Prediction of Abrasive Waterjet Machining Parameters of Military-Grade Armor Steel by Semi-Empirical and Regression Models

Rolled homogeneous armor steel (RHA) with a high tensile strength, toughness, and hardness is often used in military combat vehicles. RHA is a high-strength low alloy steel suitable for all battlefield usage in military vehicles. The present work examines the prediction output responses in the material removal rate (MRR), surface roughness (Ra), and kerf angle (Ka) for the AWJM of armor steel using regression and semi-empirical models. The AWJM trials were performed using an L27 factorial design with each process variable set to three levels, namely, the standoff distance (SOD), jet traversing speed (JT), and jet water pressure (P). A regression model was constructed using the response surface method (RSM) and data from the trials. Through dimensional analysis and with Buckingham’s π-theorem, a semi-empirical model was built using both the experimental data and material property data. Predictions made by the models were proportionate with the results of the experiments under the same conditions. Microscopic investigations on MRR and Ra were performed using a scanning electron microscope (SEM). The optimal values of the output responses of the machined armor steel plate were obtained with higher MRR = 298.92 mm3/min, lower Ka = 0.651°, and lower Ra = 2.23 µm. The present work established that semi-empirical models accurately predict the output responses in the AWJM of armor steel.


Introduction
Rolled homogeneous armor (RHA) is often utilized in many armor applications [1,2]. Because of its great energy-absorbing qualities, RHA steel is primarily utilized to build military equipment such as tanks, howitzers, and other armored combat vehicles [3,4]. RHA steel is used for ballistic protection in military and civil applications because of its high hardness and ultimate tensile strength. The ability to enhance its hardness with good toughness has been critical to its effectiveness in achieving high performance. This steel has maintained its high position due to its low price, dependability, access to manufacturing infrastructure, continuous value such as structural material, and simplicity of fabrication [5,6]. Due to their high hardness and strength, these steels cannot be turned or milled. This research attempted to cut RHA steel using an unconventional machining method called abrasive waterjet machining (AWJM) [6,7]. Due to the constraint of traditional methods, non-conventional methods are used for material removal. AWJM is an emerging non-conventional process for hard materials such as RHA steels. This is particularly due Profile cutting in armor steels is now conducted by fame-cutting. This method generates a large heat affect zone (HAZ) and poor surface quality, which require yet another operation. However, AWJM overcomes these drawbacks and produces minimal distortion in the work material. AWJM provides the following benefits over other conventional machining processes: no heat deformation; good cutting flexibility; and good surface quality. The suggested work may be developed further using this as a starting point. In this study, Buckingham's π-theorem-based dimensional analysis was used to create a semiempirical model for predicting the process parameters to identify the effect of the output response. The L27 Taguchi orthogonal array was used to measure the process parameters during these AWJM trials. ANOVA and regression modeling were used to determine the influence of the process parameters. The effect of the process parameters was investigated using a regression model fitted using RSM and ANOVA. Microscopic analysis was used to determine the surface quality under varying process parameters.

Experimental Setup
The experimental setup used in this work is shown in Figure 1.
29] developed using AWJM semi-empirical dimensional analysis-based methods to de mine the cutting depth, the angle of the kerf, and the roughness of the surface on MM Profile cutting in armor steels is now conducted by fame-cutting. This method g erates a large heat affect zone (HAZ) and poor surface quality, which require yet anot operation. However, AWJM overcomes these drawbacks and produces minimal dist tion in the work material. AWJM provides the following benefits over other conventio machining processes: no heat deformation; good cutting flexibility; and good surface qu ity. The suggested work may be developed further using this as a starting point. In t study, Buckingham's -theorem-based dimensional analysis was used to create a se iempirical model for predicting the process parameters to identify the effect of the out response. The L27 Taguchi orthogonal array was used to measure the process paramet during these AWJM trials. ANOVA and regression modeling were used to determine influence of the process parameters. The effect of the process parameters was investiga using a regression model fitted using RSM and ANOVA. Microscopic analysis was u to determine the surface quality under varying process parameters.

