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State of the art


Academic year: 2022

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Selective laser melting is a powder bed fusion process that allows the production of metal- lic pieces of high geometrical complexity. Full densification is regarded as fundamental to achieve mechanical integrity. Nevertheless, doing so for a new material requires an intensive, in time and resources, experimentation stage in order to set proper manufacturing parameters.

In this work, dimensional analysis is used to develop a general mathematical model on bulk density of SLMed components taking volumetric energy density, scanning speed and pow- der’s thermal conductivity, specific heat capacity and average grain diameter as independent variables. Strong relation between dependent and independent dimensionless products was observed. Bulk density is found to be proportional to volumetric energy density and be af- fected by scanning speed by a factor of negative two. Inconel 718 probes were produced and a particular expression, in the form of a first order polynomial, for its bulk density,in the inde- pendent dimensionless product π1 range from 3.17x10−8 to 4.6 x10−8 was obtained. In this range, better densification is achieved at lower scanning speed and lower laser power. The first is related to higher exposure time and ensuring full melt of the powder, and the second may be due to powder particle sublimation / ejection due to improperly large laser power condi- tions. An average relative density of 95.218% was measured. An average error percentage of 1.6503% between experimental and predicted bulk density (and dimensionless density) was achieved. A mathematical tool for tuning scanning speed to achieve full densification, with respect to laser power, was developed. Moreover, particular conditions for achieving so for Inconel 718 in the π1 range was provided.



achieve a desired shape. This leads to a high degree of wasted material which, even if re- utilized, represents a considerable hit to any industry’s utilities. In contrast, additive manufac- turing (AM) is based on an incremental layer-by-layer manufacturing [2]. There are multiple AM processes. However, the most relevant technologies use powder or wire as a feedstock which is subjected to a focused heat source and subsequently consolidated during a cooling stage to form a part [3]. A deeper description of AM processes is provided in the following sections.

AM has attracted attention among industrial practitioners due to the benefits it brings;

reduction of waste materials, shortening of manufacturing lead times, high flexibility, feasi- bility of complex geometry products, and shortening of product development [4]. Wohler’s Annual Worldwide Report on 3D Printing and Additive Manufacturing states that the global revenue generated by AM production and associated services grew $ 3.75 billion from 2012 to 2016 ($2.25 billion to $ 6 billion) and the forecast is to reach $ 21 billion by 2020 [5]. At early stages technology development for the AM processes was focused on the production of mock- up models and prototypes in industrial and academic environments [6]. However, according to Dehoff et al. AM is advancing towards becoming a true manufacturing platform that can



produce the form, fit and function of a component [7]. Up to 2012, only 28% of AM compo- nents were functional. Nevertheless, in 2016 this percentage went up to 34 % [5].Nowadays, it is a promising technique for industries as the aerospace, automotive and biomedical.

1.2 Problem Statement

Selective Laser Melting (SLM) is an additive manufacturing process that produces metal com- ponents from metallic powders. It uses a high intensity laser to melt and fuse, layer by layer, specific regions of the powder bed according to computer aided design data (CAD) and cross section of the three dimensional structure [8]. Because of the interaction between the laser and metal powder, SLM involves several physical and thermal phenomena (phase changes with the melting and solidification of metal powders), mechanical phenomena (residual stresses due to temperature gradients) and hydrodynamic phenomena (convection in the melt pool and Mariangoni convection) [9].

The attainable bulk density of a SLMed piece is the most important concern. Mechanical properties, and thus component performance, is highly linked to its density [10]. The objec- tive, is therefore to obtain pieces as dense as possible (minimize porosity) . The adequate selection of process parameters allows to obtain a component with full densification and thus excellent mechanical integrity.

SLM involves a wide variety of factors which will determine the outcome of the man- ufactured probe. Among these are operating parameters, powder material characteristics and atmospheric conditions. More than 50 of these may be listed (discussed in further section) and each one has an influence on the final piece’s densification. The problem rises on the appro- priate determination of these parameters. Adapting a material to a SLM productive scheme involves then an intensive experimental stage which impacts economically and in efficiency any industry.


fly cheaper. Furthermore, about 9% of greenhouse gas emissions emerge from the aviation industry and it is predicted for this percentage to grow three times by 2050. A lighter aircraft lessens fuel consumption and diminishes greenhouse emissions producing a less harsh impact on the environment [11].

As stated by Gartner’s 2018 hype cycle for 3D printing (AM) the enterprise 3D print- ing technologies and services market continues to expand. The disruption brought in by AM comes with cost-saving and revenue-generating opportunities. However, as today, its im- plementation requires intensive and lengthy testing and development in order to define the optimal operating parameters for each material to be used [12].

As a consequence, the motivation of this work is to present a mathematical model via dimensional analysis (DA) that reduces the lengthiness of said adaptive process for selective laser melting. Shortening this experimentation stage will also bring less material waste and smaller time investment. The mathematical model will provide an objective relation between parameters with physical significance and will simplify the determination and tuning of oper- ating conditions for the material of study. Reducing waste (both of time and materials) leads to a more environmentally-friendly process.

1.4 Solution Overview

In order to reduce the time it takes to determine the optimal processing parameters that ensure high bulk density resulting pieces, and achieve the contributions mentioned in the previous section, a mathematical model will be obtained. The solution will be based upon dimensional analysis. DA offers a method for reducing complex physical problems, such as SLM, to the


simplest form prior to obtaining a quantitative answer [13]. The mathematical procedure will be explained in further depth in the second chapter of this work.

Porosity on SLMed parts has been identified as one of the most important challenges to overcome. As previously stated, its application in aerospace industry demands the manu- factured pieces to have the highest mechanical integrity to ensure optimal performance. On this regard, the behaviour of bulk density will be the objective function to model. The process parameters that affects component’s density are identified and will be the factors involved in the dimensional analysis developed in this work.

An experimental validation stage is necessary to obtain the particular form for the ex- pression that will model density. This will consist in the production of several pieces via SLM varying selected process parameters included in the mathematical approach. The exper- imental design is further described in section 3. Renishaw’s AM 400 machine will be used.

The produced specimen’s density will be measured through Buoyancy method using Mettler Toledo’s density kit.

The present is a pioneer work in the research community. According to Scopus, (search of ”Dimensional analysis” AND ”SLM”) published studies on dimensional analysis for the SLM process are, as of today, only three. Institutions such as the Universit´a di Salerno (Italy) , Technichal University of Berlin (Germany), Indian Institute of Technology (India), KU Leu- ven (Belgium) and the Federal Institute for Material’s Research and Testing Berlin (Germany) have preceded this work. In specific, Van Elsen et al. presented a possible complete set of dimensionless parameters to describe the process and ease comparison between works [14].

