height is less than the initial powder layer height of 30 µm. Additionally, Marangoni convection due to surface tension differences in hot and cold regions would serve to transfer more heat to the substrate and could explain the increased depth of the experimental melt pool. If the surface tension increases with temperature, and is higher in the center of the melt pool, this causes flow of material from outer regions of low surface tension towards the center where there is increased surface tension. This will cause the formation of a narrow and deep melt pool, which could be a significant factor in the increased depth and decreased width of the experimental melt pool as compared to the simulation. Limmaneevichitr and Kou [76] showed the propensity for Marangoni convection to form such a melt pool in steels under certain conditions, especially as beam spot size decreased. Additional sources of error could include the changing absorptivity of the material as it heats up and then melts (typically liquid metals are more reflective). However, the current simulation results are at least qualitatively similar to the experimental results, indicating the validity of this model as a quick tool to predict melt size, which can be extremely useful when determining optimal process parameters (i.e. hatch spacing, layer thickness, laser power, scan speed, powder size, etc.). It should be specified here that this simulation was carried out within a few cpu hours on a MacBook Pro laptop computer. Comparatively, the massively parallelized simulations of this experiment run by Khairallah and Anderson consume on the order of 100,000 cpu hours [58]. 54

Figure 5.1: Comparison of experimental melt pool size by Khairallah and Anderson (left) [58] and simulation melt pool size depicted in red (right). Note that the ﬁgure on the left is Figure 5(a) from [58]. The experimental melt height, width, and depth are 26 µm, 75 µm, and 30 µm, respectively. The simulation melt pool dimensions are 30 µm, 85 µm, and 20 µm.

5.3 Numerical Examples

To fully illustrate this model, we consider the deposition and subsequent heating of a layer of 316L SS particles sitting on a substrate of the same material. Figure 5.2 shows screenshots depicting the deposition of these particles. The particle coordinates were randomly generated at a height of 0.2 to 0.3 mm. The particles fell into the domain and were allowed to settle for 0.1 seconds, after which they had essentially stopped moving. Next a laser was passed over in a single pass. Figure 5.3 shows screenshots of the temperature evolution of the particles and underlying substrate during the laser pass. The melt pool created is shown in Figure 5.4. The laser power is 200 W and scan speed is 2.0 m/s. All other parameters are identical to the ones provided in Tables 5.1 and 5.2. In Figures 5.3 and 5.4 notice how part of the underlying layer gets re-melted as the laser passes over (recall 1700 K is the melting temperature). This is desirable to ensure proper bonding between layers during the SLS/SLM process. Figures 5.5 and 5.6 depict how the melt pool size (depth and width) changes as the laser power and scan speed are varied, respectively. Note that the melt depth plotted refers to the depth that the underlying substrate layer that gets melted. It is necessary for some re-melting of the previous layer to occur for proper bonding between layers (indicated by the melt depth). The melt pool width can give a good indication of the optimal hatch spacing, or distance between laser passes. Again, some melt pool overlap between adjacent laser passes is necessary for proper bonding and a smoother surface ﬁnish. Finally, the laser power and scan speed can be optimized to reduce total processing time and use as little energy input 55

Figure 5.2: Screenshots depicting the deposition of a layer of 316L SS particles (600 particles total) as possible, while still producing usable parts. An additional advantage of this model over conventional continuum type models is the ability to capture inhomogeneities in the powder bed layer and the ability to simulate different powder size simulations. Table 5.3 shows how different powder size distributions can affect the density of the loose powder bed. Recall Korner et al. [60, 61] found that the loose powder bed density had the greatest effect on final part density during simulations and experiments run on selective electron beam melting. From Table 5.3 it can be seen how the loose bed density1 increases as the spread of the powder size distribution increases. This makes intuitive sense as one would expect smaller particles to fill in the voids between larger particles. Additionally, the last row of Table 5.3 shows the density of a bimodal powder distribution. In this distribution, smaller particles (with mean diameter of 8 µm) are mixed with larger particles (with a mean diameter of 30 µm). This bimodal distribution represents the largest loose bed density, as should be expected. While this bimodal distribution clearly increases the loose bed density, one would expect a bimodal distribution with even smaller particles to only further increase this loose bed density and hence yield higher final part densities. However, as the smaller particles get interspersed with larger particles, these smaller particles may start to vaporize before the larger particles melt when a laser is scanned over the bed. These vaporized particles may get trapped as gas bubbles in the final part which will decrease the final density and strength. Thus an optimal size distribution must exist to maximize loose bed density while minimizing the vaporization of smaller particles. Indeed in Figure 5.7, this exact situation is simulated for a 30 µm layer of 316L SS particles. In the left image of this figure a normal distribution of particles with mean diameter of 30 µm and standard deviation of 5 µm is used (third row of Table 5.3). 550 of these particles are deposited in a layer and a single pass of the laser is scanned over it. The phase of the particles are represented by color

1The powder bed density is calculated by adding the volume of all portions of all particles located within a 30 µm layer from the bottom substrate and then dividing by the volume of this layer (the cross section of the layer is 1000 µm x 500 µm and height is 30 µm). 56

T [K]

Figure 5.3: Screenshots showing the temperature evolution of a layer of 316L SS particles and the underlying 316L SS substrate as a laser is passed over (temperature in Kelvin). Top view on upper row. Cross-sectional view from the side on bottom row.

Figure 5.4: Screenshots depicting the melt pool (red) of a layer of 316L SS particles and the undelying 316L SS substrate as a laser is passed over. Top view on upper row. Cross-sectional view from the side on bottom row. 57

Constant Scan Speed (2 m/s) 140 Melt Width 120 ) Melt Depth

μm 100

40 Melt Pool Size ( 20

0 0 50 100 150 200 250 300 350 400 450 Laser Power (W)

Figure 5.5: Melt pool size as laser power is varied from 40 - 400 W , scan speed is constant at 2.0 m/s

Constant Laser Power (200 W) 140

120 Melt Width ) Melt Depth μm 100

40 Melt Pool Size ( 20

0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Scan Speed (m/s)

Figure 5.6: Melt pool size as scan speed is varied from 0.4 to 4.0 m/s, laser power is constant at 200 W 58

No. Particles Mean Diameter Std. Dev. Min (µm) Max (µm) Powder (µm) (µm) Density 550 30 1 10 50 46.4 % 550 30 3 10 50 48.6 % 550 30 5 10 50 51.4 % 550 30 7 10 50 52.4 % 550 30 10 10 50 54.0 % 550, 3000 30, 8 5, 2 20, 4 40, 12 55.5 %

Table 5.3: Affect of powder size distribution on loose bed density, last row represents a bimodal distribution where blue represents solid, green represents liquid, and red represents vaporized particles. Notice how none of the particles are vaporized in the scenario where a mono-modal particle size distribution is used (left image), as desired. Now look at the right image of Figure 5.7, which represents a bimodal distribution of particles. This distribution is the same as that indicated in the final row of Table 5.3. Notice here how some of the very small particles are red in color, indicating they have vaporized. Due to the rapid cooling and solidification rates present in the SLS/SLM process, these vaporized particles may be trapped as gas bubbles in the solidified part and decrease the final density. Alternatively, these ablated particles could increase the surface roughness of that layer and cause inhomogeneities in layer thickness if this issue persists over multiple layers. Obviously, one could prevent ablation of the small particles by decreasing the laser energy density (i.e. decrease power or increase scan speed); however, a decrease in input energy would prevent the underlying layer from being re-melted which would inhibit proper bonding between successive layers. Thus it can be stated that an optimal powder size distribution exists for a given material to maximize loose powder bed density while simultaneously minimizing ablation of the small particles. Additionally, it should be mentioned that all the process parameters and material properties can easily be changed in the simulation code. This can be very helpful when trying to quickly determine a window of optimal process parameters for new materials. Additionally, the DE model of the powder particles enables simulations for varying initial powder bed densities, powder sizes, and layer thicknesses. 59

Figure 5.7: Melt pool from a single laser pass over a mono-modal (left) and bimodal (right) powder distribution. Note that solid particles are blue, molten ones are green, and gaseous ones are red. Notice how some small particles are vaporized in the bimodal distribution. Laser power is 200 W and scan speed is 2.0 m/s. All parameters between the two simulations are identical other than particle size distribution. 60

Chapter 6

Residual Stress Modeling

In the previous sections a methodology was presented for (1) simulating the deposition of a layer of powder particles and (2) the subsequent heat transfer through the particles and underlying substrate as a laser is scanned over the powder bed. Such information is useful to quickly optimize process parameters during the SLS/SLM process for diﬀerent materials, lessening the need for costly and time consuming experiments. However, while the previously described simulations give information about the powder bed heating and melt pool size, no considerations were given to the potentially large residual stresses that are present after cooling. In fact, large residual stresses due to the high cooling rates (up to 106 ◦C/s) present in SLS/SLM can lead to microcracking if proper process parameters are not used [135]. In this section, background information regarding residual stress formation and modeling is presented, followed by a methodology used by the author to simulate thermal stresses that arise in a rapidly cooling part.

6.1 Background

Residual stresses refer to the stresses in a body which are not necessary to maintain equilibrium between the body and its environment. According to Withers and Bhadeshia, they may be categorized into three distinct types according to the length scale over which they act [141, 142]:

1. Type I residual stresses are macroscopic and vary continuously over large distances. They may be caused by the non-uniform plastic deformation of a bent bar or from sharp thermal gradients, such as those that may be present during welding or heat treatment operations.