Experimental Setup
The experimental setup used in this work is shown in Figure 1. AWJM trials were carried out on armor steel using M/s. DADRI International ma the DWJ FB-1313 machining center. Throughout the cutting process, the parameters of impact angle (90°), garnet abrasive size (80 mesh), nozzle diameter (0.67 mm), and ori diameter (0.25 mm) were maintained at constant. On the basis of the literature and ini experiments, the AWJM input parameters and their ranges were chosen. AWJM was u to cut a 20 mm straight linear single pass on the armor steel plate. The process paramet were varied at three levels: [P = 220, 240, 260 MPa; JT = 5, 10, 15 mm/min; and SOD = 1 3 mm]. The full factorial design was considered to conduct the experiments.
The armor steel material is mainly used in battlefield tank body structures and v ous combat applications in the defense industry. The chemical composition of armor s consists of various elements (i.e., C, 0.3301%; Mn, 0.563%; Si, 0.1937%; S, 0.0019%; 1.507%; Ni, 1.570%; Mo, 0.3857%; V, 0.119%; Al, 0.0494; and Fe, balance). Work on dev oping semi-empirical models for considering responses was expanded in this study this study, four quality requirements were chosen that significantly impact the machin of the material using the AWJM technique. The MRR for abrasive waterjet cutting depen on the penetration depth, kerf width, and JT. It was also observed that the MRR was fected only by the abrasive phase's power, and not by the water phase. Equation (1) w used to find out the MRR. The kerf characteristics were measured using ImageJ softw to find the top kerf widths (Ktw) and bottom kerf widths (Kbw). Equation (2) was used calculate the performance parameter kerf angle (Ka). AWJM trials were carried out on armor steel using M/s. DADRI International makes the DWJ FB-1313 machining center. Throughout the cutting process, the parameters of jet impact angle (90 • ), garnet abrasive size (80 mesh), nozzle diameter (0.67 mm), and orifice diameter (0.25 mm) were maintained at constant. On the basis of the literature and initial experiments, the AWJM input parameters and their ranges were chosen. AWJM was used to cut a 20 mm straight linear single pass on the armor steel plate. The process parameters were varied at three levels: [P = 220, 240, 260 MPa; JT = 5, 10, 15 mm/min; and SOD = 1, 2, 3 mm]. The full factorial design was considered to conduct the experiments.
The armor steel material is mainly used in battlefield tank body structures and various combat applications in the defense industry. The chemical composition of armor steel consists of various elements (i.e., C, 0.3301%; Mn, 0.563%; Si, 0.1937%; S, 0.0019%; Cr, 1.507%; Ni, 1.570%; Mo, 0.3857%; V, 0.119%; Al, 0.0494; and Fe, balance). Work on developing semi-empirical models for considering responses was expanded in this study. In this study, four quality requirements were chosen that significantly impact the machining of the material using the AWJM technique. The MRR for abrasive waterjet cutting depends on the penetration depth, kerf width, and JT. It was also observed that the MRR was affected only by the abrasive phase's power, and not by the water phase. Equation (1) was used to find out the MRR. The kerf characteristics were measured using ImageJ software to find the top kerf widths (K tw ) and bottom kerf widths (K bw ). Equation (2) was used to calculate the performance parameter kerf angle (Ka).
Surface roughness was measured using a Mitutoyo Corporation, Japan, SJ-401 surface tester M/s, a conical stylus with a tip radius of 5 µm, a probe measuring range of 350 µm, and a speed of 0.25 m/s. When calculating the average Ra in three regions, the sample measurement length was kept constant at 5 mm (top, middle, and bottom).