Cardaropoli et al. used dimensional analysis intending to find out an appropriate definition of a set of non-dimensional groups in order to represent the output parameters on the selective laser melting process [15]. The most recent study, performed by Khan et al., used dimensional analysis to produce a heat transfer model on the SLM process [16].

1.5 Hypothesis and Objective

It is important to remark that the mathematical model produced in this work will help produce full density components in an ideal scenario. For a real component, a post processing stage


• Determine the dependent and independent dimensionless numbers that describe the physical process of selective laser melting

• Obtain a general expression, through dimensional analysis, that describes bulk density behaviour, related to the manufacturing parameters involved, of SLMed metallic com- ponents

• Obtain a particular form of the dimensional analysis expression for the bulk density of SLMed Inconel 718 pieces

• Quantify the precision of the particular expression for bulk density of SLMed Inconel 718 pieces

• Produce a mathematical tool for reducing lengthiness, of time and resources, of deter- mining proper manufacturing parameters for a material of choice obtaining pieces with full densification

The hypothesis of this work is that the density of metallic pieces produced by SLM can be mathematically modeled through the use of dimensional analysis with high precision.

And thus, provide a tool to determine adequate (high density resulting components) operation parameters without the need of intensive experimentation stages.


State of the art

2.1 Additive Manufacturing

2.1.1 Definition and General Procedure

The technology was developed by Charles Hull in 1986 in a process known as stereolitogra- phy. This was followed by further developments such as powder bed fusion, fused deposition modelling, inkjet printing and contour crafting[17]. These processes, among contemporary developments, will be explored in the following subsection. AM uses liquid or semisolid paste, powder and solid for building products. The technology emerged as an efficient tool for rapid prototyping. However, it has evolved since to a competitive technique to produce final end near net shape pieces [18]. Fabrication of a wide range of structures and complex geometries from three dimensional model data is achieved through AM [17].

AM starts from a concept design or modification of a previous one [18].This translates into the creation of a virtual model which incorporates the data, provided by design engineers or from existing designs in reverse engineering. [19]. A virtual model ,produced via Computer Aided Design (CAD) tools, can be created by many modelling softwares such as Autocad, PTC Creo, Solidworks, Catia and Unigraphics. Models may also be created using reverse engineering. This is achieved by using scanning machines. A virtual prototype model is created through the use of UV light as the source of scanning which is emitted from the source and received back after hitting the target object to be scanned [20]. The incoming UV



3D object is performed through any AM process of choice in a layer by layer process. Figure 2.1 illustrates the previously described process.

Figure 2.1: Process flow of the general additive manufacturing process. Design data is gen- erated from a concept design or modification. Afterwards, a CAD file is either manually generated or via 3D scanning and later converted into STL format. From this, a g-code is gen- erated which provides the machine with operating instructions and trajectory paths to achieve the formation of the final piece. Finally, the three dimensional object is built via the AM process of choice.

2.1.2 AM Processes

A wide variety of AM processes have been ,and are still being developed in order to produce pieces made out of a huge range of materials. The main focus is to produce complex structures at fine resolutions with adequate mechanical properties for the function they were thought to fulfill. The 3D printing, or additive manufacturing, hype cycle as of 2018 is presented in Fig.

2.3. This will result useful for identifying in which stage the technology is currently located regarding consumption, development and expectations.

The main AM processes are fused deposition modelling (FDM), powder bed fusion (PBF), comprised by selective laser sintering (SLS), selective laser melting (SLM), and 3D printing. These main methods also include binder jetting, inkjet printing and contour crafting, directed energy deposition (DED) and laminated object manufacturing (LOM). Schematic di- agrams of FDM, inkjet printing, stereolitography (SLA) and powder bed fusion (PBF) are


presented in figure 2.2. A summary of these method’s materials used, advantages and disad- vantages are presented in table 2.1.

Figure 2.2: Schematic diagrams of four of the main methods of additive manufacturing. (a) FDM: thermoplastic polymer is extruded in the melt state through a nozzle and deposited on the print bed; (b) Inkjet printing: a ceramic stable solution is deposited drop by drop in the fabrication platform repeating the process in a layer by layer manner once solidified ; (c) SLA: photosensitive resin in the liquid state is exposed to a UV light beam which solidifies the material on the particular trajectory followed; (d) PBF: a laser heat source, with the addition or not of a binder agent, incides on a powder bed to produce a three dimensional component [17].

Fused deposition modelling

In FDM (Fig. 2.2 (a) ) a continuous thermoplastic polymer filament is heated at the nozzle un- til melt and then extruded and deposited forming the cross sectional layer of the corresponding build step[17].The main processing parameters involved in FDM are layer thickness, width and orientation of filaments and air gaps [24]. The main benefits are its low cost, high speed and process simplicity [25]. Weak mechanical properties (caused by inter-layer distortion),


Figure 2.3: 3D printing hype cycle as of 2018. Illustrates current expectations, and its ex- pected tendency in the following years, of AM processes and applications. Moreover, it in- dicates the expected time period for the specific technology or application to become main- stream [23].

surface quality and narrow range of materials to be used are the main disadvantages of FDM [26]. Material extrusion is regarded to have reached the plateau of productivity.

Powder bed fusion

PBF process consists in the heating (fusion or sintering may be achieved), with a laser as energy source , with or without the use of a binder as packing agent, of compacted powder.

The piece is built layer by layer. When fabrication is complete, the excess powder is recol- lected/removed with vacuum and processed for further re-utilization. Factors as powder size distribution and packing, and laser power and scanning speed are considered as crucial factors for the density of the manufactured parts [17]. PBF technologies are regarded to be located in the through of disillusionment and will reach the plateau of productivity in 5 to 10 years.

An schematic of the steps involved in the PBF technology’s production process is presented in figure 2.4. (1) A powder layer is deposited and distributed by a recoater, (2) a laser beam


Table 2.1: Summary of methods, materials, advantages and disadvantages of additive manu- facture

Methods Materials Applications Advantages Disadvantages

Fused deposition modelling

Continuous filament of thermoplastic polymers

-Rapid protyping -Toys

-Low cost -High speed -Simplicity

-Weak mechanical properties -Limited materials Powder bed fusion


Compacted powders:

Metals and alloys (SLS and SLM) Ceramic and polymers (3DP)

-Aerospace -Biomedical -Lattices

-Fine resolution -High quality

-Slow manufacture -Expensive Inkjet printing and

contour crafting

-Ceramic solution (Inkjet printing) -Concrete, soil (Contour crafting)

-Large structures -Construction

-Quick process -Abilitiy to produce large pieces

-Coarse resolution -Lack of adhesion between layers Stereolitography Photoactive polymer resin -Biomedical


-Fine resolution -High quality

-Limited materials -Slow process

Directed energy deposition

Metals and alloys (powders or wire)

-Aerospace -Repair -Biomedical

-Reduced processing time and cost -Excellent mechanical properties

-Controlled composition and microstructure

-Low accuracy -Low surface quality -Limitation for complex structures

Laminated object manufacturing

-Polymer composites -Ceramics


-Electronics -Smart structures

-Reduced tooling and manufacturing time -Low cost -Ability to produce large structures

-Low surface quality -Low dimensional accuracy -Limitation for complex structures

scans the powder bed and (3) the building platform is lowered and the process is repeated in a layer by layer manner.