2. Type II residual stresses are inter-granular stresses. These primarily exist in poly- crystalline materials due to mismatch of elastic and thermal properties of diﬀerently oriented neighboring grains. This is more prevalent when the microstructure contains several phases or when phase transformations occur. 61

3. Type III residual stresses are at the atomic scale and include stresses due to co- herency at interfaces and dislocation stress ﬁelds.

Typically Type I residual stresses are most commonly mentioned and reported since they are the easiest to measure and have the most significant effect on the strength of a material [141, 142]. These stresses occur due to the high cooling rates and thermal gradients present in the SLS/SLM process as molten material cools down and is constrained from thermally contracting due to surrounding solid material [122]. Many researchers have investigated techniques for measuring and/or modeling the residual stresses present in parts after the SLS/SLM process. Matsumoto et al [86] performed a 2D finite element analysis for a single layer of SLS under the plane stress assumption. They found that the first track of a laser over a powder bed shows tensile stresses along the molten and re-solidifed track with compressive stresses on the edge. Subsequent tracks relieve the compressive edge stress and ultimately form a striped pattern of tensile and compressive stresses. They also found that long scan lengths lead to more distortion. However, the pattern of compressive and tensile stresses on the top layer appears to contradict other studies and thus brings into question the validity of this plane stress assumption. Roberts et al [105] proposed a 3D finite element model taking into account multiple layers and compared simulation results against IR camera results from Fischer et al [27]. Shiomi et al [119] used an analytical model where each new layer is assumed to start at the yield stress. Residual stresses were also measured via strain gauges on the base plate. These researchers found tensile residual stresses that increase with the addition of each layer and are largest near the top and bottom layers of the final part. They found a 40% reduction in residual stresses by preheating the base plate to 160 ◦C as compared to 80 ◦C. Later Osakada and Shiomi [96] used FEM to simulate residual stress and distortions during SLM. They assumed only elastic stresses in a single layer, under the plane stress assumption. These researchers determined the residual stresses by measuring the change of strain of a reference base location during milling of the SLM part in a layer-by-layer model. For titanium powder, large regions of tension (~300 - 500 MP a) were found in the top layer. This decreased to around 0 MP a in the middle layers and then increased again up to 200 MP a at the base location (6 mm from the top of the base plate). Heat treating and laser re-melting were found to significantly reduce residual stresses by 70% and 55%, respectively. Zaeh and Branner [149] also used FEM to simulate residual stresses. Residual stresses were subsequently experimentally measured using neutron diffractometry and part deformations. The FEM model contained many simplifications to allow for whole part modeling, such as the combining of multiple 50 µm layers into individual 1 mm layers, and as a result the simulation accuracy suffered. However, from experiments they found that island laser scanning or scanning along the shortest dimension decreased residual stresses. Casavola et al [12] found very high tensile residual stresses near the top surface. However, these sharply decreased deeper in the material indicating layers well below the free surface, which had been exposed to more laser passes, were thermally relieved. They also showed that thicker specimens can cool more 62 rapidly and thus contain larger thermal gradients and higher residual stresses. The hole drilling method was used to measure residual stresses. Pal et al. [98, 99] developed a 3D FEM model based upon dynamic, adaptive mesh refinement in areas near the laser zone. This was used to predict temperatures and residual stresses during the SLS process and the temperature model was validated against measurements made using an IR camera.

6.2 Modeling Framework

As a block of material cools down after a laser is scanned during the SLS process, thermal stresses form since the heated material is not free to contract to its stress-free state due to its interactions with the surrounding material it is bonded to. Modeling the build-up of these residual stresses requires the simulation of a coupled thermal-mechanical system, which is described along with all assumptions in this section. Also, note that the finite difference method is used to model the system as it is now considered a solid block of material. Modifications involving a 2 phase material (i.e. solid metal and voids) can easily be implemented in this framework and are described in this section as well.

6.2.1 Mechanical eﬀects - balance of linear momentum To model the mechanical eﬀects, we begin with the balance of linear momentum:

∇x · σ + f = ρu¨, (6.1) where σ refers to the Cauchy stress tensor, f is the body forces acting on the object, ρ is density in the current configuration, and u¨ is the second derivative with respect to time of the displacement (i.e. the acceleration). In this problem there are assumed to be no external body forces acting on the object so f = 0. Note that the displacement is given by u = x−X, where x refers to position in the current configuration and X is the position in the reference (or undeformed) configuration. Each of these are vector quantities. For example

 u   x  u = uy , (6.2)    uz  where ux, uy, and uz refer to the components of the displacement in the x, y, and z directions, respectively. Assuming small or inﬁnitesimal deformations (x ≈ X), we can recast Equation 6.1 as

∂2u ∇ · σ + f = ρ , (6.3) X 0 ∂t2 where ρ0 is the reference (undeformed) density. It is worth noting here that the stress tensor is symmetric and thus composed of 6 independent components: 63

    σ11 σ12 σ13 σxx σxy σzx     σ = [σij] =  σ21 σ22 σ23  =  σxy σyy σyz  . (6.4) σ31 σ32 σ33 σzx σyz σzz The stress can be calculated via Hooke’s Law, following the approach in [153]:

σ = E :( − p − T )=(1 − D)E0 :( − p − T ). (6.5) In the above equation, E refers to the (4th order) elasticity tensor of the material and is the inﬁnitesimal or small deformation strain. In Equation 6.5 we decompose the elasticity tensor into E = (1 − D)E0 where D is the damage scaling parameter (0 ≤ D ≤ 1) and E0 is the undamaged material’s elasticity tensor. D = 0 refers to the material in its undamaged form and D = 1 represents a completely damaged or fractured material (described in more detail later in this section). The elastic strain is found by calculating the total strain due to deformations , the plastic strain p, and the thermal strain T . Assuming linear elasticity and inputting all this into Equation 6.3, we obtain

∂2u ∇ · (1 − D) :( − − ) = ρ . (6.6) X E0 p T 0 ∂t2 The strain due to small deformations is given by

1 = ∇ u + (∇ u)T . (6.7) 2 X X The thermal strains are deﬁned as

T = αT (T − Tref )1, (6.8) where αT is the coefficient of thermal expansion, Tref is a reference temperature (or the original starting temperature of the material), and 1 is the identity matrix indicating that thermal strains only occur in the normal directions (i.e. there are no shear thermal strains). Note that one may think it counterintuitive to experience plasticity (indicating yielding) while assuming infinitesimal or small deformations. However, for most hard materials, such as metals, yield occurs at very small strain values (0.2% is a commonly used value) which can be neglected when compared to unity. Thus many plasticity models, including the one subsequently described in this work, are based upon the infinitesimal, or small, deformation assumption [79]. The flow rule for metal plasticity is used and plastic strain only occurs when the Von Mises stress (σVM ) is greater than the yield stress (σy) of the material [79]. Following the approach by Zohdi [153], the plastic strain rate is given by  0, σVM ≤ σy ˙ = 0 (6.9) p σVM −σy σ a 0 , σ > σ  p σy kσ k VM y 64

where ap is a material parameter which may be ﬁtted to experiments and can vary with temperature. According to the ﬂow rule, the plastic strain acts in the direction of the 0 tr(σ) tr(σ) deviatoric stress, σ = σ − 3 1. Note that the value 3 is the hydrostatic stress, or pressure, experienced by the material. The Von Mises stress is given by

1 2 2 2 2 2 2 1/2 σVM = √ (σxx − σyy) + (σyy − σzz) + (σzz − σxx) + 6(σ + σ + σ ) , 2 xy yz zx (6.10) 3 1/2 = σ0 : σ0 . 2 In Equation 6.9, the plastic strain acts in the direction of the deviatoric stress. To enforce this, the deviatoric stress is divided by the norm of the deviatoric stress so as to provide directionality but not artificially enlarge the plastic strain. The norm of the deviatoric stress tensor (kσ0k) can be calculated many ways. A common one is the 2- norm which is related to the largest eigenvalue of the matrix. The 1-norm is the maximum absolute column sum of the matrix. The infinity-norm is the maximum absolute row sum of the matrix. While all of these are valid norms, the author chose to use the Frobenius norm in this work. The Frobenius norm for the deviatoric stress is calculated as v √ u 3 3 0 0 0 uX X 0 0 kσ k = σ : σ = t σijσij. (6.11) i=1 j=1 The damage parameter, D, is used to represent the initiation and growth of cavities and microcracks induced by significant deformations in metals, also called “ductile plastic damage”. A damage model following that presented by Lemaitre [71] and Kachanov [53] is used. Considering the case of isotropic damage and isotropic plasticity, the ductile damage can be characterized by the equivalent stress and the intensity of plastic shear strain rates. Similar to how the Von Mises stress is used to characterize plasticity, a damage equivalent stress (σeq) is defined to characterize damage in terms of the Poisson’s tr(σ) ratio (ν), Von Mises stress (σVM ), and hydrostatic stress ( 3 ).  1/2 2 tr(σ) !2 σeq = σVM  (1 + ν) + 3(1 − 2ν)  . (6.12) 3 3σVM The intensity of the plastic shear strain rate is taken into account with a variable p˙, defined as

2 1/2 p˙ = ˙ : ˙ . (6.13) 3 p p The damage scaling parameter growth rate can now be deﬁned as 65

 σeq 0, 1−D ≤ σD ˙ 2 s2 D = σeq σeq (6.14)  2 p,˙ > σD  2E0(1−D) s1 1−D where E0 is the undamaged material’s Young’s modulus, s1 and s2 are material coeﬃcients, and σD is the stress above which damage begins to occur (for metals damage typically begins around the stress at which necking begins, or the ultimate tensile strength) [71, 101]. Finally, assuming an isotropic material, we can represent the elasticity tensor (E) in terms of two independent parameters, the material’s Young’s modulus (E) and Poisson’s ratio (ν). Recognizing that there are 6 independent components in the symmetric stress and strain tensors, the relationship between the two can be recast as follows

