Dimensionless Analysis for Investigating the Quality Characteristics of Aluminium Matrix Composites Prepared through Fused Deposition Modelling Assisted Investment Casting

The aluminium matrix composites (AMCs) have become a tough competitor for various categories of metallic alloys, especially ferrous materials, owing to their tremendous servicing in the diversified application. In this work, additional efforts have been made to formulate a mathematical model, by using dimensionless analysis, able to predict the mechanical characteristics of the AMCs that have already been optimized and characterized by the authors. Here, the experimental and statistical data obtained from the Taguchi L18 orthogonal array and analysis of variance (ANOVA) have been used. They permit collection of the output responses and allow the identification of significant process parameters, respectively, which thereafter were used to design the mathematical model. Second order polynomial equations have been obtained from the specific output response and the relevant input parameter were incorporated with the highest level of contribution. The obtained quadratic equations indicate the regression values (R2) equal to unity, hence, proving the performances of the fit. The results demonstrate that the developed mathematical models present very high accuracy for predicting the output responses.


Introduction
In the last two decades, the rapid advancement of technology has contributed to large modification in the manufacturing sector. During this period, the demand for materials that can sustain the extreme level of service conditions increased globally. Specifically, in aerospace and automobile sectors, the requirement of materials having high strength, toughness, hardness, and prolonged service life was always a challenge. Apart from these properties, one of the major requirements is 'light weight'. Different studies have reported the needs of lighter material as one of the motivations behind the invention of reinforced materials, commonly, referred to as metal matrix composites [1][2][3]. Figure 1 presents the methodology adopted in this research. By using a Fish bone diagram (see details on Figure 2), we highlight the main process parameters associated to the IC which can affect the quality features of the IC components. The number of IC slurry layer (N SL ) has been judicially selected as an input parameter due to its significance highlighted in the literature [38][39][40]. The following are the procedural steps followed to obtain AMCs, which refer to the original Taguchi L18 orthogonal array (Table A1), as given in the Appendix A:

Materials and Methods
• The alternative feedstock filaments (F P ) have been prepared using PA, Al 2 O 3 , and Al in different %wt. proportions with the help of single screw extrusion process.

•
The formed filaments were used for the development of sacrificial patterns of cubical shape with three different volumes (V P ), such as 17,576 mm 3 , 27,000 mm 3 , and 39,304 mm 3 . They were produced at low, high, and solid density of FDM process (D P ) by using uPrint-SE system of Stratasys Inc. (Edina, MN, USA). In the works, reported previously, it has been seen that the change in the in-fill density affects the mechanical and tribological performances of the developed AMCs [1][2][3]. The prime reason behind the selection of FDM technology is due to its affordability and suitability for hybridization within the IC process [23,24,41]. Further, the selection of the process parametric levels from previous studies has been judicially selected, based on the pilot studies.