In SLS full melt is not performed. The elevated local temperature on the surface of the grains results in solidification at a molecular level, also known as solid state diffusion.

On the other hand, powders are fully melted and fused together in selective laser melting.

When using a liquid binder (3D Printing) the most important factors are the composition and rheological properties of the binder along with the size and shape of powder particles and deposition speed [28].

The main advantages of PBF are the achievable piece’s fine resolution, high quality and ease of removal from the build plate. On the otter hand, the main disadvantages include the lengthiness of the process and high investment cost to get the process going [17].

Inkjet printing and contour crafting

Inkjet printing (Fig. 2.2 (b) ) is mainly used for producing complex ceramic structures.

Droplets of stable ceramic suspension are deposited in a substrate forming a continuous pat- tern layer by layer [29]. Contour crafting uses large nozzles at high pressures to deposit


Figure 2.4: Schematic of the steps involved in powder bed fusion process. (1) a powder layer is deposited and distributed by a recoater, (2) a laser beam scans the powder bed and (3) the building platform is lowered and the process is repeated in a layer by layer manner [27].

concrete paste or soil [30]. Inkjet printing is fast and efficient while coarse resolution and lack of adhesion between layers are its main challenges. Particle size distribution, viscosity and solid content of the ceramic suspension , as well as extrusion rate and nozzle size are the main factor which determine the quality of the piece[31]. According to Wohler’s additive manufacturing hype cycle, this method will be regarded as mainstream in a threshold of 2 to 5 years.


SLA (Fig. 2.2 (c) ) works with the radicalisation of a UV-active monomer solution via UV light or electron beam. This method offers high-quality parts yet is slow and expensive. The main factors involved in the process are the energy source for curing and its exposure time [32]. SLA is entering the slope of enlightenment regarding the hype cycle for additive manu- facturing and its expected for it to reach the plateau of benefit in 2 to 5 years.


Directed energy deposition

DED (figure 2.5) uses an energy source (laser or electron beam) to melt feedstock material (powder or wire) which is deposited on the substrate and solidified. In this technique, the feedstock is melted before deposition and no powder bed is used [33]. DED is commonly used for low complexity bulky components, reduces manufacturing time and cost while producing parts with excellent mechanical properties and providing the ability to control component’s microstructure and chemical composition. Nevertheless, it produces low surface quality and is not applicable for complex structures [17]. Currently, DED is considered to be entering the trough of disillusionment and is expected to achieve mainstream use in 5 to 10 years.

Figure 2.5: Schematic of the directed energy deposition process. Deposition and fusion of the metallic powder is performed simultaneously producing pieces with excellent mechanical properties but low geometrical resolution [34].

Laminated object manufacturing

LOM is based on the successive formation of cross sectional layers by cutting (with laser or mechanical cutter) and (bonding) of sheets of materials. The process may be performed in the contrary order depending on the ease of fabrication. Materials such as polymer composites, ceramics and paper are used in this method [35]. The main advantages of LOM are the reduction of tooling cost and manufacturing time. However, pieces produced via this method have lower surface quality and dimensional accuracy compared to PBF [17]. LOM is, as of 2018, in the peak of inflated expectations and it is forecasted to reach mainstream usage from 5 to 10 years.


Figure 2.6: Schematic of the laminated object manufacturing process. Sheets of material covered with a binding agent are piled layer by layer and cut to the desire shape with a laser beam forming thus the final object [36].

2.1.3 Selective Laser Melting

SLM involves heating, melting and solidification of an alloy powder by a moving heat source such as a laser or an electron beam in a layer by layer manner [37]. Each scanning process by the power source produces a thin layer of the final product form. The final component is therefore completed in an iterative process of depositing feedstock, power scanning, melting the feedstock, and undergoing a solidification lapse [38]. Once the laser repetitive process is completed, loose powders are removed from the building chamber and the produced piece is separated from the substrate plate either manually or by electrical discharge machining as required [8]. The technology is used to produce complex structures that are previously designed on a CAD-CAM system. Said design is afterwards converted to the STL file format where all surfaces of the design are approximated by polygons. This STL file is then sliced into numerous cross-sections of a layer thickness pre-defined by the user [39]. Fig. 2.7 illustrates the previously described process.

AM for metals is categorized depending on the type of melting power source and/or its feedback deposition. The most popular ones are SLS and SLM [38]. The main difference between both technologies is the point of temperature on which either processes operate.

While SLM was initially only applied to pure metals and achieved melting of the powders, SLS processes did not heat the powers to such temperature. However, they may be used nowadays interchangeably in practice as SLS systems now reach melting temperatures. They


Figure 2.7: Schematic of the SLM additive manufacturing process. A metallic powder, located at a feed container, is homogeneously spread in a building platform by a roller/recoater. A laser incides on the powder bed melting the material in an inert environment (N2 is here referred as shielding gas agent). After a cross section of the three dimensional object is melt and solidified, the roll displaces the remaining powder to an overflow aperture. The build cylinder is lowered and the procedure is repeated [14].


are both referred to as powder bed fusion processes [27]. Nevertheless, they are commonly referred interchangeably, the main difference between them is the temperature point they both reach. SLM achieves full melt of the material while SLS heats the material up to a point of sintering between powder particles in a solid state diffusion process. SLM fully melts the powder material, producing components of high density and near net-shape without the need of post- processing aside of the removal of parts and supports from the substrate plate. On the other hand, SLS binds powder via solid state sintering or melting of binding agents. This leads to parts with high porosity and low strength. Post processing stages as heat treatment and material infiltration are commonly used to enhance the SLSed component’s mechanical integrity. SLM is therefore a superior AM process compared to SLS as it has no need for binder materials and post processing is not mandatory [8]. Nevertheless, a post processing stage is commonly applied as well to the SLMed component in order to ensure low porosity and therefore excellent mechanical integrity.


a determined exposure time (in the range of µs) in a point by point manner distanced by a determined length. As well, distribution of energy in time is not the same in both processes.