 σ   1 − ν ν ν 0 0 0     xx   xx         σyy   ν 1 − ν ν 0 0 0   yy         σzz  (1 − D)E0  ν ν 1 − ν 0 0 0   zz  =   . σ  0 0 0 1 − 2ν 0 0   xy  (1 + ν)(1 − 2ν)    xy         σyz   0 0 0 0 1 − 2ν 0   yz       σzx  0 0 0 0 0 1 − 2ν  zx  (6.15)

6.2.2 Thermal effects - balance of energy The temperature can be solved for via the first law of thermodynamics, or the balance of energy. Starting from a continuum formulation, the first law of thermodynamics is [129]

ρw˙ = σ : ∇xu˙ − ∇x · q + H, (6.16) where w is the stored energy per unit mass, q is the heat flux, and H is a source term or the rate of energy absorbed. Note that the first term on the right-hand side of Equation 6.16 represents the energy generated due to deformations, or the stress power. Assuming infinitesimal deformations, the stored energy in the referential frame is 1 ρ w = ( − − ): :( − − ) + ρ cT, (6.17) 0 2 p T E p T 0 where c is the constant pressure specific heat capacity and T is temperature. The first term on the right-hand side of Equation 6.17 represents the strain energy of the body. Taking the derivative with respect to time, and recognizing the commutativity of the double inner product for matrices, we obtain

1 ρ w˙ = (˙ − ˙ − ˙ ): :( − − ) + ( − − ): ˙ :( − − ) + ρ cT.˙ (6.18) 0 p T E p T 2 p T E p T 0 66

Now we assume that heat transfer only occurs due to conduction, as would be expected within a solid block of metal. Using Fourier’s Law, we can write

q = −K∇X T, (6.19) assuming an isotropic material. Plugging Equations 6.18 and 6.19 into Equation 6.16 and simplifying, the ﬁrst law becomes

1 ρ cT˙ = σ :(˙ + ˙ ) − ( − − ): ˙ :( − − ) + ∇ · (K∇ T ) + H. (6.20) 0 p T 2 p T E p T X X Finally, it should be noted that all material parameters may vary as a function of space and time depending on material and temperature.

6.2.3 Numerical solution scheme Clearly this is a coupled thermal-mechanical system where Equations 6.3 and 6.20 must be solved simultaneously. A centered-difference finite difference scheme is employed to calculate all spatial derivatives and a fourth order explicit Runge-Kutta (RK-4) time marching scheme is utilized, similar to the ones previously employed in the heating simulations. In this section, a detailed description of the code is provided so that this simulation can easily be recreated. Using a centered difference scheme, the infinitesimal strain (given by Equation 6.7) can be calculated via a second order accurate, O(∆x)2, finite difference scheme as follows [72]:

 ∂ux ∂ux ∂ux  ∂x ∂y ∂z  ∂uy ∂uy ∂uy  ∇X u =   , (6.21)  ∂x ∂y ∂z  ∂uz ∂uz ∂uz ∂x ∂y ∂z ∂u u (x + ∆x, y, z) − u (x − ∆x, y, z) x ≈ x x . (6.22) ∂x 2∆x The other derivatives can be calculated in an analogous manner. The thermal strain (Equation 6.8) is calculated explicitly by

T = αT (T (x, y, z, t) − T (x, y, z, t = 0))1. (6.23) Second derivatives are calculated in a manner identical to the method used in Equation 4.30. More details are available in Appendix B. Additionally it should be noted that the double inner product for matrices is calculated as

3 3 X X σ : = σijij. (6.24) i=1 j=1 67

˙ ˙ Integration in time is done via an explicit RK-4 scheme for the variables ˙p, D, T , and u¨. This is done in the same manner as described in Equation 4.33. All programming was performed in Fortran 90. A basic pseudocode is presented in Algorithm 6.1.

6.3 Numerical Examples

Many simulations were carried out with diﬀerent initial conditions to demonstrate the features of this methodology for calculating residual stresses. The two main types of simulations shown in this section are (1) cooling of a small block with a prescribed temperature and varying porosities and (2) cooling of a single laser scan using the temperature output from the heating simulations carried out in the previous chapter. In all situations the following boundary conditions were used, as illustrated in Figure 6.1:

∂u ∂T • Left and right faces (i.e. when x = 0 or x = Lx): ∂x = 0 and ∂x = 0, ∂u ∂T • Front and back faces (i.e. when y = 0 or y = Ly): ∂y = 0 and ∂y = 0,

• Bottom face (i.e. when z = 0): u = 0 and T = T0 ,

∂u cond conv rad ∂T • Top face (i.e. when z = Lz): ∂z = 0 and q = q + q =⇒ k ∂z = 4 4 hconv(T (x, y, z) − T0) + σSB(T (x, y, z) − T0 ) where hconv is the convective heat transfer coefficient, is the material emissivity, σSB is the Stefan-Boltzmann constant, and T0 is the temperature of the ambient environment, or sintering chamber. To implement the non-Dirichlet (or non-fixed) boundary conditions, a finite difference stencil of the form shown in Equation 4.31 was used to calculate derivatives on the boundaries.

6.3.1 Cooling of a solid block First a set temperature proﬁle is prescribed to a block of 316L SS and residual stresses upon cooling are calculated. The initial temperature distribution quadratically decreases with height and is given as

z 2 T (x, y, z, t = 0) = (1700 − T0) + T0, (6.25) Lz such that the temperature is equal to the melting temperature of steel at the top (1700 K) and the bottom face is ﬁxed at T0 = 373 K. The initial displacements and stresses were set equal to 0, and the stress and temperature ﬁelds were calculated as the block cools down. Figure 6.2 shows the initial temperature distribution of the block. The material properties of 316L SS used in this simulation are the same as those provided in Table 5.1. Additional properties necessary for the residual stress calculations are given in Tables 6.1 and 6.2. 68

Algorithm 6.1 Residual stress calculation pseudocode 1. Deﬁne all parameter and variable values

2. Set initial conditions

3. Begin time loop (for t < tend)

(a) Calculate strain due to deformations (b) Calculate thermal strains

(d) If σeq > σD, where σeq is from the previous time step i. Calculate damage rate (e) Calculate new total stress (using plastic strain from previous time step) (f) Calculate acceleration using the balance of linear momentum (g) Calculate divergence of the conductive heat ﬂux (h) Calculate thermal strain rate (i) Calculate stress power (j) Calculate rate of temperature change (k) Update plastic strain, damage parameter, velocity, displacement, and temperature using RK-4 method (requires 4 iterations within each time step) (l) Apply boundary conditions as necessary after each time step (m) Update material properties

4. Output necessary data to plotting ﬁles 69

qcond=qconv+qrad ∂T/∂y=0 z ∂u/∂y=0 y ∂u/∂z=0 x

∂T/∂x=0 ∂T/∂x=0 ∂u/∂x=0 ∂u/∂x=0

∂T/∂y=0 T=T ∂u/∂y=0 0 u=0

Figure 6.1: Boundary conditions for residual stress calculations

Figure 6.2: Initial temperature distribution (temperature in K) 70

Temperature, Yield Strength, UTS, CTE, −6 T (K) σy (MP a) σD (MP a) αT (10 /K) 293 290 627 - 366 - - 15.1 478 241 558 15.8 589 214 538 16.4 700 190 524 16.7 811 165 483 17.1 922 145 393 17.5 1033 124 241 18.0 1144 110 165 18.5 1255 - - 19.1

Table 6.1: 0.2% Yield strength, ultimate tensile strength, and instantaneous coeﬃcient of thermal expansion for 316L SS [124, 48]

Parameter Symbol Value Poisson’s ratio ν 0.26 W 2 Heat transfer coeﬃcient hconv 40 /m K Material emissivity 0.33 Reference temperature T0 373 K −3 Plasticity rate parameter ap 10 Damage parameter, denominator s1 1.25 MP a Damage parameter, exponent s2 1.0

Table 6.2: Additional material properties used in this simulation [101, 124] 71

The first simulation was for a solid block of 316L SS. The block is 1 mm x 1 mm x 1 mm in size and the mesh is 41 x 41 x 41 nodes, yielding a grid spacing of ∆x = ∆y = ∆z = 25 µm. With a grid spacing of this size, the required time step using the RK-4 method is ∆t = 2.0 ∗ 10−9 s, or 2 ns. This time step was determined by continuously refining the time step in half until the results stopped changing within a tolerance of 1 % after 1 µs. Note that if the time step is too large, then the solution will explode and produce non-physical or non-real results due to the accumulation of tiny errors over the course of many (potentially millions of) time steps. The total simulation time for this cooling simulations was 1 ms, yielding a total of 500,000 time steps. OpenMP parallelization was used to speed up simulation time. However, even with parallelization across 4 cores, total simulation time was on the order of 1 day on a Macbook Pro laptop computer with an Intel i7 processor. Figures 6.3, 6.4, and 6.5 show the average temperature, Von Mises stress, and equivalent strain of the entire domain as a function of time. These average temperature was calculated by summing the temperature of each nodal points and then dividing by the total number of nodes (Ntotal = NxNyNz where N refers to number of nodes). Average stress and strain were calculated in an equivalent manner. Note that the equivalent strain (eq), or Von Mises equivalent strain, is given by s 2 = 0 : 0, (6.26) eq 3 0 0 tr where is the deviatoric strain tensor defined as = − 3 1. In these plots it is apparent that the temperature, stress, and strain throughout the domain have started to level off towards their steady state value. However, it should be noted that this leveling off curve will extend for quite a while and the final average temperature after 1 ms is slightly over 500 K. After an infinite amount of time, the whole domain will be at 373 K since the bottom surface is fixed at this temperature and the top surface is convecting and emitting radiation to an ambient environment also at 373 K. Additionally, it is worth observing how the metal cube will yield and experience damage in some parts of the domain under these conditions. Figures 6.6, 6.7, and 6.8 depict the evolution of the temperature, Von Mises stress, and equivalent strain throughout the domain at t = 0, t = 0.5, and t = 1 ms. In Figures 6.7 and 6.8 it can be seen that the highest stresses and strains occur near the boundaries, especially at the top boundary as this is initially the hottest region. This is similar to experimental studies which have shown the highest stress regions to be located near the top surface [96, 12, 149]. It is worth noting that the top layers are actually slightly cooler than the layers just underneath them due to the heat loss experienced via convection and radiation on the top surface. 72