•
Prior to shell moulding, the barrel finishing (BF) process was performed on the samples, for the refurbishment of resulted surface finish [31]. Here, barrel finishing time (BF T ) and barrel finishing media weight (BF W ) have been selected as input process parameters. • Then, the IC moulds were prepared by coating the trees (consisting of riser, pouring basin, gating, and also the FDM printed sacrificial pattern) with refractory layers of silica. The number of IC slurry layers (N SL ) has also varied in accordance to      The castings manufactured were tested for surface hardness, dimensional accuracy and surface roughness by using HVS-1000BVM hardness tester (HV0.01 scale; ASTM-E384, Laizhou, China), Vernier Caliper (Mitutoyo: least count 0.01mm, Takatsu-ku, Kawasaki, Japan) and Mitutoyo SJ-210 (Japan, ISO: 1997) surface roughness tester, respectively. For microstructural evaluation, the Scanning Electron Microscopy (SEM, JEOL, Peabody, MA, USA) analysis has been performed on the casting manufactured in the experiment #16, #17 and #18 associated to Table A1. It has been seen that the Al 2 O 3 particles presented in Al matrix allow to enhance the quality characteristics of the castings, especially the hardness on the surface. Figure 3 shows the SEM micrographs and their associated Energy Dispersive Spectroscopy (EDS) spectrums (JEOL, USA). The measurements indicate the presence of Al, O, Si, Fe, and C-peaks, which confirm the existence of alumina. These elements identified on the EDS measurements (i.e., Al, O, and C) are the common sign of alumina surface [42]. They were noted as well as the presence of elements Fe and Si, which denote some small impurity. • Prior to shell moulding, the barrel finishing (BF) process was performed on the samples, for the refurbishment of resulted surface finish [31]. Here, barrel finishing time (BFT) and barrel finishing media weight (BFW) have been selected as input process parameters. • Then, the IC moulds were prepared by coating the trees (consisting of riser, pouring basin, gating, and also the FDM printed sacrificial pattern) with refractory layers of silica. The number of IC slurry layers (NSL) has also varied in accordance to Table A1 in the Appendix. • Autoclaving and baking were performed in one step at 1150 °C (by maintaining the pouring sprue in a vertical up position so that the Al2O3 filler particles could be arrested within the cavity only). At this range of temperature, the matrix of the sacrificial patterns evaporates, immediately, without causing mould cracks. • Finally, pouring of molten Al-6063 has been carried out.
The castings manufactured were tested for surface hardness, dimensional accuracy and surface roughness by using HVS-1000BVM hardness tester (HV0.01 scale; ASTM-E384, Laizhou, China), Vernier Caliper (Mitutoyo: least count 0.01mm, Takatsu-ku, Kawasaki, Japan) and Mitutoyo SJ-210 (Japan, ISO: 1997) surface roughness tester, respectively. For microstructural evaluation, the Scanning Electron Microscopy (SEM, JEOL, Peabody, MA, USA) analysis has been performed on the casting manufactured in the experiment #16, #17 and #18 associated to Table A1. It has been seen that the Al2O3 particles presented in Al matrix allow to enhance the quality characteristics of the castings, especially the hardness on the surface. Figure 3 shows the SEM micrographs and their associated Energy Dispersive Spectroscopy (EDS) spectrums (JEOL, USA). The measurements indicate the presence of Al, O, Si, Fe, and C-peaks, which confirm the existence of alumina. These elements identified on the EDS measurements (i.e., Al, O, and C) are the common sign of alumina surface [42]. They were noted as well as the presence of elements Fe and Si, which denote some small impurity.

Dimensionless modelling: Buckingham Pi Approach
Dimensionless modelling of the experimental data is considered an efficient method in order to formulate analytic mathematical functions that are out of a highly complex experimental system associated to numerous process parameters [43]. The concept of dimensionless analysis helps to reduce the influence of variables by means of physical equations [44][45][46]. To date, dimensionless modelling with the help of Buckingham Pi approach has been extensively investigated for a wide range of scientific and engineering applications including fluid dynamics [47], energy [48], electronics [49], heat transfer [50] and others. According to the Buckingham approach, any practical problem containing "n" factor sand further "m" dimensions, then the subtraction of n and m will result the counts of independent factors, which could be assumed. Presently, "n" and "m" are 7 and 3, respectively. Therefore, the problem will consist of π1, π2, π3 and π4 that are the dimensional magnitudes. Furthermore, the mathematical formulae derived for the assumed independent parameters help to develop the dimensional relationships by following a set of standard steps [51,52]. Standard quantities of the same physical nature (mass, length, and time) are used based on fundamental units. Consequently, it can be said that these systems belong to the same class. To generalize, a set of systems of units that differ only in the magnitude (but not in the physical nature) of the fundamental units are called a class of systems of units [53]. Unlike other statistical approaches, the mathematical modelling in the case of Buckingham' Pi approach could be very tedious if a proper set of producers is not considered. Based on [53], following are the step-by-step descriptions of the modelling process adopted in the present work: i.
First of all, the units of the input and the output process parameters have been unified and converted into physical quantities (such as M, L, and T). Further, it is of utmost importance to highlight that any kind of categorical parameter, either input or output, is not suitable for the modelling. Moreover, upon such conversions, it should be considered that the replacement could be represented in-terms of M, L, and T formats. Therefore, in present work, the original Table A1 in the Appendix has been modified in order to balance the units, as well as to convert the qualitative parameters into quantitative. For instance, the parameter "filament proportion" has been quantified in-terms of its tensile strength; density of the FDM pattern has been considered in terms of mass and volume; mould wall thickness has been converted from a number of layers to thickness of the wall, etc. Table 1 is the final prepared modified version of Table A1.
The obtained dimensions of input and output parameters would be: Dimensional accuracy as L, Surface roughness as L, Filament proportion (P) in-terms of tensile strength of filament as MLT −2 , Figure 3. SEM micrograph and EDS spectrum of experiment #16, #17, and #18.