CW lasers apply energy with an average value of laser power which is constant throughout the trajectory. PW power output is not continuous. Constant drops and peaks in power are observed as transitions between energy input and traslational operations occur. If a PW laser is modulated, the wave is square in shape and therefore is similar to the average power value that of PW lasers.

As an SLMed component may be thought as a pile of laser weld beads, the process presents similarities to laser assembly processes. Nevertheless, they differ in several aspects.

First of all, laser weld beads dimensions are, as the minimum, a few millimeters wide. On the other hand, the laser trajectory line thickness may be, as a maximum, around a hundred microns. Scan and cooling speeds are much lower in laser welding [40]. In both processes a melt pool is produced. However, the environment for each melt pool are quite different.

In laser welding, melting occurs continuously within one or more materials in the solid state [41]. On the other hand, in the SLM process the weld trajectory may be in contact with the powder, with already solidified material or with a combination of the two [1]. Both cases are depicted on figures 2.8 and 2.9.

During the SLM process, the building chamber is filled with a gas agent that provides an inert atmosphere to protect the built against oxidation. Nitrogen and Argon are usually used.

In the same way, certain SLM machines may provide pre-heating either to the substrate plate or the whole building chamber. Layer thickness is to be determined as to achieve a balance between fine resolution and good powder flowability [42]. Powders with large grain size result in poor resolution while the opposite tend to agglomerate easily due to Van der Waals forces


Figure 2.8: Physical phenomena during laser welding. A flux of laser produces a fusion zone.

Melting occurs continuously withing one or more materials in the solid state. Scanning speed and cooling conditions are slow compared than to those of SLM [41].


which result in poor powder flowability [8].

SLM may be reduced to a heat transfer process where a laser transfers energy to a pow- der bed.The powder is melted and a solidification period is allowed. Conduction, convection and radiation related heat transfer occurs in the SLM process. When the laser beam incides the powder bed a part of the energy is absorbed and the remaining is emitted to the surround- ings. Moreover, the absorbed energy causes the material to melt and heat conduction occurs in different directions. Heat conduction takes place from the melt pool to the surrounding powder, from the powder to the substrate, from the substrate to the machine and within the powder itself. Moreover, convective heat flow is also present in the interface of the top layer and the atmosphere in the direction of gas flow [1]. The resulting properties of the piece will depend upon said heat transfer conditions among with sub-processes such as laser systems and optics, energy absorption, phase changes, fluid flow as Mariangoni convection , sublimation and ejection of particles.


Figure 2.9: Heat transfer conditions in melt pool formation and solidification in the SLM process. An incident laser beam creates a melt pool in the powder bed. A fraction of the energy input is absorbed by the powder and the rest is radiated and reflected. Changes in phases and interaction between them are present. Conductive, convective and radiative heat flow occurs [1].


2.1.4 SLM Process Parameters and Characteristics

SLM involves a series of process parameters such as laser power, scanning speed, hatch spac- ing, and layer thickness. These, among the intensive list presented as follows, are adjusted such that a single melt line can fuse completely with the neighbouring melt trajectories and the preceding layer [8]. The key parameters of SLM are presented in detail by Spears. In his work, the author divides these parameters into four categories: (1) laser and scanning param- eters (table 2.2) , (2) powder material properties (table 2.3) , (3) powder bed properties and recoat parameters (table 2.4) , and (4) build environment parameters (table 2.5). Of the 50 parameters listed, only twelve are directly modifiable during the process. [1].


Table 2.2: SLM process parameters related to Laser and Scanning [1]

Parameter Description Controlled or

predefined Laser and Scanning parameters

1. Average power Measure of total energy output of a laser Controlled

2. Mode Continuous or pulsed wave Predefined

3. Peak power Maximum power in a laser pulse Predefined 4. Pulse width Length of a laser pulse when operating in pulsed



5. Frequency Pulses per unit time Predefined

6. Wavelength Distance between crests in laser electromagnetic waves


7. Polarization Orientation of electromagnetic waves in laser beam


8. Beam quality

Related to intensity profile and

used to predict how well beam can be focused and determine minimum theoretical spot size (equal to 1 for a Gaussian)


9. Intensity profile Determines energy input at a specific locations


10. Spot size Length and width of elliptical spot (equal for circular spots)


11. Scan velocity Velocity at which laser moves across build plate


12. Scan spacing Distance between neighboring laser paths Controlled 13.Scan strategy Pattern in which the laser is scanned across the

powder bed (hatches, zig-zags, spirals, etc.)


Laser, depending on its mode, may be categorized as of continuous wave (CW) or pulsed wave (PW). PW emission results in an intermittent release of energy during the process [43].


the resulting piece. Other important characteristics of the laser that may be enlisted are its wavelength, polarization and intensity profile.

Table 2.3: SLM process parameters related to Powder Material [1]

Parameter Description Controlled or

predefined Powder material properties

14. Theoretical density Material density, limits maximum density of final component

Predefined 15. Thermal conductivity Measure of material’s ability to conduct heat Predefined 16. Heat capacity Measure of energy required to raise the

temperature of the material


17. Latent heat of fusion Energy required for solid-liquid and liquid-solid phase change

Predefined 18 Melting temperature Temperature at which material melts Predefined

19. Boiling temperature

Temperature at which material vaporizes;

may only be important in certain process conditions


20. Melt pool viscosity Measure of resistance of melt to flow Predefined 21. Coefficient of thermal


Measure of volume change of material on heating or cooling


22. Surface free energy Free energy required to form new unit area of solid-liquid interfacial surface



Table 2.3 continued from previous page

Parameter Description Controlled or

predefined 23. Vapor pressure Measure of the tendency of material to


Predefined 24. Heat (enthalpy) of


Energy associated with a chemical reaction of the material


25. Material absorptivity

Measure of laser energy absorbed

by the material, as opposed to that which is transmitted or reflected


26. Diffusivity Important for solid state sintering, not as critical for melting

Predefined 27. Solubility Solubility of solid material in liquid melt Predefined 28. Particle morphology Measures of shape of individual particles

and their distributions

Predefined 29. Surface roughness Arithmetic mean of the surface profile Predefined 30. Particle size


Distribution of particle sizes Predefined

31. Pollution

Ill-defined factor describing change in properties of powder due to reuse

as dust and other particles added to powder


The interaction between the laser beam and powder is closely related to its thermo- dynamic properties. Theoretical density will be the target and limit of densification for a particular material. Thermal conductivity is an important factor as heat conduction occurs as previously discussed. Heat capacity is a quantification of the energy required to raise the temperature of the material (with melting temperature as target) and thus is closely related to the sufficient energy input which achieves melting. Powder starts from solid state, transitions to liquid when melt and solidifies back. Latent heat of fusion is a measurement of the energy required for such phase changes to occur. Boiling temperature along with vapor pressure are


as it influences the required energy input. Particle size distribution and morphology are con- sidered as relevant powder characteristics. Pollution of the powder occurs when reused and/or mistreated.