Figure 6.3: Average temperature of the fully dense 1 mm x 1 mm x 1 mm block of 316L SS

Figure 6.4: Average Von Mises stress of the fully dense 1 mm x 1 mm x 1 mm block of 316L SS 73

Figure 6.5: Average equivalent strain of the fully dense 1 mm x 1 mm x 1 mm block of 316L SS

Figure 6.6: Temperature distribution (in K) of the fully dense 1 mm x 1 mm x 1 mm block of 316L SS at t = 0 (left), t = 0.5 ms (middle), and t = 1 ms (right) 74

Figure 6.7: Von Mises stress distribution (in MP a) of the fully dense 1 mm x 1 mm x 1 mm block of 316L SS at t = 0 (left), t = 0.5 ms (middle), and t = 1 ms (right)

Figure 6.8: Equivalent strain distribution of the fully dense 1 mm x 1 mm x 1 mm block of 316L SS at t = 0 (left), t = 0.5 ms (middle), and t = 1 ms (right) 75

Parameter Symbol Value Density ρ 5.77 kg/m3 Thermal conductivity K 0.0165 W/m∗K −9 −1 Coeﬃcient of thermal expansion αT 10 K Speciﬁc heat capacity c 521 J/kg∗K 10 Yield stress σy 10 P a (doesn’t yield) 10 Damage stress σD 10 P a (doesn’t damage) Young’s modulus E 1700 P a Damage parameter, exponent s2 0.0 (doesn’t damage)

Table 6.3: Argon material properties [77]

6.3.2 Cooling of a porous block While the previous simulations depicts the cooling and stress build-up in a fully dense block of material, simulations were also run for more porous materials. These simulations can more accurately represent how the residual stresses build up in a part created using SLS, which may not have fully densified. To mimic the effects of porosity, the same finite difference method for calculating stress and temperature is used, except now considering a 2-phase material. The first phase is composed of solid 316L SS and uses the same material properties as given previously. Phase 2 is argon gas, as this is typically the gas present in the sintering chamber to prevent oxidation of the metal powders. The pores within the solid are considered to be trapped argon. Each node in the finite difference mesh is assigned properties of either 316L SS or argon depending on whether that node is considered to be solid or a void. Table 6.3 shows the material properties for argon used in these simulations. Two simulations were run to simulate a porous block. First, a porosity of 30% (meaning 70% dense) was assigned to a 1 mm x 1 mm x 1 mm cube. A 41 x 41 x 41 node mesh was used, identical to the previous simulation of a fully dense cube. In this simulation, nodes were randomly assigned to be either phase 1 (316L SS) or phase 2 (argon), such that the block was composed of 70% 316L SS and 30% argon. The phase distribution of the cube is shown in Figure 6.9. The initial temperature distribution is the same as in the previous simulation and is given by Equation 6.25. The time step size is ∆t = 2 ns and the total simulation time is 1 ms again. To minimize numerical error due to the sharp interface in material properties [38], a Laplacian interface smoothing technique is employed [152]. Here the material properties are “smoothed” out by averaging each node’s material properties with that of its closest neighbors. For example, for the thermal conductivity (using the finite difference stencil shown in Figure 4.8), is given by

1 K = (K + K + K + K + K + K ). (6.27) i,j,k 6 i−1,j,k i+1,j,k i,j−1,k i,j+1,k i,j,k−1 i,j,k+1 76

Figure 6.9: Phase distribution of 70% dense 1 mm x 1 mm x 1 mm block of material. Red points are phase 1 (316L SS) and blue are phase 2 (argon gas)

Similarly, the heat capacity, density, Young’s modulus, Poisson’s ratio, coefficient of thermal expansion, yield stress, and damage stress were smoothed across all nodes. In addition to improving the numerical performance of this FD scheme across sharp interfaces in material properties, Laplacian smoothing has a physical basis in that there is very rarely an abrupt transition from one phase to another in a given material. Rather there is typically some sort of transition region which is represented via this smoothing technique [152]. Figures 6.10, 6.11, and 6.12 show the average temperature, Von Mises stress, and equivalent strain of the entire domain as a function of time. In Figure 6.10 it is worth noting how the final temperature is still slightly higher than in the case of the fully dense block after 1 ms (approximately 540 K vs. 520 K). This can be explained by the lower thermal conductivity of argon, which will make the block take longer to cool down. Figures 6.13, 6.14, and 6.15 depict the evolution of the temperature, Von Mises stress, and equivalent strain throughout the domain of the 70% dense block at t = 0, t = 0.5, and t = 1 ms. Similar trends are shown in these figures as compared with the simulations for a fully dense block. However, the temperature, strain, and stress distribution is not completely symmetric as before due to the random addition of voids in the material. Note that there is not a larger difference between the fully dense and 70% dense simulations due to the random nature of the void addition and due to the Laplacian smoothing technique employed, which averaged out the material properties of argon with those of neighboring 316L SS. Next, a simulation is run to represent actual particles lying in the domain, as would happen during SLS. To simulate the microstructure after SLS particle data points are read in from the particle deposition algorithm described in Chapter 4. In this scenario, particles with a mean diameter of 30 µm and standard deviation of 7 µm are deposited in 77

Figure 6.10: Average temperature of the 70% dense 1 mm x 1 mm x 1 mm block of 316L SS

Figure 6.11: Average Von Mises stress of the 70% dense 1 mm x 1 mm x 1 mm block of 316L SS 78

Figure 6.12: Average equivalent strain of the 70% dense 1 mm x 1 mm x 1 mm block of 316L SS

Figure 6.13: Temperature distribution (in K) of the 70% dense 1 mm x 1 mm x 1 mm block of 316L SS at t = 0 (left), t = 0.5 ms (middle), and t = 1 ms (right) 79

Figure 6.14: Von Mises stress distribution (in MP a) of the 70% dense 1 mm x 1 mm x 1 mm block of 316L SS at t = 0 (left), t = 0.5 ms (middle), and t = 1 ms (right)

Figure 6.15: Equivalent strain distribution of the 70% dense 1 mm x 1 mm x 1 mm block of 316L SS at t = 0 (left), t = 0.5 ms (middle), and t = 1 ms (right) 80

Pores: Argon

Particles: 316L SS

Figure 6.16: Finite difference mesh of a layer of powder particles a domain. The minimum particle diameter is 10 µm and the maximum is 50 µm. Once these particles have settled, a small section of the domain is sampled and used in this residual stress simulation. The domain is meshed up with a finite difference grid. All nodes within a particle are assigned material properties of 316L SS and nodes outside of particles are assigned argon properties as shown in Figure 6.16. The selected domain size in this simulation is 100 µm x 100 µm x 100 µm. Again, 41 x 41 x 41 nodes are used given a grid spacing of ∆x = ∆y = ∆z = 2.5 µm. This grid spacing was chosen such that there were enough nodal points to adequately capture even the smallest particles (4 grid points for a 10 µm particle). Note that the overall simulation density in this simulation is 57%; however, it is not completely randomly dispersed as in the previous simulation. This is too give a more accurate representation of the possible microstructure during SLS where very little consolidation takes place if making a porous part. The phase distribution of the selected domain using a pre-deposited powder layer is depicted in Figure 6.17. Due to the smaller grid spacing in this simulation, a much smaller time step is required. The time step in this simulation is ∆t = 10−12 s. The total simulation time (due to computational expense) is 1 µs, which requires 106 time steps. This simulation required approximately 2 days to run. Laplacian interface smoothing is again used to smooth out material properties at interfaces. The initial temperature distribution is given by Equation 6.25. Figures 6.18, 6.19, and 6.20 show the average temperature, Von Mises stress, and equivalent strain of the entire domain as a function of time. It is clear that the domain is still undergoing rapid cooling, which will lead to rapid stress and strain increases, at the end of the simulated time. However, general trends regarding the temperature, stress, and strain behavior throughout the domain are still easily visualized in Figures 6.21, 6.22, and 6.23, respectively. By comparing the temperature distribution in 6.21 with the phase distribution in Figure 6.17, it can be seen that areas with particles cool down faster. This is to be expected due to the higher thermal conductivity of the steel particles as compared 81

Figure 6.17: Phase distribution of the SLS simulation with using pre-deposited particles, sampled from a 100 µm x 100 µm x 100 µm domain. Red points are phase 1 (316L SS) and blue are phase 2 (argon gas) with argon gas. Similarly, areas where particles are present clearly have experience higher stresses as can be seen in Figure 6.22. Again, this is due to the much higher stiﬀness of the steel particles.