Dimensionless modelling: Buckingham Pi Approach
Dimensionless modelling of the experimental data is considered an efficient method in order to formulate analytic mathematical functions that are out of a highly complex experimental system associated to numerous process parameters [43]. The concept of dimensionless analysis helps to reduce the influence of variables by means of physical equations [44][45][46]. To date, dimensionless modelling with the help of Buckingham Pi approach has been extensively investigated for a wide range of scientific and engineering applications including fluid dynamics [47], energy [48], electronics [49], heat transfer [50] and others. According to the Buckingham approach, any practical problem containing "n" factor sand further "m" dimensions, then the subtraction of n and m will result the counts of independent factors, which could be assumed. Presently, "n" and "m" are 7 and 3, respectively. Therefore, the problem will consist of π1, π2, π3 and π4 that are the dimensional magnitudes. Furthermore, the mathematical formulae derived for the assumed independent parameters help to develop the dimensional relationships by following a set of standard steps [51,52]. Standard quantities of the same physical nature (mass, length, and time) are used based on fundamental units. Consequently, it can be said that these systems belong to the same class. To generalize, a set of systems of units that differ only in the magnitude (but not in the physical nature) of the fundamental units are called a class of systems of units [53]. Unlike other statistical approaches, the mathematical modelling in the case of Buckingham' Pi approach could be very tedious if a proper set of producers is not considered. Based on [53], following are the step-by-step descriptions of the modelling process adopted in the present work: i.
First of all, the units of the input and the output process parameters have been unified and converted into physical quantities (such as M, L, and T). Further, it is of utmost importance to highlight that any kind of categorical parameter, either input or output, is not suitable for the modelling. Moreover, upon such conversions, it should be considered that the replacement could be represented in-terms of M, L, and T formats. Therefore, in present work, the original Table A1 in the Appendix A has been modified in order to balance the units, as well as to convert the qualitative parameters into quantitative. For instance, the parameter "filament proportion" has been quantified in-terms of its tensile strength; density of the FDM pattern has been considered in terms of mass and volume; mould wall thickness has been converted from a number of layers to thickness of the wall, etc. Table 1 is the final prepared modified version of Table A1. Then, it is mandatory to find out the significance level of the input process parameters for the measured outcomes. In the present case, ANOVA has been implemented with the help of MINITAB-17 based statistical software in order to identify the significance and contribution of input parameters. Table 2 shows the contribution percentage of input process parameters for surface hardness, dimensional accuracy, and surface roughness. iii.
Before starting to formulate the π equations (let us say 'x'), it is necessary to identify the 'x − 1' top performing input parameters. For instance, in the case of surface hardness, when 'x' is equal to 4 that allows to develop 4 π-equations, three top performing input parameters have to be identified. iv. Now, the top performing input parameters and the output parameters being analyzed represent the π equations. v.
After calculating the π equations, the π1 (related to the output parameter) is solved as a function of other πs (π2, π3, and π4, consisted of input parameters). vi.
Once the step-v is completed, a constant 'K' has been considered whose value has been driven from a second order quadratic equation of the fitness curve that connect the output response and the most contributing input parameter. vii.
Further, the fitness curve should be plotted between the measured output values and the corresponding values of the most significant input parameter, while keeping the rest of the parameters constant. Alternatively, in the present case, the plots have been drawn between the three levels of the input process parameters and the average of the corresponding output result. For instance, in case of Figure 4, the average of hardness for experiment #1, #4, #7, #10, #13, and #16 has been plotted against first level of F D (5.12 × 10 −6 N/mm 3 ) and the average of hardness for experiment #2, #5, #8, #11, #14, and #17 has been plotted against second level of F D (7.63 × 10 −6 N/mm 3 ). Similar procedure has been adopted for the third level of the F D . viii.
Noticeably, the regression (R 2 )~1 indicates the best fitness of the data.

