(a) (b)

Figure 4.2: Used equipment for bulk density determination of produced Inconel 718 cubes (a) Mettler Toledo’s XPR analytical weighing machine and (b) Mettler’s Toledo XPR/XRS-Ana density measurement kit

Figure 4.3: SEM image of Praxair’s TruForm In718 powder. The image was taken at a energy level of 10 kV, working distance of 7.5 mm and magnification of 100X. An overall spherical particle shape is observed. However, it is not homogeneous throughout the sample. From this image, a PSD is obtained (and presented in figure 4.4) and average powder particle is determined to be 26.56 microns.

1080 nm and laser focus diameter of 70 µm). The process was conducted under an argon
atmosphere. Layer thickness (Lt), hatch spacing (hs) and point distance (pd) were set to 60
µm, 70 µm and 70 µm respectively. Laser power (P ) and exposure time (t_{on}) were varied
from 360 W - 400 W and 35 µs - 40 µs respectively. Scanning speed (v) was approximated as
(eq. 4.2), referring to the work of Tiwari et al. [44] , the quotient between point distance and
exposure time.

A summary of the parameters used, along with volumetric energy density (γ), are listed in table 4.2. In total, 27 probes were manufactured ;3 probes per experimental condition. The results presented are the average of said 3 specimens.

Figure 4.4: PSD of Praxair’s TruForm In718 powder. A normal distribution of sizes is ob-served with a standard deviation of 11.05 microns. The average particle diameter is deter-mined to be 26.56 microns.

Table 4.1: TruForm Inconel 718 Metal Powder typical chemical composition reported by the provider

Element Ni Cr Fe Nb+Ta Mo Ti Al Co Mn Si Cu

Composition(*Max) 50-55 17-21 15-21 4.75-5.5 2.8-3.3 0.65-1.15 0.2-0.8 1* 0.35* 0.35* 0.3*

v = pd

t_{on} (4.2)

After fabrication, bulk density of the built specimens was measured with Mettler Toledo XPR Analytical Balance equipped with its density measuring kit. 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 powder 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. The procedure of measurement

Table 4.2: Summary of fabrication parameters ID Laser power

(W )

Exposure time (µs)

Scanning speed (m/s)

Volumetric energy density
(J/m^{3}) (1 × 10^{9})

11 360 35 2 42.86

12 360 38 1.84 46.53

13 360 40 1.75 48.98

21 380 35 2 45.24

22 380 38 1.84 49.11

23 380 40 1.75 51.70

31 400 35 2 47.62

32 400 38 1.84 51.70

33 400 40 1.75 54.42

comprises the weighing of the probes in environments of air and an auxiliary liquids (dis-tilled water). Illustration of this process is presented in figures 4.5a and 4.5b respectively.

Temperature of distilled water was also read to be 24° C.

(a) (b)

Figure 4.5: Bulk density measurement procedure; The solid sample is first (a) weighed in air environment and then (b) weighed while submerged in auxiliary liquid (distilled water).

(a) (b)

Figure 5.1: SLMed Inconel 718 cubes in (a) the build chamber after construction and (b) a close up of the build plate.Good surface finish is observed.

Bulk density results for the fabricated specimens are reported in table 5.1 along with
relative density. Relative density is a quantification of densification with respect to a
the-oretical achievable density which is defined by the material. The greatest relative density
achieved is 96.082% and is obtained under a laser power of 400 W and exposure time of
40 µs (a scanning speed of 1.75 m/s) and a volumetric energy density of 54.42 x10^{9} J/m^{3}.

51

The probe with smaller relative density (94.274%) was produced at a laser power of 400 W ,
exposure time of 35 µs (a scanning speed of 2 m/s), and volumetric energy density of 47.62
x10^{9} J/m^{3}. In average, a densification of 95.218 % was obtained. Resulting bulk density is
presented as a function of volumetric energy density in figure 5.2 labeling each point with its
respective specimen ID. A positive relation between volumetric energy density and
densifica-tion is observed. However, excepdensifica-tions like specimens 23 and 32 are observed. The effect of
volumetric energy density on densification of pieces produced by selective laser melting has
been thoroughly studied and, even though an overall positive relation between them has been
found, discrepancies, as the ones found in this work, have been identified by several authors
as Mishurova and Scipioni [68] [67]. This leads to the conclusion that VED is not to be used
by itself as a design parameter for the SLM process.