6.3.3 Cooling of a single laser scan The residual stresses upon cooling after a single laser pass are described in this section. The initial temperature profile is taken from the results obtained in the heating simulations carried out in Chapter 5. The results of the simulation relating to row 4 of Table 5.3 is used. Note that only the output results of the substrate were considered in an effort to reduce computation time. The temperature of the substrate material modeled via the FD method is taken from this heating simulation and used as the initial condition. This is in an effort to simulate the stress build up after a single laser pass over a powder bed. All points above 1100 K at the end of the simulation are considered to have at least partially melted and are thus considered solid 316L SS. Points which did not melt are considered to remain powder. Unlike the previous simulations, the non-solid nodal points are considered to be powder as opposed to argon gas. As such, homogenized material properties are assigned to the powder nodal points, similar to the techniques used by Shapiro et al [116] and Matsumoto et al [86]. Phase 1 (above the 1100 K after the laser pass) is considered to be solid 316L SS and phase 2 (below 1100 K) is considered powder material. Homogenized powder material properties are given in Table 6.4. The density of powder is 50% of the density of bulk 316L SS since the powder bed density is approximately 50%. Similarly the coefficient of thermal expansion is half that of the 82

Figure 6.18: Average temperature of the 100 µm x 100 µm x 100 µm domain

Figure 6.19: Average Von Mises stress of the 100 µm x 100 µm x 100 µm domain 83

Figure 6.20: Average equivalent strain of the 100 µm x 100 µm x 100 µm domain

Figure 6.21: Temperature distribution (in K) of the 100 µm x 100 µm x 100 µm domain at t = 0 (left), t = 0.5 µs (middle), and t = 1 µs (right) 84

Figure 6.22: Von Mises stress distribution (in MP a) of the 100 µm x 100 µm x 100 µm domain at t = 0 (left), t = 0.5 µs (middle), and t = 1 µs (right)

Figure 6.23: Equivalent strain distribution of the 100 µm x 100 µm x 100 µm domain at t = 0 (left), t = 0.5 µs (middle), and t = 1 µs (right) 85

Parameter Symbol Value Density ρ 3820 kg/m3 Thermal conductivity K 0.25 W/m∗K −6 −1 Coeﬃcient of thermal expansion αT 9 ∗ 10 K Speciﬁc heat capacity c 568 J/kg∗K 10 Yield stress σy 10 P a (doesn’t yield) 10 Damage stress σD 10 P a (doesn’t damage) Young’s modulus E 1.05 GP a Damage parameter, exponent s2 0.0 (doesn’t damage)

Table 6.4: Powder material properties [77, 116, 86, 124] bulk material. The specific heat is the average of bulk 316L SS and argon. The Young’s modulus is the same used in [86] and the thermal conductivity of the powder bed is reported by [116]. The powder material is assumed not to yield or experience damage. Laplacian interface smoothing is again used to smooth out material properties The initial temperature distribution and phase information is shown in Figures 6.24 and 6.25. The domain size in this simulation is 300 µm x 500 µm x 300 µm. The grid spacing is the same as used in the heating simulations so ∆x = ∆y = ∆z = 5 µm, yielding a mesh of 61 x 101 x 61 nodes. The time step size is ∆t = 5 ∗ 10−11 s and the total simulation time is 10 µs. Figures 6.26, 6.27, and 6.28 show the average temperature, Von Mises stress, and equivalent strain of the entire domain as a function of time. Again, the material is still undergoing rapid cooling during the time frame simulated. In Figures 6.29, 6.30, and 6.31 the temperature, stress, and strain distributions throughout the domain are depicted. It is interesting to note that the middle of the block (which is hotter) is actually at a lower temperature than some of the outer points in Figure 6.29. The reason for this is because of the cut-off temperature of 1100 K when deciding whether a node was solid 316L SS or still powder. Those powder points just below this cut-off temperature cool much slower in the simulation due to their much lower thermal conductivity. In Figures 6.30 and 6.31, it is apparent the highest stresses while cooling occur along the center of the laser pass. This is to be expected as these areas are initially at the highest temperature and thus will cool, and subsequently want to thermally contract, the most. Finally, it should be observed that it is difficult to compare these results to experiments in the literature since experiments typically measure residual stresses when the part has cooled down to room temperature. However, it was not feasible to simulate for a long enough time frame to allow complete cooling of the domain to occur with the available resources. Such a simulation would require access to more powerful computing clusters and parallelization across many more cores. However, the basic trends that are to be expected can be observed in these simulations. 86

Figure 6.24: Initial temperature distribution of the 300 µm x 500 µm x 300 µm powder- steel mixture after a single laser scan. All nodes over 1100 K are assumed to have molten and be solid 316L SS.

Figure 6.25: Phase distribution of the 300 µm x 500 µm x 300 µm powder-steel mixture after a single laser scan. Red points represent solid 316L SS and blue represents unmolten powder. 87

Figure 6.26: Average temperature of the 300 µm x 500 µm x 300 µm powder-steel mixture

Figure 6.27: Average Von Mises stress of the 300 µm x 500 µm x 300 µm powder-steel mixture 88

Figure 6.28: Average equivalent strain of the 300 µm x 500 µm x 300 µm powder-steel mixture

Figure 6.29: Temperature distribution (in K) of the 300 µm x 500 µm x 300 µm powder- steel mixture at t = 0 (left), t = 5 µs (middle), and t = 10 µs (right) 89

Figure 6.30: Von Mises stress distribution (in MP a) of the 300 µm x 500 µm x 300 µm powder-steel mixture at t = 0 (left), t = 5 µs (middle), and t = 10 µs (right)

Figure 6.31: Equivalent strain distribution of the 300 µm x 500 µm x 300 µm powder-steel mixture at t = 0 (left), t = 5 µs (middle), and t = 10 µs (right) 90

Chapter 7

Conclusions and Future Extensions

Many topics related to the SLS/SLM process (and AM in general) were discussed in this work. Computational models were presented for simulating the deposition and laser heating of a layer of powder particles. Subsequently a model for calculating thermal stresses upon cooling was presented. Three dimensional numerical examples were provided. This chapter will summarize this work, discuss its limitations and future extensions, and conclude with a discussion of new AM techniques and how modiﬁcations to the models presented in this paper could be applied to such techniques.