Table 2.4: SLM process parameters related to Powder Bed properties and Recoat parameters [1]

Parameter Description Controlled or

predefined Powder bed properties and recoat parameters

32. Density Measure of packing density of powder particles, influence heat balance

Predefined 33. Thermal


Measure of powder bed’s ability to conduct heat


34. Heat capacity Measure of energy required to raise the temperature of the powder bed


35. Absorptivity Measure of laser energy absorbed,

dependent on Ab and state of powder bed

Predefined 36. Emissivity Ratio of energy radiated to that of black body Predefined 37. Deposition


Recoater velocity, pressure, recoater type, dosing parameters


38. Layer thickness Height of a single powder layer, limiting resolution and impacting process speed

Controlled 39. Powder bed

temperature Temperature of the powder bed Controlled


Powder bed characteristics are related to, but different, to those of the powder material.

This difference depends upon the packing density (which is a function of particle size and shape distribution) of particles in the powder bed. Free space may significantly alter thermo- dynamic properties, such as thermal conductivity, heat capacity, absorptivity and emissivity, of the material. Moreover, layer thickness has been deemed to have a considerable influence on densification.In the same way, deposition system parameters, such as recoater velocity, pressure, recoating type and dosing parameters, are characteristics to take into account when producing an SLMed part.

Table 2.5: SLM process parameters related to Build Environment parameters [1]

Parameter Description Controlled or

predefined Build environment parameters

40. Shield Gas Usually Ar or N2, but may also be He Predefined

41. Oxygen level

Probably most important environmental parameter; oxygen can lead to oxide formation in metal, change wettability, and energy required for welding


42. Shield gas molecular weight

Influences heat balance, diffusivity into/out of part Predefined 43. Shield gas


May influence free surface activity of melt pool, convective heat balance

Predefined 44. Thermal


Term in heat balance Predefined

45. Heat capacity of gas

Term in heat balance Predefined

46. Pressure Influence vaporization of metal as well as oxygen content

Controlled 47. Gas flow


Influences convective cooling, removal of condensate



50. Surface free energy

Between liquid and surround gas influence melt pool shape


The fourth category is related to build environment parameters. Shield gas, usually Ar,N2 or He, is used to prevent oxidation (minimize oxygen level) and other chemical reac- tions in the powder bed. Some of its important characteristics are its molecular weigh, viscos- ity, thermal conductivity, heat capacity and convective heat transfer coefficient. Convection is deeply influenced by the gas flow velocity. Ambient pressure influences vaporization of metal as well as oxygen content.

To these 50 parameters, Galy remarks the importance of considering the following pa- rameters as well [9];

• Flowability of powder: poor flowability may lead to poor powder dispersion when dis- tributed

• Hydrometry of powder: high humidity may also lead to poor dispersion

2.1.5 SLM research and design parameters

Research on SLM has focused on the influence of the process parameters on the produced metallic piece characteristics. Tiwari et al. studied the laser power variation effect on AlSi10Mg / graphene composite’s porosity and hardness [44]. As well, Darvish et al. evaluated grain solidification orientation of a Co-Cr-Mo alloy over a range of laser power [45]. Morgan et al. studied the effect of scanning speed and scan spacing on stainless steel 316L [10] density.

Wang et al. investigated the influence of laser scanning speed on micro-segregation in SLM


of binary Fe-C alloy [46]. The effect of laser power and scan speed on the relative density, melt pool depth and Vickers hardness of SLMed maraging steel was evaluated by Suzuki et al. [47]. Pleass et al. investigated Particle Size Distribution (PSD) and flowability of Inconel 625 powder on mechanical properties and microstructure [48]. Similarly, Zhang et al. studied the influence of particle size on the powder-to-laser absorptivity [49] on tungsten. Morphol- ogy of TiB2/316L stainless steel composite powder was evaluated as a factor for hardness and microstructure[50]. Bajaj et al. developed normalized process maps of energy density vs hatch spacing identifying processability windows. Process parameters such as hatch spacing, layer thickness, exposure time and point distance are optimised for density in the manufacturing of molybdenum and aluminium components [51]. Mechanical, fractographic and compositional properties of Co-Cr metallic components were evaluated regarding layer thickness [52]. The influence of laser power, scan speed and scan spacing on porosity of AlSi10Mg alloy was studied by Read et al. [53]. Alfaify et al. studied the effect in densification of laser power, exposure time, point distance and hatching distance independently on [54].

Energetic input has been explored as a key concept in controlling the SLM process. This concept relates input power by the laser, scanning speed and process characteristic lengths such as hatch spacing and layer thickness, which are the most commonly used. Other charac- teristic lengths used in previous works are the laser spot size, average particle diameter, and point distance. Figure 2.10 presents an illustration for such parameters.Energetic density has been defined by several authors per unit length (eq. 2.1) , area (eq. 2.2) or volume (eq. 2.3).

κ = P

v (2.1)

ϕ = P

vCL1 (2.2)

$ = P

vCL1CL2 (2.3)

Where κ (J/m), ϕ (J/m2) and $ (J/m3) are the energy density per unit length, area and volume respectively, v is the scanning speed, and CL1, CL2 are process characteristic lengths such as hatch spacing, layer thickness,laser spot size and average particle diameter. As ob- servable, the energy density increases when laser power increases and decreases when hatch


energy per unit area and its influence on mechanical properties of 18Ni300 [58]. Work on the microstructure and phase evolution of SLMed Ti-Si-C composites were studied regarding energy density per unit area by Chen et al. [59]. Moussaoui et al. varied VED on Inconel 718 specimens and studied its microstructure, porosity and microhardness [60]. Koutiri et al.

performed a parameters optimization process focusing on the variation of VED on Inconel 625 parts regarding its densification, fatigue life, and spatter contamination [42]. Tucho et al. investigated densification and hardness of SS316L regarding VED [61]. Jadhav et al.

evaluated the effect of laser power, scanning speed and hatch spacing independently and in the form of VED on pure copper porosity [62]. Jing et al. studied the influence of VED in the densification of 300M steel and how the melt pool shape changes with respect to laser power [63]. Studies were performed on Ti-6Al-4V regarding the effect of VED and building orientation on its tensile properties by Pal et al. [64]. VED dependence of texture anisotropy and mechanical properties of SLMed Inconel 718 was studied by Liu et al. [65]. Ghayoor et al. evaluated the role of VED on the microstructural evolution, texture and mechanical properties of 304L stainless steel parts produced via selective laser melting [66].