Table 5.1: Bulk and relative density of the fabricated specimens
ID Bulk density results (Kg/m^{3}) Relative density

11 7745.66 0.94575

12 7788.83 0.95102

13 7807.09 0.95325

21 7781.87 0.95017

22 7832.60 0.95636

23 7848.19 0.95826

31 7721.01 0.94274

32 7791.15 0.95130

33 7869.09 0.96082

The C and α fitting parameters, for the selective laser melting of Inconel 718, from
eq. 3.10 are determined to be 4910.4 and 1.34 respectively for a independent dimensionless
variable (π_{1}) range from 3.17x10^{−8}to 4.6x10^{−8}. Refer to appendix B for a detailed process of
determination of C and α. As a result; the particular expression for the bulk density modelling
of SLMed In718 in a first order polynomial function (or power law model) is presented in eq.

5.1. In the dimensionless causal relationship it takes the form of eq. 5.2. From eq. 5.1 it is observed that dimensional analysis has provided an insight on the attained bulk density for the physical process of selective laser melting. It is observed that bulk density will be proportional to volumetric energy density (γ) and inversely proportional to velocity (v) to the second power. The independent dimensionless product π1 is found to directly influence bulk

Figure 5.2: Graph of experimental bulk density as a function of volumetric energy density la-beled with specimen ID. An overall positive influence of VED is observed in the densification of the manufactured probes. However, exceptions to this manifest, in specific specimens 23 and 32. These results are in agreement with those found by authors like Mishourova [68] and Scipioni [67] where it is concluded the following; Even though it has been found that VED has a positive influence on densification, it is not a consistent design parameter as exceptions occur and therefore should not be used by itself as such.

density with a power of 1.34. It is important to highlight that the individual factors involved
in π1are not to be considered independently, but as a whole. Bulk density is influenced by the
whole independent dimensionless product π_{1}and not by its components under consideration.

ρ = 4910.4 · (γ

v^{2})( κv

γC_{p}φ)^{1.34} (5.1)

ρv^{2}

γ = 4910.4 · ( κv

γC_{p}φ)^{1.34}; π_{0} = 4910.4 · π_{1}^{1.34} (5.2)
Dimensional analysis has made possible to reduce the variables in which the physical

process of selective laser melting is described. Following Buckingham’s π-theorem, the
prob-lem, firstly described by n factors (bulk density (ρ), volumetric energy density (γ), scanning
speed (v), heat conductivity (κ), specific heat capacity (C_{p}) and average powder particle
di-ameter (φ) ) is now described by d = n − k (6-4=2) ,where k is the number of fundamental
dimensions and number of elements of the chosen dimensionally independent set (γ, v, ρ, C_{p}),
dimensionless π-products. In specific, a dependent (may also be referred to as dimensionless
density, π_{0} presented in eq.3.7) and a independent (π_{1} presented in eq. 3.8) dimensionless
variables. Dimensionless density product π0 incorporates density, scanning speed and
volu-metric energy density. The two latter are closely related to the energy input received by the
powder bed. Independent dimensionless product π1 includes energetic input factors as
scan-ning speed and volumetric energy density, thermal factors such as thermal conductivity and
specific heat capacity and powder characteristics in the form of the average powder particle
diameter.

The graph of dependent π0 and independent π1 dimensionless numbers along with the prediction provided by eq. 5.2 is presented in figure 5.3. A strong dependency between experimental π0 and π1 is observed. This is a proof that dimensional analysis has succeeded at describing the physical problem of selective laser melting with the dimensionless products.