7.1 Summary of this Work

This manuscript began by introducing the basics of additive manufacturing, including the most commonly used techniques such as stereolithography, fused deposition modeling, inkjet printing, laminated object manufacturing, and selective laser sintering. The applications of these AM technologies, with a focus on SLS, was discussed in regards to the tooling, biomedical, aerospace, automotive, energy, and consumer products industries (although AM certainly is not just limited to these applications). Next a discussion was presented into the ever growing climate change problem, of which manufacturing plays a key role. As manufacturing accounts for nearly 20% of total energy use in the US and over 80% of industrial CO2 equivalent emissions [109], improving the efficiency and sustainability of manufacturing processes is of vital importance. Life cycle analyses comparing the use of AM technologies with conventional machining were discussed, showing that AM has the significant benefits of reduced material consumption, less required tooling, and the elimination of often toxic cutting fluids. While such direct impacts were measured in previous studies, firm conclusions regarding energy use in AM vs. conventional machining were unclear due to a lack of literature on the subject and the fact that energy measurement methodologies often significantly varied by researcher. However, many indirect benefits of AM were discussed, which clearly show the potential environmental benefits of widespread AM use. Some of these benefits include lighter and 91 more resource efficient parts and a reconfigured supply chain. Lighter parts can offer significant energy savings in the automotive and aerospace industries, while simultaneously reducing fabrication time and cost. A more lean supply chain is possible due to the ability to locally print parts on-demand, rather than needing to stockpile parts in a warehouse. These indirect environmental benefits of AM also lead to societal benefits by moving away from large centralized manufacturing facilities in developing countries where labor is cheap but safety regulations are lacking. UN projections have shown that more efficient manufacturing can not only lead to decreased emissions but also the creation of more jobs for the ever-growing population [102]. Once AM as a whole and its potential impacts were discussed, Chapter 3 onwards narrows the focus specifically to SLS/SLM. Full details of the SLS process are provided along with the main process parameters: laser power, scan speed, spot size, hatch spacing, scan strategy, powder material, powder size distribution, layer thickness, and surrounding gas atmosphere. These parameters need to be optimized depending on the material and desired final part characteristics. Otherwise problems could arise regarding density of the final part and microcracking due to high residual stresses. SLS processed materials tend to be more brittle than conventionally forged materials due to the high cooling rates. Post- processing can be used to increase densification and change the microstructure, though this adds time and cost to the manufacturing process. Additionally final parts may suffer from dimensional accuracy and surface roughness issues due to adhering powder sticking to the part, though this can be alleviated with laser re-melting and/or post-machining. Warping and other effects such as balling may also occur if proper process parameters are not used. In order to ensure proper processing of materials, computational models of the process are necessary. Many researchers have attempted this and the models generally fall into three categories: (1) empirical models, (2) continuum based models, or (3) particle- scale models. Empirical models are very material and process specific; continuum models fail to capture powder bed inhomogeneities; and particle scale models are often extremely computationally expensive. This work presents a reduced order coupled discrete element - finite difference model that has the ability to relatively quickly calculate the deposition and laser heating of a layer of powder particles. In Chapter 4 the powder deposition and laser heating model is described in detail. The discrete element method is used to simulate the powder bed and the finite difference method (FDM) for simulating the underlying solid substrate. Individual particle are represented as thermally and mechanically interacting spheres. Each sphere is assumed to be smooth and without any internal temperature gradient. Hertzian contact is used to model contact forces between particles. Friction, environmental, and gravity forces are also considered. The heating of these particles is governed by the 1st law of thermodynamics (balance of energy). Particle to particle conduction is taken into account along with convection and radiation for particles on the top surface, exposed to the ambient environment. Heat energy from the Gaussian laser beam is modeled using the Beer-Lambert law where laser intensity decreases exponentially as a function of penetration depth. Material properties for each particle are updated as a function of temperature and phase. Heat 92 transfer to the underlying substrate is modeled via the FDM. Conduction from the bottom layer of particles to the top layer of the substrate is applied as a boundary condition. An RK-4 time stepping scheme is used to solve the mechanical system, while a Forward Euler scheme is used for the thermal system. A binning algorithm is used to speed up computation time such that all simulations can be run in the order of minutes to hours on a laptop computer. Further parallelization could easily be implemented if necessary. Chapter 5 presents model validation and numerical results for the powder deposition and laser heating model. Validation is performed by comparing the simulated melt pool size after a single laser pass with experimental results found in the literature. Pretty good agreement (less than 15% error) is found. Additionally, this approach was able to run the simulation in approximately 3 hours on a laptop computer, as opposed to the 100,000 cpu hours required to run the same simulation by the authors of the paper results were compared against [58]. This shows the current model’s potential to be used as a quick tool for optimizing process parameters for a given material. After validation, three dimensional numerical results are shown of the deposition of a layer of steel powder particles and the temperature distribution of the powder and underlying substrate as a laser is scanned over it. Parameter studies are run by varying the laser scan speed and power and looking at the resultant melt pool dimensions. Such studies can be useful in figuring out optimal processing conditions such that the scan speed for a given power may be as high as possible to reduce fabrication time, while simultaneously ensuring enough melting occurs for proper bonding. Next a parameter study is performed by varying the powder size distribution and calculating the loose powder bed density, which has been shown to have a significant effect on final part density [60, 61]. By increasing the spread of the particle size distribution, a more dense unsintered powder bed can be produced since smaller particles fit inside the interstitial spaces between larger particles. A bimodal powder distribution offered the highest loose bed density. However, heating simulations performed using this bimodal distribution showed that some of the smallest particles evaporated before larger ones melted. This can be problematic as these gas bubbles may get trapped in the final part, increasing porosity. This indicates that an optimal size distribution exists for different particles to ensure maximum loose bed density and powder flow without too much ablation of the smallest particles. Chapter 6 deals with the thermal stresses that build up within a part after cooling. Background information is provided regarding how residual stresses form, and other researcher’s modeling attempts are discussed. Next a framework for simulating the residual stresses in a multi-phase cooling body is presented. A finite difference scheme is used as the material is assumed to be one solid piece after melting under a laser. Mechanical effects are governed by the balance of linear momentum. Stress is calculated via Hooke’s Law assuming an isotropic, linear elastic material. Thermal and plastic stresses are also considered. A ductile plastic damage model is used to simulate microcracks that may occur due to large stresses on the material. Thermal effects are governed via a balance of energy, where heating due to deformations and strain energy is also considered. Only conductive heat transfer is assumed to occur within the body, while convection and radi- 93 ation serve as a boundary condition for the top surface. An explicit RK-4 time stepping scheme is again used to solve this coupled thermal-mechanical system. Different initial conditions are prescribed and numerical simulations are shown. Multi-phase materials (either steel-argon or steel-powder) are simulated by assigning different material properties to each FD node. Due to the extremely small time step needed to resolve the mechanical field (due to the high cooling rates), final residual stresses could not be fully calculated in a reasonable simulation time. However, the thermal stresses that built up during cooling were calculated and trends could be observed. The highest stresses occurred on the top face of the material, which agrees with results in the literature and makes sense as the top face is initially the hottest, and thus will undergo the most thermal contraction upon cooling. More porous materials cooled more slowly, and the stresses were borne by the solid phase. The thermal stresses produced indicated yielding would occur, especially on the boundaries and top surface. The middle layers experienced significantly lower stresses.

7.2 Model Limitations and Future Extensions

In the formulation of this (or any) model certain assumptions are made which affect the overall accuracy. The key to an effective model is finding the right balance between required accuracy and computational expense. This work presented a reduced order model to rapidly simulate the deposition and laser heating of a layer of powder particles. In the laser heating and deposition code each particle was assumed to be at a uniform temperature such that it was observed that a particle was either fully molten or not molten; partial melting of an individual particle was not simulated. Additionally, thermal expansion of the spherical particles was neglected along with gas gap conduction between the particles. Solid and molten particles followed the same governing equations, just with different material properties and a reduced stiffness for liquid particles. Thus, this model was not able to simulate the fluid flow and densification upon melting. Simulating fluid flow and densification could be done in a couple of ways: (1) break each molten particle up into many smaller fluid particles with different governing equations or (2) use a continuum type method to simulate fluid flow when a particle melts. Method (2) could involve the use of the Lattice Boltzmann method (LBM) as used by Korner et al. [60, 61] in the simulation of selective electron beam melting. However, this would also require a unique DEM-LBM coupling mechanism, the feasibility of which has been demonstrated by Galindo-Torres [28] as shown in Figure 7.1. However, conversion of DEM particles to LBM continuum nodes upon melting and additionally solving for temperature is not as simple as the coupling mechanism for only fluid flow demonstrated in [28]. For the residual stress model, it was not feasible to simulate for a long enough time frame to fully resolve the residual stresses once the block had cooled to an equilibrium temperature. While access to highly parallelized computing clusters would allow for simulations to run for the requisite time, there may be other work arounds rather than increased computing power (though this would be the most accurate). One could try im- 94

Figure 7.1: Fluid flow modeled by the Lattice Boltzmann method around a Discrete Element sphere, demonstrating couple feasibility (color represents fluid velocity) [28] posing a step temperature change and then resolving the mechanical fields. Alternatively, the thermal problem could be solved and the mechanical problem only periodically solved as the mechanical field was the limiting factor in time step size. Additionally, this work demonstrated the need for a multi-scale approach to build a full part level model, which would be the ultimate goal of such computational methods. A full part level model could possibly be constructed as follows:

• DEM to predict particle deposition and temperature distribution

• LBM to simulate fluid flow and densification upon melting

• FDM/FEM to predict stresses and potentially warpage upon cooling

Parallelization and access to increased computing power could then allow for full part-level simulations. Of course the most difficult part of such a multi-scale approach would be the coupling mechanisms between the various methods and scales present. Information from the particle scale would need to be read in and homogenized over larger scales for use in the FDM/FEM continuum methods in order to simulate a larger domain. Additionally, DEM- LBM coupling where DEM particles become LBM nodes after melting has not been done to the author’s knowledge. Of course, other approaches would also work for full-scale part level models. However, regardless of specific modeling scheme, a unique approach would need to be developed where none currently exists for a fully resolved multi-scale model of SLS/SLM. Finally, as with all models, experimental results would also be necessary to determine appropriate material parameter coefficients and offer validation. The pursuit of such models would be extremely useful as a software package to go along with physical 95

Figure 7.2: DMG Mori LASERTEC 65 laser deposition welding and milling machine. Outer case (left) and close-up of a part being built (right) [6]

SLS machines and as such there are many research labs and companies working on exactly this [58, 98]. Finally, while the numerical methods presented in this work were specifically aimed at modeling the SLS process, such methods are certainly not limited to SLS. The coupled DEM-FDM method described could be applied to other AM technologies as well. For example, DMG Mori has recently commercialized a machine which combines a conventional CNC mill with laser deposition welding (see Figure 7.2 of the LASERTEC 65). In this machine powder particles are shot onto the object being fabricated and simultaneously melted with a laser. Milling tool heads are also present to quickly create smooth surface finishes or drill any necessary components. Certainly discrete elements could be used to model the powder particles which are ejected and melted via a laser. A continuum method (such as FDM/FEM) could be used to model the solid part being built. This could aid in reducing the sputtering that occurs and ensure better part quality and bonding of the molten particles. This work has explored many different aspects of AM in general and outlined a reduced order model of the SLS process which can be used as a guide for quick optimization of process parameters. I would like to conclude this manuscript with a quote from esteemed physicist Richard Feynman at a talk given to the APSociety at the California Institute of Technology: “But I am not afraid to consider the final question as to whether, ultimately - in the great future - we can arrange the atoms the way we want; the very atoms, all the way down!” Certainly, this future is approaching us and will continue to get closer as advances in 96

AM are made. Computational models such as the one presented here will be a key aid in the development of such new technologies, which can have a disruptive impact on the manufacturing industry and beneﬁt society as a whole. 97

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Appendix A

Economics and Projected Growth of Additive Manufacturing

Additive manufacturing, or 3D printing, originated in various forms around the late 1980s. In recent years its use has rapidly grown, with a recent Economist article even describing the digitization of manufacturing as the third industrial revolution due to its potential to radically change the manufacturing industry [85], a thought that has since been echoed by many others [90]. In 2012, the global market for AM products and services experienced a compound annual growth rate (CAGR) of 28.6% to $2.2 billion (up from $1.7 billion in 2011) [143]. This CAGR is expected to increase in future years according to studies by Canalasys, Wells Fargo, and others [14]. This can be reflected in the sales growth of industrial AM systems worldwide from 1988 to 2012, which can be seen in Figure A.1. Note that industrial systems refer to those that sell for greater than $5,000. The sales shown in Figure A.1 exclude those of low-cost personal systems, which may retail for on the order of hundreds of dollars for desktop size FDM machines. Also note that the dip in total sales from 2008 to 2009 corresponds to the height of the global economic crisis. For reference, personal 3D printer sales were believed to have hit 23,265 units in 2011 (growth of 289%) [143]. The increase in the use of AM can be applied across all industries. Figure A.2 depicts revenue forecasts for AM from 2014 to 2020 in various industries. It is clear in this figure that revenue does not just come from system sales. Rather, significant growth potential exists for AM service providers and the introduction of new materials into the AM market. One of the main limitations for AM lies in the limited number of materials that can be 3D printed. Currently only hundreds of materials are able to be 3D printed, compared to thousands available for conventional manufacturing processes [131]. As the number of functional materials able to be processed via AM increases, revenue and expansion of the market will also increase. Another opportunity for increased AM use in industry lies in the production of end- use products, rather than just prototypes. Although only 24% of the AM market was 111