Hardness
In the present study, hardness is considered a function of all input process parameters that is expressed by Equation (1). So, Based on the Table 2; the least significant parameters for this particular parameter are BF cycle time, BF media filament proportion, and weight that will directly go in "π" groups. The "π" eqns. for hardness can be written as: After substituting the decided dimensions in the "π" groups, Equations (6), (8), (10), and (12) are formed. Now, in order to solve these further, the resulted equations are equated to zero. For instance, the π1 will be solved as follows: Equating the basic dimensions to zero: M: 1 + a 1 + c 1 = 0 L: −1 − a 1 = 0 T: −2 −2a 1 + b 1 = 0 We get, a 1 = −1, b 1 = 0 and c 1 = 0 So, Equation (2) can be re-written as: Similarly, on solving π 2 ; Similarly, equating the basic dimensions to zero: We get, a 2 = −3, b 2 = −6 and c 2 = 2 So, Equation (3) can be re-written as; On solving π 3 ; The Equation (4) for π 3 can be re-written as; Solving π 4 ; The Equation (5) for π 4 can be re-written as; The final relationship between all four Equations of "π" can be assumed as; π 1 = f(π 2 ,π 3 and π 4 ) Or H/F = ρ F 3 t 6 , lFt 2 W and VF 3 t 6 /W 3 The above expression can be written as: Here, "K" is the proportionality constant. Experimentally, it has been found that a correlation between the hardness and " " exists (refer Table 2). Hence, it was taken as representative factors to develop the mathematical model. The average values of the hardness obtained at different levels of "ρ" (throughout the Table 1) has been plotted (see details in Figure 4). In this case, a regression equation (R 2 = 1) with a second order has been determined. Based on the obtained linear equation, the final mathematical model that includes the hardness is given:  Here, "K" is the proportionality constant. Experimentally, it has been found that a correlation between the hardness and "ρ" exists (refer Table 2). Hence, it was taken as representative factors to develop the mathematical model. The average values of the hardness obtained at different levels of "ρ" (throughout the Table 1) has been plotted (see details in Figure 4). In this case, a regression equation (R 2 = 1) with a second order has been determined. Based on the obtained linear equation, the final mathematical model that includes the hardness is given:

Dimensional Accuracy
In a similar way, dimensional accuracy is considered as a function of all input process parameters that is expressed by Equation (16). Δd = ( , , ρ, t, W, l)

