A novel mathematical model for predicting a sustainable selective laser melting and controlled densification

Figures & data

Table 1. Units and dimensions of the physical quantities involved in the mathematical model for the prediction of density through the dimensional analysis of selective laser melting.

Physical quantity	Symbol	Units	Dimensions
Laser power	$\mathit{P}$	W	ML^2 T^−3
Hatch distance	$\mathit{h}$	m	L
Scanning speed	$\mathit{v}$	m/s	LT^−1
Specific heat capacity	$C_p$	J/kgK°	L^2T^−2Θ^−1
Heat conductivity	$\mathit{\kappa }$	W/mK°	MLT^−3Θ^−1
Bulk density	$\mathit{\rho }$	Kg/m^3	ML^−3
Layer thickness	$L_t$	m	L

Figure 1. SEM images of Inconel alloy 718 (In718-0405) on a) at 200 X and b) at 500 X magnifications. The powder exhibits a spherical morphology.

Gray-scale Scanning Electron Microscope (SEM) images of Inconel Alloy 718. At 200X magnification and Figure 2b at 500X magnification. The powder exhibits a spherical morphology with similar diameters randomly distributed against a black background.

Table 2. In718 (In718–0405, Renishaw, Monterrey, México) metal powder chemical composition (Renishaw 2017.).

Element	Ni	Cr	Fe	Nb + Ta	Mo	Ti	Al	Co	Mn	Si
Composition (*Max)	50–55	17–21	15–21	4.7–5.5	2.8–3.3	0.65–1.15	0.2–0.8	1*	0.35*	0.35*
Element	Cu	C	N	O	P	S	Ca	Mg	Se	B
Composition (*Max)	0.3*	0.02–0.05	0.03*	0.03*	0.015*	0.015*	0.01*	0.01*	0.005*	0.005*

*Maximum percentage of chemical composition.

Table 3. Experimental data for the validation of the mathematical model hereby developed.

Author	Material	Reference	Data table
Estrada et al.	In718	This work	S1
Enneti et al.	W	(Enneti, Morgan, and Atre 2018)	S2
Read et al.	AlSi10Mg	(Read et al. 2015)	S3
Shi et al.	Ti6Al4V	(Shi et al. 2016)	S4
Wang et al.	SS316L	(Wang et al. 2017)	S5

Figure 2. Influence of the independent dimensionless products a) $\mathit{\pi }$₁ and b) $\mathit{\pi }$₂ in the dimensionless form of density, $\mathit{\pi }$₀, respectively.

Two scatter charts in logarithmic scale illustrating the relationship between the influence of dimensionless products π1 and π2 with π0. Each chart shows five sets of data for all studied materials. An almost proportional relation is observed for all sets in π1, while for π2, sets for distinct materials are polarised in different areas of the chart.

Table 4. Fitting parameters (C, α and β) values determined through the non-linear least-squares method alongside the R-squared value and the independent dimensionless groups (π₁ and π₂) working range.

Material	C	α	β	R-squared	π1 range	π2 range
In718	296700	1.501	−0.4507	0.9991	6.73x10^−09-1.44x10^−08	0.85–1.2
W	141.1	1.425	−0.5172	0.9998	8.61x10^−09 − 8.43x10^−07	1.00–2.00
AlSi10Mg	9.517	1.136	−0.6748	0.9771	2.96x10^−08 − 2.65x10^−07	0.25–1.00
Ti6Al4V	457600	1.469	−0.5014	0.9998	2.34x10^−11 − 1.47x10^−10	0.25–1.00
SS316L	334500	1.502	−0.4991	0.9999	6.03x10^−10 − 2.67x10^−09	0.42–1.25

Figure 3. Contour plot indicating isolines of SLEC, shown in solid blue lines, with its associated value of laser power, represented by dashed black lines. The plots illustrate the selection of scanning speed, and hatch distance, depending on a chosen value of power or SLEC for the SLM manufacturing of highly dense components in a) In718, b) W, c) AlSi10Mg, d) Ti6Al4V, and e) SS316L.

Five contour plots (each for a material) indicating isolines of Sustainable Laser Energy Consumption (SLEC), represented by solid blue lines, and its associated value of laser power, P, depicted by dashed black lines. The plots explain the selection of hatch distance and scanning speed, depending on a chosen value of power or SLEC for the SLM manufacturing of highly dense components.

Table 5. SLEC (ε), scanning speed (v), hatch distance (h), laser power (P) and area scanning velocity (v_A) associated with the A, B and C points in Figure 3(b).

	Point A	Point B	Point C
ε (J/mm^2)	1.6	1.6	1.6
v (m/s)	1.21	0.85	0.69
h (μm)	15.44	23.44	30.22
P (W)	30	32	33.23
vA (mm^2/s)	18.68	19.92	20.85

Figure 4. Processed SEM images of metallic powder material of (a) In718, (b) W, (c) AlSi10Mg, (d) Ti6Al4V and (e) SS316L for the calculation of fractal dimension and lacunarity through the differential box counting method.

Five greyscale Scanning Electron Microscope (SEM) images display powders of the five studied materials. Each image covers a square area approximately 500 µm per side. The images shown are digitally processed to enhance sharpness and contrast for the calculation of fractal dimension and lacunarity.

Figure 5. Fractal dimension (in blue), lacunarity (in red) and average relative densification attained (in black) for W, AlSi10Mg, In718, Ti6Al4V and SS316L SLMed metallic alloy powders.

The plot illustrates three-bars chart representing the fractal dimension, lacunarity, and densification achieved for the studied materials in the following order: W, AlSi10Mg, In718, Ti6Al4V, and SS316L metallic alloy powders. In this order, the fractal dimension and density show an ascending tendency, while lacunarity exhibits a decreasing trend.

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Data availability statement

The authors confirm that the data supporting the findings of this study are available within the article and its supplementary materials.