Even though VED has been thoroughly studied on a wide variety of materials, several authors have stated that such is not able to fully describe the SLM physical process and there- fore should be used with caution as a design parameter. Working with SS316L, Scipioni et al.

produced single tracks of metal deposition and compared its density results with previously published work, which should produce pieces with full densification. The authors concluded that VED fails to capture melt pool physics hence its suitability as design parameter is limited to a narrow band of applicability [67]. Mishurova et al. evaluated near surface residual stress and porosity of SLMed Ti-6Al-4V characteristics regarding VED. However, it states that this


parameter is by no mean exhaustive and further investigates the influence of laser de-focusing and sample positioning inside the build chamber [68].

Besides the experimental design methodologies, computational algorithms and tools are used to study the process [69]. Tawfik et al. evaluated the influence of the laser scanning speed on AISI 3014 stainless steel part distortion. A 3D finite element model (FEM) was developed to simulate the deposition process and predict thermal gradients [70]. Conti et al.

developed a Finite Element Model (FEM) evaluating the effect of process parameters such as laser power, scanning speed and overlap percentage (degree of overlap of melting trajectories) on heat distribution and residual stresses in components fabricated via SLM [71]. Sun et al. predicted the temperature profile and melting of an aluminum alloy via 3D FEM [72].

Adaptive meshing techniques were used by Zeng et al. to determine thermal gradients near the laser incidence points [73]. Moser et al. presents a part scale continuum model predicting thermal stresses which incorporate thermal, laser and mechanical properties for stainless steel 316 [74] Ahmadi et al. investigated the response of 316L stainless steel to the melt pool size, overlapping melt pools, texture, process-induced defects and the orientation of layers with respect to the loading direction regarding mechanical properties through a computational model [75]. Gu et al. developed a discrete element method in order to model multi-track, multi-layer and multi-material SLM process [76]. Rifolfi et al. produced a numerical tool for designing processing windows with the objective of manufacturing fully dense parts [77]. A three-dimensional finite element heat transfer simulation tool to estimate the size of melt pool cross section with respect to a heat source and powder size was developed by Tran et al. [78].

Mishra et al. developed a numerical model of selective laser melting which incorporates a volumetric heat source [55].

2.1.6 Applications

AM has evolved over the years and has the ability to to transform manufacturing and logistics processes. It has been already applied in a wide variety of industries. One of the main drivers for the increase of accessibility to said technologies is the expiry of earlier patents. This has given manufacturers the ability to further develop new technologies and improve the existing ones.


Figure 2.10: Schematic of the factors involved in the energy density parameter. This may be defined in terms of energy per unit length, area or volume. It is generally defined by the quotient of laser power and the product of scanning speed and a product of characteristic lengths of the process. Said may be layer thickness or hatch spacing [8].

Through SLM it is possible to produce metallic parts with intricate geometries in high three-dimensional accuracy [79]. The sectors where SLM has found its most significant con- tributions are in high value-added industries, such as aerospace, automotive and medicine [8].

Nickel based super alloy abrasive turbine blade tips where produced, by Das et al., through SLM where gas leakage was minimized and gas turbines efficiency was improved [80]. Alu- minum based metal matrix composites that exploit the light-weight, high specific strength and thermal conductivity of aluminium were studied by Dadbakhsh et al. and directed towards aerospace and automotive industries[81]. A proposal for potential porous implant or drug delivery system was given by Shishkovskii et al. via HA processed with NiTi (known for its high strength, high corrosion resistance, biocompatibility, and shape memory effect) [82].

Inconel 718 is a popular alloy for additively manufactured components.Applications are strong in aircraft, gas turbines, turbocharger roots,nuclear reactors, liquid fuelled rockets and several corrosive and structural applications involving high temperatures [83]. Ni-Cr based super alloy is chosen for its strength, creep resistance, good weldability fatigue life at


temperatures up to 700° C [84]. Aircraft engine components such as critical rotating parts, aerofoils, supporting structures, and pressure vessels (which sum up approximately 30 of its total weight percentage have been produced with Inconel 718 [85]. In figure 2.11a and 2.11b two relevant applications of SLM in the aerospace industry are presented. The firs one is a nickel based super alloy abrasive turbine blade tips produced by Das et al., through SLM [80]

and the latter is an SLMed protective structure from Mars specimens re-entering Earth from expedition [36].

(a) (b)

Figure 2.11: Relevant applications of SLM in the aerospace industry: (a) Nickel based super alloy abrasive turbine blade tips produced by Das et al., through SLM [80] and (b) SLMed protective structure for Mars specimens re-entering Earth from Mars [36].

2.1.7 Post processing in SLM

The as-built pieces generally have a considerable degree of porosity which is undesirable.

Therefore, post processing for selective laser melting is crucial for ensuring proper mechan- ical integrity. The main objective of implementing a post processing stage is to diminish piece’s porosity and achieve a relative density as close as possible to one (which indicates full densification). It has been observed that post processing promotes microstructural evolution which may be detrimental or beneficial to the component’s performance [6].


Laser processing

The main objective of laser processing techniques are to improve the surface quality of SLMed pieces as it may be remelted and relocation in liquid phase produces a smoother surface [88].

Through ablation, a defective layer of the component is removable. In general, methods of this nature are used to improve surface roughness, wear resistance and porosity. A microstructural evolution is observed as well [89].

Hot isostatic pressing

In the hot isostatic pressing (HIPping) technique, a great pressure is applied isostatically on the component at high temperature. The main objective is to reduce porosity of as-built additevily manufactured piece. The huge pressure (which ranges between the 1000 and 2000 bar) and high temperature (which ranges from 920°C to 2000°C, depending on the material) plastically deforms the material and collapses the pores promoting contact between them. Creep and solid state diffusion are the responsible for reducing porosity. Moreover, the process does not compromise surface quality [90].HIPping is always sugested after production of an SLMed component since it has been demonstrated that the technique almost completely erradicates porosity and promote densities above 99.99% [91].

2.1.8 Economic and Industrial Development

During the last 30 years, additive manufacturing has gained significant scientific and industrial importance. It has experienced an annual growth rate of approximately 30 percent in the last


five years and in 2015 the AM industry accomplished a revenue value of 5.1 billion USD [92].

In the researching community, it is a rapidly expanding topic. The 143 edition edito- rial of ”Materials Characterization” reports that looking at Scopus records in 2010, 96 papers treated that specific topic while in 2017, 860 papers were reported.The most common mate- rials are metallic alloys such as aluminum and titanium. However, new materials are fastly emerging. Without a question, SLM is a complex process which requires further research to be fully understood [93].