Figure A.1: Worldwide sales of industrial AM systems since 1988 [143]

Figure A.2: AM Revenue forecasts through 2020 [82] 112

Figure A.3: Metal AM powder demand forecasts by industry [30] involved the direct manufacture of end-use products as of 2011, this is up from 3.9% in 2003 [143]. This is the industry’s fastest growing segment with a 60% annual growth rate [13]. One key to the increased number of end-use products has been improvements in the additive manufacturing of metal products. Figure A.3 depicts forecasts of the metal powder demand in AM by industry from 2014 to 2023. The dominant industries driving this demand are aerospace, automotive, and medical (including dental) [30]. Research in especially the aerospace industry has been instrumental in solving some of the current material and material property limitations associated with AM of metal parts, including advances in the printing of compositionally graded metals (metals with different properties at different locations) [43]. These advances then propagate to other industries and markets. Additionally, it should be noted here that AM metal powder demand constitutes only 1% of total metal powder demand [22]. Thus the price of metal powders used in AM is artificially inflated to a degree due to low demand and proprietary manufacturing processes by OEMs. As demand increases, metal powders for AM become standardized, and more aftermarket suppliers enter the market, the price for metal powders used in AM should decrease. Additional cost declines can occur as new AM technologies incorporate the use wire metal as feedstock [22]. However, even though AM clearly shows massive growth potential, it is important to realize that it will never never completely replace conventional manufacturing techniques. In terms of cost and energy use, casting still significantly outperforms AM for producing large quantities of the same part [45]. Additionally, limitations still exist in AM regard- 113 ing size, imperfections, performance, materials, and cost. As these issues are gradually addressed due to the vast research into AM technologies, the global economic impact of AM will continue to increase. In fact, the McKinsey Global Institute suggests that 3D printing could have an impact of up to $550 billion per year by 2025 [13]. 114

Appendix B

Numerical Derivatives

As it is often impossible to take analytical derivatives of the non-linear equations found in most real world engineering problems, numerical approximations to these derivatives are necessary. This section will focus on the techniques used in this manuscript, and thus is certainly not inclusive of the many different numerical techniques that exist for solving coupled partial differential equations (PDEs) and/or ordinary differential equations (ODEs). All of the information found in this section may be found in LeVeque [72] or most other books on numerical methods, unless otherwise stated.

B.1 Spatial Derivatives using Finite Diﬀerences

The ﬁnite diﬀerence (FD) method is a numerical technique used to approximate spatial derivatives by dividing a domain up into grid points, as shown in Figure B.1. Consider a spatially varying function u(x, y, z) and say we want to approximate the partial derivative ∂u ∂x |(x,y,z). One obvious choice would be to approximate this as u(x + ∆x, y, z) − u(x, y, z) D u(x, y, z) = , (B.1) x+ ∆x or

u(x, y, z) − u(x − ∆x, y, z) D u(x, y, z) = . (B.2) x− ∆x ∂u In these situations, as ∆x → 0, we obtain the true derivative ∂x |(x,y,z). Both of these one-sided approximations are considered ﬁrst order, or O(∆x), accurate (indicating that the size of the error is roughly proportional to ∆x). A better approximation is the 2nd order, O(∆x)2, accurate centered diﬀerence scheme

u(x + ∆x, y, z) − u(x − ∆x, y, z) 1 D u(x, y, z) = = (D u(x, y, z) + D u(x, y, z)). x0 2∆x 2 x+ x− (B.3) 115

z L y z ∆z ∆y L x y ∆x

Figure B.1: Illustrative mesh of a domain used in the ﬁnite diﬀerence method

To illustrate the accuracy of these schemes, let us consider a Taylor expansion about the point (x, y, z)

∂u (∆x)2 ∂2u (∆x)3 ∂3u 4 u(x+∆x, y, z) = u(x, y, z)+(∆x) + 2 + 3 +O(∆x) . ∂x (x,y,z) 2 ∂x (x,y,z) 6 ∂x (x,y,z) (B.4) Similarly,

∂u (∆x)2 ∂2u (∆x)3 ∂3u 4 u(x−∆x, y, z) = u(x, y, z)−(∆x) + 2 − 3 +O(∆x) . ∂x (x,y,z) 2 ∂x (x,y,z) 6 ∂x (x,y,z) (B.5) If we subtract these we can ﬁnd that

u(x + ∆x, y, z) − u(x, y, z) ∂u

Dx+u(x, y, z) = = + O(∆x), (B.6) ∆x ∂x (x,y,z) and

u(x, y, z) − u(x − ∆x, y, z) ∂u

Dx−u(x, y, z) = = + O(∆x). (B.7) ∆x ∂x (x,y,z) However, for the centered diﬀerence scheme, we obtain 116

u(x + ∆x, y, z) − u(x − ∆x, y, z) ∂u 2 Dx0u(x, y, z) = = + O(∆x) (B.8) 2∆x ∂x (x,y,z) since the first order terms cancel out; thus proving second order accuracy of the centered difference scheme. To take the derivative of a flux at point (x, y, z), we can use the following 2nd order accurate stencil,

∂u ∂u ! A − A ∂ ∂u ∂x (x+ ∆x ,y,z) ∂x (x− ∆x ,y,z) 2 2 A ≈ , ∂x ∂x (x,y,z) ∆x 1 " ∆x ! u(x + ∆x, y, z) − u(x, y, z)!# ≈ A x + , y, z (B.9) ∆x 2 ∆x 1 " ∆x ! u(x, y, z) − u(x − ∆x, y, z)!# − A x − , y, z , ∆x 2 ∆x where we can approximate

∆x ! 1 A x + , y, z ≈ (A(x + ∆x, y, z) + A(x, y, z)) , (B.10) 2 2 and

∆x ! 1 A x − , y, z ≈ (A(x, y, z) + A(x − ∆x, y, z)) . (B.11) 2 2 Here A(x, y, z) is a spatially dependent material property. This can be derived by using the centered difference scheme around the components in the first derivative. To take the cross derivative of a flux at point (x, y, z), we can use this 2nd order accurate stencil,

! ! ∂ ∂u ∂ u(x + ∆x, y, z) − u(x − ∆x, y, z)

A ≈ A(x, y, z) , ∂y ∂x (x,y,z) ∂y 2∆x 1 ≈ (A(x, y + ∆y, z)[u(x + ∆x, y + ∆y, z) − u(x − ∆x, y + ∆y, z)] 4∆x∆y − A(x, y − ∆y, z)[u(x + ∆x, y − ∆y, z) − u(x − ∆x, y − ∆y, z)]). (B.12)

Again, this can be derived by using the centered diﬀerence scheme around the components in the ﬁrst derivative. 117

Clearly for such methods, appropriate boundary conditions must be applied. Depend- ing on the nature of the equation to be solved, other forms of the FD scheme can be used, such as upwind or downwind methods. The interested reader is referred to LeV- eque [72] for more details about such schemes. Note that many other methods, such as finite elements or finite volumes (among others), can also be employed to calculate spatial derivatives depending on the specific problem.

B.2 Time Marching Schemes

While the FD scheme is used for calculating spatial derivatives for a boundary value problem, many diﬀerent time marching schemes exist for solving an initial value problem. In this section let us consider a diﬀerential equation of the form u0(t) = f(u(t), t) where u is an unknown function (scalar or vector) that is a function of time, t. The rate at which u changes is represented by u0(t). The rate of change of u is a function of time and the n variable itself. Let U represent the approximate solution at the current time (tn) step n+1 and U represent the approximate solution at the future time step (tn+1). The time step size is ∆t.

B.2.1 Euler Methods The simplest methods for evaluating u0(t) are known as Euler’s methods. The explicit 0 n Forward Euler methods is given by estimating u (tn) ≈ D+U as in Equation B.1, U n+1 − U n f(U n) = . (B.13) ∆t Given an initial value, it is simple to calculate the value at a future time step by

U n+1 = U n + ∆tf(U n). (B.14) Using the same Taylor series expansions as derived previously for the FD scheme, we can show that this Forward Euler scheme is 1st order, O(∆t), accurate in time. Forward Euler is considered an explicit scheme because it only requires evaluation of the function u0(t) = f(u(t), t) at the current time step. 0 n+1 The Backward Euler method is given by estimating u (tn+1) ≈ D−U as in Equation B.2,

U n+1 − U n f(U n+1) = . (B.15) ∆t The value at a future time step is then given by

U n+1 = U n + ∆tf(U n+1). (B.16) 118

This method is again 1st order, O(∆t), accurate in time. However, Backward Euler is considered an implicit method because it requires knowledge of the function u0(t) = f(u(t), t) at a future time step. This can be approximated using iterative methods (such as Newton’s method), but adds to the number of calculations required for each time step. Another implicit method is the trapezoidal scheme, which is obtained by averaging the Forward and Backward Euler methods, 1 1 U n+1 = U n + ∆t f(U n) + f(U n+1) . (B.17) 2 2 This method is 2nd order, O(∆t)2, accurate in time, which can be seen from the Taylor series expansion.