The expression obtained from dimensional analysis and further experimentation, in the form of a first order polynomial, proves to be a precise fit.

Figure 5.3: Graph of experimental dependent π_{0} and independent π_{1} dimensionless numbers
along the particular form model produced. A strong positive relation between π0 and π1 is
observed. This is an indication that dimensional analysis has succeeded at describing the
physical process. Moreover, the particular form obtained is observed to be a good prediction
of experimental values.

Moreover, experimental bulk density and dependent dimensionless product, along with its prediction and error percentages are presented in table 5.2. An average error percentage of 1.6503 % is achieved. The biggest discrepancies were found at experimental conditions that of specimens 13 and 31 with 3.7119 % and 3.0542 %. However, high precision was achieved with error percentages as little as 0.1611 % and 0.1617% in specimens 22 and 11 respectively.

Table 5.2: Bulk density and dependent dimensionless product error percentages for each man-ufactured specimen and average between experimental results and predicted

ID π_{1} Exp. π_{0} Mod. π_{0} Exp. ρ
(Kg/m^{3})

Mod. ρ
(Kg/m^{3})

Error ptg.

11 4.60E-08 7.23E-07 7.24E-07 7745.6633 7758.1974 0.1617%

12 3.91E-08 5.68E-07 5.81E-07 7788.8333 7965.0737 2.2626%

13 3.53E-08 4.88E-07 5.06E-07 7807.0933 8096.8901 3.7119%

21 4.36E-08 6.88E-07 6.73E-07 7781.8700 7616.8824 2.1202%

22 3.70E-08 5.41E-07 5.40E-07 7832.6033 7819.9905 0.1611%

23 3.34E-08 4.65E-07 4.71E-07 7848.1867 7949.4059 1.2896%

31 4.14E-08 6.49E-07 6.29E-07 7721.0067 7485.1977 3.0542%

32 3.52E-08 5.11E-07 5.04E-07 7791.1500 7684.7944 1.3652%

33 3.17E-08 4.43E-07 4.40E-07 7869.0900 7811.9724 0.7259%

Avg: 1.6503%

A surface plot of bulk density with respect to laser power and scanning velocity is pre-sented in figure 5.4. Bulk density is observed to increase with lower scanning speed and to diminish with increasing laser power. Higher scanning speed implies less exposure time of the laser to the powder bed which is related to energetic input and therefore to densification.

However, bulk density behaviour with relation to power is counter intuitive. It is important
to remark that the fitting parameters of C and α determined are valid, with certainty, only
for a independent dimensionless π_{1} product range from 3.17x10^{−8} to 4.6x10^{−8}. This means
that said behaviour may change for different manufacturing conditions. Moreover, high laser
power values may be related to powder particle sublimation and ejection negatively affecting
densification [8] [14].

Figure 5.4: Graph of density as a function of Laser power (P ) and Scan velocity (v). From the
graph, it is observed that better densification conditions are obtained at lower scanning speeds
and laser power. The first observation is intuitive. A lower scanning speed implies larger
exposure time of the laser with the powder bed and is related to proper melt and therefore
better densification. The second observation is counter intuitive. However, this phenomena
is attributed to the fact that improperly high laser power conditions are associated to particle
sublimation / ejection, which thus leads to higher porosity in the resulting piece. Moreover, it
is important to remark that the behaviour presented in the graph is exclusive for Inconel 718
and a independent dimensionless product π_{1} range from 3.17x10^{−8} to 4.6x10^{−8}.

Incorporating volumetric energy density definition, provided in eq. 3.2, into eq. 5.1 and
using theoretical density (ρT H) instead of bulk density, an expression for defining the proper
scanning speed conditions with respect to laser power to achieve full densification is obtained
and presented in eq. 5.3. This expression is valid for any independent dimensionless product
range π_{1} as it emerged from the dimensional analysis itself.

v =

P^{1−α}·

C

ρ_{T H}ht

· kht
C_{p}φ

α_{3−2α}^{1}

(5.3) Likewise, an expression to determine laser power with respect to scanning speed pre-ceding from eq. 5.3 is developed in eq. 5.4.