Gartner states that the impact of AM will reach every industry sooner or later. This effect implies cost-saving and revenue-generating opportunities. However, the implementation of AM requires intensive and lengthy testing and development. Said consideration is taken care of with the present work. The four market growth drivers for additive manufacturing are;

1. Cost savings: beneficial financial impacts come form cutting production, inventory and manufacturing costs of low-volume products. Volkswagen has been saving $ 300,000 per year by additively manufacturing jigs and fixtures used for their vehicles.

2. Business transformation: mass customization and quite specific niche service offerings are achievable.

3. Agility: the added value to the products transform existing customer relationships and promotes new ones.

4. Customer intimacy: Shift from focusing on design for ideal manufacturing to manufac- turing for the ideal design.

A priority matrix provides an insight of the time stamp when a technology will achieve a degree a profit, based on the hype cycle. The priority matrix for AM is presented in table 2.6. PBF technologies, to which SLM belongs, is located at a threshold of 5 to 10 years to be adopted as mainstream with a transformational benefit. Among the applications forecasted to become mainstream in a 2 to 5 years threshold with high benefit are surgical implants, tooling, jigs and fixtures, printed electronics, automotive and medical devices. AM in bioprinting human tissue, aerospace, defense and manufacturing operations offers high benefit and will be fully adopted in a 5 to 10 years threshold.


Transformational -3D printing of dental devices

-3D printed wearables -3D printing in

retail -Macro 3D

printing -Powder bed


-4D printing


-3D print creation software -3D printing service

bureaus -Material extrusion

-3D printed surgical implants -3D printed tooling,

jigs and fixtures -3D printing in

automotive -3D printing of medical devices -3D scanners -Enterprise 3D

printing -Material jetting -Printed electronics


-3D bioprinted human tissue -3D printing in aerospace and defense

-3D printing in manufacturing operations

-3D printing in oil and gas -3D printing in

supply chain -Consumer 3D

printing -Directed energy

deposition -IP protection in

3D printing

-3D bioprinted organ transplants -3D printing with bound materials

-Blockchain in 3D printing -Nanoscale 3D printing

Moderate -Binder jetting

-3D printed presurgery anatomical models -3D printing of consumable

personal products -3D printing workflow

software -Classroom 3D printing -Managed 3D print services

-Sheet lamination

-3D printed drugs



2.2 Dimensional Analysis

2.2.1 Definition and background information

The onset of dimensional analysis dates back to the nineteenth century. In 1822, Fourier was the first to extend the concept of dimension from geometry to physics. In 1871, Lord Rayleigh used dimensional analysis to progress towards a scientific description of the color of the sky.

He also analyzed the tune emitted from strings by wind. In 1883, Reynolds applied the method to fluid flow arriving to the most known Reynolds Number which relates viscous and inertial forces in a fluid. In 1914, E. Buckingham proved that physical law may be expressed in dimensionless quantities, noted as π. Bridgman identified Buckingham’s methodology as the π- theorem. In 1940, Taylor developed the propagation law of explosion waves for the first atomic bomb using dimensional analysis [94].

In natural and engineering sciences, the common way to measure is by comparison to some agreed standard. By specifying a numerical value and the applied unit, the considered physical quantity is uniquely described. Depending on the applied unit, this value will change.

Even if physical phenomena is independent of the unit of measure, it is mandatory to pick some unit for comparative purposes. A way to overcome this dilemma is to define a physical quantity through its ratio on an appropriate scale. This provides not only numerical value, but it also becomes independent of the unit system of choice. The comparison between both leads to meaningful conclusions. Moreover, in a mathematical sense, the number of variables that describes a problem is reduced simplifying it and facilitating founding a solution to said problem. In the same way, the number of required experiments to reveal the complete physical conduct is reduced [95].

Dimensional Analysis (DA) comprises the precise formulation of the previously de- scribed approach along with its scientific reasoning and generalization. DA brings huge bene- fits as physical problems are considerably eased. It removes unessential information from the regarded problem, reducing the number of variables and therefore providing a sharper insight to the essential physical interactions between factors. It is important to be noted that a deeper understanding and insight into the physical process is necessary [96]. Physical processes are then not described by dimensional, but rather by non-dimensional quantities.


physical manipulation of physical quantities of the same property. Along with the previously mentioned operations, subtraction between base quantities of the same kind, multiplication and division by a pure number are defined. A base quantity is measured with a chosen unit along with a numerical value[13].

Figure 2.12: Illustration of the comparison and addition operations of base quantities.In this case, length is used as such. In the left section, the comparison operation is depicted. Lengths of A and B segments are compared concluding that they are equal. The right section of the figure illustrates the addition operation. The length of segment C is the sum of the lengths of segments A and B.

Derived quantities are obtained by inserting base quantities into a mathematical formula [95]. The nature of each physical quantity its described by its dimension, or combination of dimensions as a power law formula. For any derived physical quantity, this power law is described by the generalized dimension formula, also known as Bridgman equation (eq.2.4);

[x] =




Xiai (2.4)

where x is a derived physical quantity, Xi is the ith base quantity of the fundamental dimen- sional system consisting of m number of fundamental dimensions, and ai is the dimension


exponent. If all dimension exponents of a physical quantities are zero, eq. 2.4 takes the following form;

[x] =




Xi0 = 1 (2.5)

and a dimensionless variable is obtained. A dimensionless variable does not change regarding its numerical value when the base units are changed.

A system of units is defined by (1) a complete set of base quantities, (2) the base units and (3) relevant derived quantities. A derived quantity’s dimension will depend on the chosen unit systems, and therefore may complicate or simplify the analysis. Nevertheless, the physi- cal significance of the analysis will remain the same, if the independent variables chosen were the same.

Physical processes are then described by physical quantities. The problem arises as a change of units modifies the numerical values of physical quantities. Therefore, dimensional analysis provides a tool for the analysis of a physical process in terms of dimensionless quan- tities, denoted by the π symbol.

Following Spurk’s derivation, a physical quantity p1 is expressed as a function of the n number of physical variables p2, p3, · · · , pn, which described the phenomenon. This task takes the following form

p1 = f (p2, p3, · · · , pn) (2.6) , in the implicit function form respectively

f (p1, p2, p3, · · · , pn) = 0 (2.7)

Looking to describe the physical process independently of the chosen system of units, the previous is reduced to a dimensionless form of the following form

f (π1, π2, π3, · · · , πd) = 0 (2.8)

where d is the number of dimensionless products. The relation between n and d is given by Buckingham’s π-theorem and will be further explored in the following section.


of the magnitudes of the base units [96].

Assuming an interest in some particular physical quantity Q0 that is a ”dependent vari- able” we have that:

Q0 = f (Q1, Q2, ..., Qn) (2.10) As a whole, Qii=1,2,...,n is a complete set of independent quantities. A set is complete if the factors completely describe the physical phenomena and independent if each member may be adjusted without affecting another. Defining these factors is the first and most important step in dimensional analysis[13].