B.2.2 Runge-Kutta Schemes Runge-Kutta (RK) methods are multistage methods where intermediate values of the solution and/or its derivative are calculated and used within a single time step. As an example, let us ﬁrst consider a simple two-stage explicit RK method. A 2nd order, O(∆t)2, accurate RK method is given as follows 1 U ∗ = U n + ∆tf (U n, t ) , 2 n

∆t! U n+1 = U n + ∆tf U ∗, t + . (B.18) n 2 ∗ ∆t In this scheme U represents the solution at time tn + 2 if a Forward Euler scheme is used. This information is then fed back into the function u0(t) = f(u(t), t) to obtain the solution at time tn + ∆t. Higher orders of accuracy can be obtained by the use of more intermediate stages. A very commonly used RK scheme is the following 4th order, O(∆t)4, accurate explicit RK method (frequently referred to as RK-4)

n Y1 = U , 1 Y = U n + ∆tf (Y , t ) , 2 2 1 n 1 ∆t! Y = U n + ∆tf Y , t + , 3 2 2 n 2 ∆t! Y = U n + ∆tf Y , t + , 4 3 n 2 ∆t " ∆t! ∆t! # U n+1 = U n + f (Y , t ) + 2f Y , t + + 2f Y , t + + f (Y , t + ∆t) . 6 1 n 2 n 2 3 n 2 4 n (B.19) 119

Many other RK schemes exist, of varying orders of accuracy. Implicit and diagonally implicit RK schemes are also frequently used depending on the nature of the problem. A general r-stage Runge-Kutta method has the form

r n X Y1 = U + ∆t a1jf(Yj, tn + cj∆t), j=1 r n X Y2 = U + ∆t a2jf(Yj, tn + cj∆t), j=1 . .

r n X Yr = U + ∆t arjf(Yj, tn + cj∆t), j=1 r n+1 n X U = U + ∆t bjf(Yj, tn + cj∆t). (B.20) j=1 Consistency constraints require that

r X aij = ci, i = 1, 2, . . . , r, j=1 r X bj = 1. j=1 For simplicity, often these schemes are represented using a so-called Butcher tableau:

c1 a11 . . . a1r ...... cr ar1 . . . arr b1 . . . br

For example, the explicit RK-4 method given in Equation B.19 can be concisely represented via the following Butcher tableau

0 1/2 1/2 1/2 0 1/2 1 0 0 1 1/6 1/3 1/3 1/6 120

B.2.3 Other Schemes All of these methods are known as one-step methods since U n+1 is obtained only from knowledge of U n and not any other previous values of U. Multi-step methods exist, along with other multistage methods (in addition to Runge-Kutta). In this work purely explicit schemes are utilized. Implicit schemes oﬀer increased stability regions, allowing for larger time steps to be used (especially for so-called stiﬀ problems). However, implicit schemes also require iterations within each time step or solving a system of coupled equations, which increases the number of computations required per time step. Time step size is determined by the nature of the governing equation, the scheme used, and the physics of the problem. Adaptive time stepping schemes can be employed, such as the explicit method of Dormand and Prince [20] or the implicit adaptive schemes frequently used by Zohdi [152, 155, 156]. For more details about the described schemes, and others, the interested reader is referred to LeVeque [72] and other books discussing numerical solutions to ODEs and/or PDEs. 121

Appendix C

Basic Parallelization Techniques

Many techniques exist for speeding up simulations through parallel processing. Only the basic ones used in this work are described here. More details on other eﬃcient parallelization techniques can be found in the work by Hager and Wellein [37] and other works on high performance computing.

C.1 Binning Algorithm

To speed up computation time for discrete element simulations, a binning algorithm is often used. A basic check for interactions between N particles that could potentially be interacting with any other particle would require O(N 2) operations (check each of N particles to see if they are interacting with N other particles). Binning enables one to reduce the number of computations necessary from O(N 2) to O(N). This works by decomposing the domain into grid boxes, or bins (see Figure C.1). Particle interactions only have to be checked within the same or neighboring bins. In three dimensions this equates to checking interactions within 27 bins (including the particle’s own bin). While this may seem like a lot, it is considerably less than having to check every particle. Clearly as the bin size decreases the number of checks for each particle also decreases. However, care must be taken to not let the bins get too small, otherwise some interactions may be missed. One problem that arises when decomposing the domain into bins is the issue of particles moving from one bin to another as system time evolves. Thus it is necessary to recompute each particle’s location and corresponding bin at the beginning of each time step. However, if large enough bins are used then it may only be necessary to “re-bin” the domain every few time steps, rather than every time step. A balance between using larger bins to eliminate the need for “re-binning” every time step and using smaller bins to reduce the number of particle checks must be reached. This often is a function of the dynamics of the problem at hand. A linked list methodology is used in this work to program the binning algorithm, 122

Figure C.1: Illustration of binning algorithm. Only particles in the shaded boxes will be checked for contact with the red particle in the center box. following the approach used by Queiruga [103]. With this methodology a grid array is created that points to the ﬁrst particle in each bin, respectively. A separate array, the linked list, then points to subsequent particles within each bin. A location in the grid array or linked list is given a value of “-1” if no further particles occupy that bin. This is the way of telling the computer to stop search within that bin. A visual depiction of how the linked list works for an example 2D situation is shown in Figure C.2. A full pseudocode of the implementation/programming of this binning algorithm is shown in Algorithm C.1.

C.2 OpenMP

To further decrease computational time, parallelization across multiple cores can be utilized. In the majority of this work, simulations could be run within a matter of minutes to hours on a laptop computer since the program was written in a high performance language (Fortran 90 in the present case). However, in a few instances of larger simulations (those described in Chapter 6), OpenMP parallelization was employed to utilize all available cores on the laptop computer or available computing cluster. OpenMP parallelization is easily implemented for discrete element and ﬁnite diﬀerence schemes using explicit time-stepping, as done in this work. OpenMP works by having a shared memory across all processors, though each individual processor performs its set of operations independently. Shared memory allows for immediate access to all data without explicit communication between processors. A detailed discussion of OpenMP is beyond the scope of this section and the interested reader is referred to Hager and Willein [37]. This section is only designed to allow the reader to 123

Domain with particles and bins Linked list 1 8 2 4 1: 4 7 3 5 9 2: -1

6 3: -1 4: -1 2D grid array 5: 7

1 8 2 6: -1 7: 9 3 5 -1 8: -1 -1 6 -1 9: -1

Figure C.2: Linked list binning for a 2D example problem 124

Algorithm C.1 Binning algorithm set-up and contact check pseudocode 1. Deﬁne bin size (bin size must be ≥ maximum particle diameter)

2. Assign grid numbers to each bin

3. Loop through all N particles and store bin coordinates for each particle based on location of particle center

4. Create grid array that is the same size as the number of bins in each dimension. Each entry in the grid array is the number of the ﬁrst particle that lies within that bin. A value of “-1” is entered if there are no particles in that bin.

5. Create linked list, a vector array of size N. Each entry in the linked list points to the number of the subsequent particle in the same bin. A value of “-1” is entered if no further particles exist within that particle’s bin.

6. Check for particle-to-particle contact: loop through all N particles (current particle denoted by i)

(a) Loop through all neighboring bins for particle i (27 total in 3 dimensions) i. Ensure that bin does not go out of bounds of domain, if so then skip ii. Get particle j from grid array (corresponding to ﬁrst particle in that particular bin) iii. While j/ = −1 A. Check distance between particle i and particle j B. If distance < combined radii of particles, then contact exists and perform appropriate calculations C. Get next particle j from linked list

7. Re-bin as necessary depending on dynamics of the problem. 125 understand the OpenMP basics necessary to parallelize the codes used in this work. Note that the code snippets given below are Fortran commands, as this was the programming language used by the author. Similar commands exist for C/C++. First, a command must be placed at the top of the ﬁle to indicate use of the OpenMP library: use omp_lib

The number of cores used can be set in the terminal window:

$ export OMP_NUM_THREADS=4 $ . / a . out

This sample terminal script will run the program “a.out” using 4 cores. Note that many modern laptop computers have between 4 - 8 cores. The MacBook Pro used in this work contains 8 cores. If the number of cores to be used is not speciﬁed, the computer will default to using all available cores. Altering a pre-written serial code (of the type used in this work) so that it can run in parallel requires very few commands. The section to be parallelized must indicated. Additionally, a barrier command is required if synchronization of the processors is required before proceeding further in the code. An example is seen below:

!$OMP PARALLEL ! indicate beginning of parallelized section !$OMPDO ! begin parallel do loop do i = 1,N ! do loop for particles or FD nodes

! lines of code in this section end do !$OMP END DO

!$OMP BARRIER ! forces cores to syncronize before continuing

!$OMP DO do i = 1 ,N

! lines of code in this section end do !$OMP END DO 126

!$OMP END PARALLEL ! end parallelized section of code

In this example code anything within the “do loops” will be parallelized across the set number of cores. The information from the cores will be synchronized due to the barrier command “!$OMP BARRIER”. The “!$OMP END PARALLEL” command ends the parallelized section and also acts as a barrier forcing the cores to synchronize before continuing. Note that excessive use of barriers should be avoided as this adds to the over- head time required to run the program. These basic commands are enough to parallelize the explicit codes used in this work, where all variable values are updated at the end of each time step.