P =

v^{2α−3}·

C

ρ_{T H}ht

· kht
C_{p}φ

α_{α−1}^{1}

(5.4)

For the particular case presented in this study, the behaviour of scanning speed to achieve
full densification with respect to laser power is presented in figure 5.5. This behaviour is valid,
as previously stated, for the π_{1}range from 3.17x10^{−8}to 4.6x10^{−8}.

Figure 5.5: Scanning speed with respect to laser power to achieve full densification of SLMed
Inconel 718 pieces. This graph, originated from equation 5.3, is useful for tuning scanning
ve-locity with respect to laser power for the SLM of Inconel 718 in an independent dimensionless
product range π_{1} from 3.17x10^{−8} to 4.6x10^{−8}.

six factors; bulk density (ρ), volumetric energy density (γ), scanning speed (m/s), powder’s heat conductivity (κ), specific heat capacity (Cp) and average powder diameter (φ) where bulk density was regarded as a dependent variable which is a function of the rest of independent variables. Through dimensional analysis, and applying Buckingham’s π-theorem, process complexity was reduced from n number of variables to n − k = d where n is the total number of factors (dependent and independent), k is the number of elements of the chosen dimensionally independent set (which is equal to the number of fundamental dimensions) and d is the number of dimensionless π-products which will now describe the process. In other words, the process transitioned to be described by 6 variables, to 2. This eases the task of describing the physical process of selective laser melting. Moreover, as dimensionless products are independent of the choice of unit systems, it is now possible to compare works under different experimental conditions and unit system used.

Applying dimensional analysis to the selective laser melting process concluded that it could be described by two dimensionless π products, one dependent and one independent.

Independent dimensionless product π_{1} incorporated thermal factors as specific heat capacity
and thermal conductivity, energetic input parameters as scanning speed and volumetric
en-ergy density and powder shape characteristics in the form of average particle diameter. It is

59

important to remark that dimensional analysis indicates that the physical process must have a dependency on the whole π1 product and not by its elements alone. Moreover, the fact that it involves energetic and powder’s thermal and morphology factors is a proof of the com-plexity of selective laser melting and that parameters of different nature must be considered.

Dependent dimensionless product π_{0}(otherwise referred to as dimensionless bulk density)
in-volves energetic input factors as scanning speed and volumetric energy density. This indicates
the importance of supplied energy from the laser to the powder bed in material’s
densifica-tion. Moreover, isolating bulk density from the dimensionless causal relationship expression
provides physical meaning to the process; bulk density will be directly proportional to the
volumetric energy density and affected by a factor of minus two by scanning speed. This is an
intuitive and coherent conclusion as higher volumetric energy density indicates higher energy
supplied to the powder bed ensuring proper melting of the powder and better densification.

Moreover, if scanning speed is greater, laser will incide in the powder bed for less time which may lead to higher porosity.

The specimen built in this work for determining the C and α fitting parameters was of cu-bic geometry. This work is aimed at being applied in industry sectors such as aerospace. Even though a real size end component was not built, the physical relation between operating pa-rameters and powder characteristics remain the same independently of the built component’s geometry. Dimensional analysis is based upon thermodynamic and energetic interactions.

As proved by Buckingham in the year 1914, physical laws remain valid in the dimension-less space. Moreover, the technique was first used by Lord Rayleigh, applying the so-called Rayleigh’s dimensional analysis, in 1871 to describe the color of sky and heat transfer phe-nomena. These processes and occurrences have their basis on interactions between physical quantities. Therefore, the physical conclusions drawn from the present dimensional analysis for SLM remain valid independently of geometrical feature and construction parameters such as scanning strategy and built orientation.