The fundamental dimensions involved are to be defined. These will constitute the system type for the analysis. Dimensions of dependent and independent variables are expressed with respect to such system type as a power law. For a M (mass), L (length), T (time) and Θ (temperature) system the following form is observed.

[Qi] = MmjLljTtjΘθj (2.11)

where the exponents, mj, lj, tjandθj are real numbers defined from each physical quantity’s definition. A complete dimensionally independent subset Q1, ..., Qkis chosen from the com- plete independent set previously described Q1, ..., Qn. For it to be dimensionally indepen- dent, the dimensions of one of its member can not be formed with the dimensions of the remaining. It is complete if the dimensions of the rest of dependent and independent variables Qk+1, ..., Qnmay be expressed in terms of the dimensions of the subset Q1, ..., Qk[13]. Hav- ing chosen a complete, dimensionally independent subset the dimensions of the dependent and remaining independent quantities are expressed in terms of the dimensions of the subset.


From these, the dimensionless numbers are constructed.

Buckingham-π theorem of dimensional analysis transforms Eq.2.9 into:

f (π1, π2, ..., πd) = 0 (2.12)

where each of πii=1,2,...,n are variables in the non-dimensional form which are obtained from the product of some or all of the variables in Eq. 2.9 [96].

Buckingham’s π-theorem (Theorem 1 for which proof is provided in appendix A) gives us that the physical phenomena will be now described by n − k = d number of variables instead of n where n is the complete set, k is the chosen dimensionally independent set, and d is the resulting amount of dimensionless numbers.

Buckingham’s π- theorem 1. When a complete relationship between dimensional physical quantities is expressed in dimensionless form, the number of independent quantities that ap- pear in it is reduced from the original n to n − k, where k is the maximum number of the originaln that are dimensionally independent. [13]

In the dimensionless causal form of dimensional analysis we have that;

π0 = f (π1, π2, ..., πn−k) (2.13)

The particular form of the function in dimensionless causal relationship is not provided by Buckingham’s π-theorem and should be determined experimentally. Domains of independent variables are divided and a power law (Eq. 2.14) is adopted to fit the results.

π0 = c · π1απ2β · · · πn−kδ (2.14) where c is a proportionality constant, and α, β,...,δ are real numbers.

A summary of the previously described mathematical procedure for the implementation of dimensional analysis is presented in figure 2.13.



2.3 Characterization techniques

2.3.1 Scanning Electron Microscopy

A scanning electron microscope (SEM) images a sample by scanning it with a high-energy beam of electrons in a raster scan pattern. Images with very high resolution of a sample surface are attainable with this method due to to magnifications from 20X-30000X with spatial resolution of 50-100 nm[97].

Under vacuum, electrons generated by a source (electron gun) are accelerated (by the anode) in a field gradient. Electromagnetic lenses focus the passing beam onto the specimen to be analyzed. The bombardment causes the specimen to emit different types of electrons, from which the secondary are caught by a detector. An image of the sample’s surface is constructed by comparing intensities of secondary electrons and the primary scanning electron beam [98].

An schematic of the components of a scanning electron microscope is presented in figure 2.14.

2.3.2 Buoyancy method

Density measurements for the specimens was carried out through buoyancy method which is based upon Archimedes’ principle. Moreover, the procedure stated by ASTM B692 standard for density measurement was followed with little modification as it was constructed for pow- der metallurgy sintered pieces and not for additively manufactured ones. The main deviation from this method was the omission of using an oily agent to cover the specimen.


Figure 2.14: Scanning electron microscope schematic. An electron gun produces an elec- tron beam which is accelerated by the anode. Afterwards, the laser beam is accelerated by magnetic lenses and scans the specimen of study. Secondary beam electrons are gathers by a dedicated collector. Finally, a surface image is generated by comparing intensities of sec- ondary electrons and the primary scanning electron beam [97].

Archimedes’ principle states that a body immersed partially or fully experiences a buoy- ant force acting upwards on it. The magnitude of this force is equivalent to the weight of the fluid displaced by the body. Buoyancy method for density determination is based upon this principle.

In the method, the solid is weighted in air, and then again in auxiliary liquid (commonly distilled water) with a known density. Density of the solid can therefore be calculated as;


WA− WBL− ρ0) + ρ0 (2.15) where WAand WBare the weight of the sample in air and in the auxiliary liquid respectively, and ρSOLID, ρLand ρ0 are the density of the solid, of the auxiliary liquid and air respectively.

This method is suitable for solids and liquids (with glass sinkers) and proves to be a quick, flexible process.However, it is important to consider the temperature of liquid as it may


Model Development

3.1 Factors Involved

The application of dimensional analysis to SLM has not been studied thoroughly in the re- search community. Van Elsen et al. presented a possible complete set of dimensionless pa- rameters to describe the process and ease comparison between works [14]. Cardaropoli et al. used dimensional analysis intending to find out an appropriate definition of a set of non- dimensional groups in order to represent the output parameters on the selective laser melting process [15]. The most recent study, performed by Khan et al., used dimensional analysis to produce a heat transfer model on the SLM process [16].

The independent variables that best relate to the bulk density of metallic pieces produced by selective laser melting and are therefore included in the dimensional analysis are volumet- ric energy density (γ which is particularly defined by eq.3.2 ), average particle diameter (φ), scanning speed (v), specific heat capacity (Cp) and heat conductivity (κ). Densification of the SLMed metallic piece has been identified as the most important characteristic impacting its mechanical properties. For this reason, density will be regarded as the dependent physical quantity. This previous statement may be summarized in expression 3.1.

ρ = f (γ, v, κ, Cp, φ) (3.1)

For the independent factors involved in the present dimensional analysis, the first is



γ = vht (3.2) where P is the laser power (W ), v is the scanning speed (m/s), h is the hatch spacing (m) and t is the layer thickness (m).

3.2 Selective laser melting dimensional analysis

The fundamental dimensions involved in the dimensional analysis developed are time (T ), length (M ), mass (M ) and temperature (Θ). The independent variables that best relate to the bulk density of metallic pieces produced by selective laser melting and are therefore included in the dimensional analysis are volumetric energy density (γ), average particle diameter (φ), scanning speed (v), specific heat capacity (Cp) and heat conductivity (κ). Definition of funda- mental dimensions along with symbol, units and dimensions of factors are presented in tables 3.1 and 3.2 respectively.

Table 3.1: Symbol and unit of the fundamental dimensions involved in the dimensional anal- ysis

Dimension Symbol Unit

Time T s

Length L m

Mass M Kg

Temperature Θ K


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