Dimensional Accuracy
In a similar way, dimensional accuracy is considered as a function of all input process parameters that is expressed by Equation (16).
From Table 2, the least significant parameters are BF cycle time, number of IC slurry layers, and filament proportion, that will directly go in "π" groups. The "π" equation for dimensional accuracy can be written as: The same set of mathematical iterations has been repeated for dimensional accuracy and the relationship between the all four "π" equations is given in Equation (21) as below: On solving the above expression, we get: BF media weight, which is the most significant parameter (refer to Table 2) with regards to dimensional accuracy, of the casted composites, has been taken as the representative parameter to develop the mathematical model. For this, the average values of the dimensional accuracy obtained at different levels of "BF W " (throughout the Table 1)  From Table 2, the least significant parameters are BF cycle time, number of IC slurry layers, and filament proportion, that will directly go in "π" groups. The "π" equation for dimensional accuracy can be written as: The same set of mathematical iterations has been repeated for dimensional accuracy and the relationship between the all four "π" equations is given in Equation (21) as below: On solving the above expression, we get: BF media weight, which is the most significant parameter (refer to Table 2) with regards to dimensional accuracy, of the casted composites, has been taken as the representative parameter to develop the mathematical model. For this, the average values of the dimensional accuracy obtained at different levels of "BFW" (throughout the Table 1)

Surface Roughness
Further, Equation (24) represents the surface roughness, as a function of all input process variable:

Surface Roughness
Further, Equation (24) represents the surface roughness, as a function of all input process variable: Based on the Table 2; density of FDM pattern, filament proportion, and BF media weight are the least significant parameters for surface roughness that will directly go in "π" groups. The "π" equation for dimensional accuracy can be written as: Now, repeating the same set of mathematical operations the final expression that describes the relationship between all the four "π" is given as Equation (29): Equation (29) can be written as: Similar to the dimensional accuracy, volume of FDM reinforced pattern which is the most significant parameter (refer to Table 2) with regard to surface roughness, of the casted composites, has been taken as the representative parameter to develop the mathematical model. For this, the average values of the surface roughness obtained at different levels of "V P " (throughout Table 1) has been plotted; refer to Figure 6. Then, a regression equation (R 2 = 1) with a second order has been determined. From the obtained linear equation, the final mathematical model for surface roughness is given as: These results obtained in the present work are found to be in-line with the observations presented in the literature [42,45]. Based on the Table 2; density of FDM pattern, filament proportion, and BF media weight are the least significant parameters for surface roughness that will directly go in "π" groups. The "π" equation for dimensional accuracy can be written as: Now, repeating the same set of mathematical operations the final expression that describes the relationship between all the four "π" is given as Equation (29): Equation (29) can be written as: Similar to the dimensional accuracy, volume of FDM reinforced pattern which is the most significant parameter (refer to Table 2) with regard to surface roughness, of the casted composites, has been taken as the representative parameter to develop the mathematical model. For this, the average values of the surface roughness obtained at different levels of "VP" (throughout Table 1) has been plotted; refer to Figure 6. Then, a regression equation (R 2 = 1) with a second order has been determined. From the obtained linear equation, the final mathematical model for surface roughness is given as: Ra = [(−2E − 6V 2 + 0.2101V +1846.5)]·(t · l · ρ )/F 1/3 ·W 2/3 (31) These results obtained in the present work are found to be in-line with the observations presented in the literature [42,45].

Conclusions
In this work, Vashy-Buckingham's π-theorem was employed successfully for the development of the mathematical models related to the hardness, dimensional accuracy, and surface roughness of AMCs; material that was produced through FDM assisted by the IC process. The ANOVA simulation were embedded in the present methodology in order to generate a standard database and to recognize the significance process parameters, respectively. Further, all three mathematical models developed are of second order polynomial equations, with a regression value equal to 1, which prove the reliability of the models.
Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflict of interest. Where, F P , V P , D P , BF T , BF W , NS L , Ra, ∆d, HV, and S/N represent the filament proportion, volume of the pattern, density of the pattern, barrel finishing time, barrel finishing media weight, number of IC slurry layers, surface roughness, dimensional accuracy/deviation, Vickers hardness, signal/noise, respectively. Further, C1 and C2 are the compositions of PA x /Al 2 O 3y /Al z (where x is 60% by wt.; y is 10% and 12% by wt., respectively; and z: 28% and 30% by wt., respectively).