Dimensional analysis does not provide a particular form of an expression. This is where an experimental stage enters the discussion. A first order polynomial form was used for doing so and through an experimental stage, the fitting parameters C and α were determined for a

C and α parameters were determined to be 4910.4 and 1.34 respectively for the bulk
density behaviour of SLMed Inconel 718 pieces in a π_{1} range from 3.17x10^{−8} to 4.6 x10^{−8}.
Therefore, a particular expression was found. A strong relation between dependent π0 and
independent π_{1} dimensionless products was observed. This is an indication that dimensional
analysis has succeeded at describing the physical process of selective laser melting through
said dimensionless numbers in stead of physical quantities. Moreover, the method proved
to be precise as an average error percentage between experimental bulk density, and
dimen-sionless bulk density, and the predicted values by the first order polynomial was of 1.6503 %
(MIN=0.1611 %, MAX=3.7119 %).

From the surface plot of bulk density with respect to laser power and scanning speed
two interesting conclusions may be drawn. First, it is observed that bulk density is inversely
affected by scanning speed. With higher scanning speed, laser incides less time on the powder
bed possibly leading to poorer densification. On the other hand, a counter intuitive conclusion
was drawn from the relation of bulk density to laser power. It was observed that higher laser
power leads to higher porosity. This may be attributed to the fact that improperly high laser
power may lead to powder particle sublimation and/or ejection causing improper
densifica-tion. It is also important to remark, as previously stated, that the C and α fitting parameters
were determined for a specific π_{1} range. The previously discussed behaviour of laser power
and density may be different for a different range.

A general expression for the tuning of scanning speed to achieve full densification with respect to operating laser power, powder material’s characteristics (theoretical density, spe-cific heat capacity and heat conductivity and average particle diameter), commonly and easily

modifiable manufacturing parameters (hatch spacing and layer thickness) and fitting param-eters (C and α) was obtained. Moreover, a graph was provided for the tuning of scanning speed with respect to laser power for the full densification of SLMed Inconel 718 pieces.

Even though more than 50 factors affecting selective laser melting were listed in the theoretical framework, only six of them were included in the dimensional analysis. The fact that high precision, a coherent physical significance of factors on the influence of bulk den-sity, and strong dependency between independent and dependent dimensionless products were achieved is an indicator that dimensional analysis has succeeded at describing the physical process of selective laser melting with these six factors. The first step in dimensional analysis is to define, out of all the variables that involve a process, the factors that have the biggest influence. Incorporating more factors does not imply that dimensional analysis will be more accurate. On the contrary, it may introduce unwanted noise to the model.

The competitive advantage from the present work with respect to previous publications is on several aspects. First, the physical process of SLM was successfully described by di-mensional analysis with only five independent physical quantities. Comparing these to the 50+ factors listed, it is a significantly more practical practical approach which produces a highly precise model. Moreover, this thesis possess a high industrial focus. An expression for the adjustment of laser power and scanning speed to obtain fully dense pieces through the selective laser melting process was developed. This work is thus directly scalable to the industrial level offering significant benefits regarding the lessen of material waste and ease of adaption of a new material to the SLM productive scheme. From the five independent phys-ical quantities involved in the SLM dimensional analysis presented, two of them are directly controllable in the manufacturing process (scanning speed and volumetric energy density).

The rest of the independent variables are related to powder characteristics. In the industrial world, a provider delivers the material that passes certain property which value is among a cer-tain acceptable range. This deviation from reference value may produce discrepancies within the expected and obtained density results. However, the resulting piece will be of high bulk density nonetheless and the benefits of applying the model are still valid.

1.6503 % was achieved. Finally, a mathematical tool for determining proper scanning speed to achieve full densification, with respect to laser power, was developed proving useful to reduce lengthiness, of time and resources, of the process of adapting a new choice of mate-rial. Through this pioneer work, dimensional analysis has been successfully applied to the selective laser melting process for tuning the proper operation parameters in order to produce fully dense pieces of Inconel 718. Therefore, the hypothesis of this work has been confirmed;

the bulk density of metallic pieces produced by selective laser melting can be mathematically modeled through the use of dimensional analysis.