Buckingham's "π"-Theorem
Dimensional analysis is a technique used to gather a specific set of data on a certain occurrence. A full set of dimensionless parameters may be obtained after the dimensional analysis. The dimensional analysis method is mostly used to make a physical problem easier to solve by cutting down on the number of factors. This may not significantly affect the current problem. Buckingham's π-theorem can integrate all parameters in a given issue into a dimensionless product (π n ). Algebraic expressions, called terms, are used to produce the necessary mathematical expression related to all the variables in the problem. In this section, a predictive model was established to calculate the performance characteristics including the MRR, Ra, K tw , and kerf angle. These models were constructed using dimensional analysis and were based on appropriate process parameters such as P, JT, and SOD in the AWJM process. The performance characteristics evaluated in this study are expected to vary depending on the machining parameters used and the material and thermal properties of the workpiece. Table 1 lists the parameters used in the development of the semi-empirical models and their dimension formulas.

RSM Modeling
The regression analysis method includes experiments, mathematical tools, and statistical analysis. The present investigation's experimental results given in Table 2 are thoroughly examined. The output response variables MRR, RA, and Ka, are predicted using MINITAB 18 software utilizing a multilinear stepwise regression analysis. The regression model is written as follows with a sample size of n observations for the dependent variable Y (output response) [16,29]: where Y = output response (Ka, MRR, Ra); Ao = co-efficient; A aa = quadratic co-efficient; A ab = interaction co-efficient; A a , A b = dimensionless coded independent variables. The output responses may be described as a function of the AWJ machining parameters such as (JT), (P), and (SOD) through a second-order polynomial regression equation. Equations (4)-(6) can be reformulated as follows to find a connection between the AWJ process parameters and the output responses:

Experimental and Modeling of MRR
The machining process parameters and the physical and thermal characteristics of the workpiece material were taken into account while creating the models. According to Equation (7), the MRR performance was calculated. The dimensional analysis was used to determine the relationship between the performance measure MRR and the parameters.
The "n" variables are represented as (n − p) non-dimensional groups of the variables, referred to as "π" groups using this approach. This dimensional analysis method is based on Buckingham's π-theorem. Mass (M), length (L), time (T), and temperature (θ) are the primary dimensions in this present study. As a consequence, since there are 10 variables and four fundamental dimensions, n = 10 and r = 4, therefore, n − r = 6 according to the present study. Therefore: f (π 1 , π 2 , π 3 , π 4 , π 5 , π 6 ) = 0 (8) Dimensional analysis was used to find the six terms and Equation (8) may be rewritten as Equation (9). The details of the calculated terms are as follows: By replacing a 1 , b 1 , c 1 , d 1 with a, b, c, d; rearranging and rewriting the above phrase as a consequence, the following expression is generated.
According to the power indices on both sides, the dimensionless coefficients are shown in the following way: For Thus, Equation (10) is developed by replacing the calculated value of the power index of the π 1 term.
Similarly, the values of the following π terms are calculated using the same techniques as for the first term. Table 3 shows the obtained equations. The equation derived by substituting all of the acquired π terms in Equation (9) is provided in Equation (10).
Equation (8) is given as Equation (6) by rearranging the terms: The machining process parameters and the thermal-physical characteristics of the work material are considered in the semi-empirical MRR model, as stated in Equation (12). In Equation (12), the coefficients (α) and the power indices (a, b, c, d, and e) must be determined. Therefore, an experiment was carried out. The coefficients and power indices of Equation (12) were calculated using the nonlinear estimation.
From the ANOVA results in Table 4, it is evident that Jt 85.38% had the greatest effect on MRR compared to the other two process parameters. At A × B 0.31%, A × 0.03%, and B × C 1.56%, the influence of the interaction terms was insignificant (0.31%, 0.03%, and 1.56%, respectively). The impacts of all of the process parameters were obtained by maintaining the performance parameter, one variable constant which is kept constant.  Figure 2 illustrates the contour plots of the MRR. It is the most influencing parameter in the MRR compared to P and SOD. It was observed that when JT increased, the MRR increased. Abrasive waterjet particles in the waterjet stream erode high-strength materials such as armor steel, resulting in material loss. Thus, increased kinetic energy increases the erosion rates and material removal rates. The machining rate increases with a higher traverse speed, removing more material from the workpiece. In turn, the material removal rate is mainly determined by the speed of the traverse, which is consistent with previous research [30,31]. The semi-empirical modeling findings were compared with the experimental data and a regression model. Figure 3 illustrates the trends for semi-empirical modeling, the experimental, and regression model results for each observation number. In the semi-empirical model, the findings demonstrate a good agreement between the experimental and predicted values. The semi-empirical modeling was also more accurate and precise than the regression model. The percentage error variance between the experimental and predicted values was less than 5%. The MRR range changes depending on the JT. With increasing JT and P, the MRR value rises. experimental data and a regression model. Figure 3 illustrates the trends for semi-empirical modeling, the experimental, and regression model results for each observation number. In the semi-empirical model, the findings demonstrate a good agreement between the experimental and predicted values. The semi-empirical modeling was also more accurate and precise than the regression model. The percentage error variance between the experimental and predicted values was less than 5%. The MRR range changes depending on the JT. With increasing JT and P, the MRR value rises.  The SEM image of the materials machined at a JT of 5, 10, and 15 mm/min is shown in Figure 4. Figure 4a illustrates the JT at a speed of 5 mm/min. At this speed, more micro voids are detected. The micro-cut and plastic deformation were observed at a JT of 10 mm/min, as illustrated in Figure 4b. Few micro gaps were found at 15 mm/min than at 5 mm/min in Figure 4c. The present study showed that a higher MRR 298.92 mm 3 /min was achieved by increasing the JT to 15 mm/min, with a SOD = 2 mm and P = 260 MPa. experimental data and a regression model. Figure 3 illustrates the trends for semi-empir ical modeling, the experimental, and regression model results for each observation num ber. In the semi-empirical model, the findings demonstrate a good agreement between th experimental and predicted values. The semi-empirical modeling was also more accurat and precise than the regression model. The percentage error variance between the exper imental and predicted values was less than 5%. The MRR range changes depending on the JT. With increasing JT and P, the MRR value rises.  The SEM image of the materials machined at a JT of 5, 10, and 15 mm/min is shown in Figure 4. Figure 4a illustrates the JT at a speed of 5 mm/min. At this speed, more micro voids are detected. The micro-cut and plastic deformation were observed at a JT of 1 mm/min, as illustrated in Figure 4b. Few micro gaps were found at 15 mm/min than at mm/min in Figure 4c. The present study showed that a higher MRR 298.92 mm 3 /min wa achieved by increasing the JT to 15 mm/min, with a SOD = 2 mm and P = 260 MPa. The SEM image of the materials machined at a JT of 5, 10, and 15 mm/min is shown in Figure 4. Figure 4a illustrates the JT at a speed of 5 mm/min. At this speed, more micro voids are detected. The micro-cut and plastic deformation were observed at a JT of 10 mm/min, as illustrated in Figure 4b. Few micro gaps were found at 15 mm/min than at 5 mm/min in Figure 4c. The present study showed that a higher MRR 298.92 mm 3 /min was achieved by increasing the JT to 15 mm/min, with a SOD = 2 mm and P = 260 MPa.

Experimental and Modeling of Ra
The modeling of Ra was determined in a similar manner to that of MRR. The π1 term varies and is generated by semi-empirical modeling. The performance parameter Ra is expressed as: Ra = f (JT, SOD, P, AF, ρ, Cp, K, MT, Hv) As in MRR, the solution for Ra is found in the form of a product of six independent dimensionless ( n) terms, as provided in Equation (11).
Dimensional analysis is used to find the six terms and Equation (14) is redefined as in Equation (15).
The terms in Equation (12) are rewritten to calculate the 1 term as follows:

Experimental and Modeling of Ra
The modeling of Ra was determined in a similar manner to that of MRR. The π 1 term varies and is generated by semi-empirical modeling. The performance parameter Ra is expressed as: Ra = f (JT, SOD, P, AF, ρ, C p , K, MT, H v ) As in MRR, the solution for Ra is found in the form of a product of six independent dimensionless (π n ) terms, as provided in Equation (11). f (π 1 , π 2 , π 3 , π 4 , π 5 , π 6 ) = 0 Dimensional analysis is used to find the six terms and Equation (14) is redefined as in Equation (15).
The terms in Equation (12) are rewritten to calculate the π 1 term as follows: The power indices were calculated in the same manner as explained in Section 4.1 for the MRR modeling formulation. The dimensionless coefficients (a, b, c, and d) had values of {0, 1, 0, and 0}. Thus, by replacing the derived values of the power indices for the Ra model, the produced π1 term is provided in Equation (16).
Rearranging the terms and rewriting Equation (14) is provided in Equation (15).
The power indices β, a, b, c, d, and e, are to be determined. Therefore, an experiment was carried out, and the coefficients and power indices of Equation (19) were calculated using the nonlinear estimation. The ANOVA findings in Table 5 show that JT has the most significant effect on Ra compared to the other two machining factors.  Figure 5 shows the contour plot of P, JT, and SOD in Ra. When increasing the SOD from 1 mm to 2 mm, Ra gradually increases from 2.326 µm to 4.516 µm and after that decreases to 2.94 µm at 3 mm. Ra increases from 3.14 µm to 3.28 µm when P is increased from 220 MPa to 260 MPa. The roughness of the surface is directly related to the JT. From 2.85 to 3.58 µm, the Ra is directly proportional to JT. P increases the particle's kinetic energy and substantially increases its MRR, while decreasing Ra [32,33].
In this study, Ra is initially low as P increases. But the Ra increases as the P increase further. It is an identity that increasing the P (up to 240 MPa) reduces the Ra. Minimizes the chances of plastic deformation and micro-cutting. Increases in P and JT will alter the roughness of the surface. It is due to the size and type. The P and its interaction do not affect the roughness of the character in this case. When P is increased, abrasive fragments is also increased along with their enhanced rate. Due to fragmentation, the size of impacting abrasive particles is reduced. Similarly, when P is increased, the kinetic energy of the water jet also increased, affecting the Ra. The JT, second-order terms of P, and JT are identified as significant process parameters for Ra, based on the experimental results. The standoff distance does not have any physical effect on the roughness of the surface, as previously stated in the literature [34,35]. With a large SOD, more contact P is produced than with a small SOD. It makes a better machining surface [36,37]. Ra is predicted to be reduced by increasing JT since the number of abrasive particles that are imposed on the workpiece gets reduced by increasing JT.  Figure 5 shows the contour plot of P, JT, and SOD in Ra. When increasing the SOD from 1 mm to 2 mm, Ra gradually increases from 2.326 µm to 4.516 µm and after that decreases to 2.94 µm at 3 mm. Ra increases from 3.14 µm to 3.28 µm when P is increased from 220 MPa to 260 MPa. The roughness of the surface is directly related to the JT. From 2.85 to 3.58 µm, the Ra is directly proportional to JT. P increases the particle's kinetic energy and substantially increases its MRR, while decreasing Ra [32,33]. The semi-empirical modeling is validated by evaluating it to Ra's experimental data and regression model. The variations for each observation number for semi-empirical, observed, and regression model Ra values are shown in Figure 6. The results from the semi-empirical modeling indicates a strong level of correlation between experimental and predicted values. According to Figure 6, semi-empirical models are more precise than regression models. The difference between the observed and predicted values is less than 5%. In this study, Ra is initially low as P increases. But the Ra increases as the P increase further. It is an identity that increasing the P (up to 240 MPa) reduces the Ra. Minimizes the chances of plastic deformation and micro-cutting. Increases in P and JT will alter the roughness of the surface. It is due to the size and type. The P and its interaction do not affect the roughness of the character in this case. When P is increased, abrasive fragments is also increased along with their enhanced rate. Due to fragmentation, the size of impacting abrasive particles is reduced. Similarly, when P is increased, the kinetic energy of the water jet also increased, affecting the Ra. The JT, second-order terms of P, and JT are identified as significant process parameters for Ra, based on the experimental results. The standoff distance does not have any physical effect on the roughness of the surface, as previously stated in the literature [34,35]. With a large SOD, more contact P is produced than with a small SOD. It makes a better machining surface [36,37]. Ra is predicted to be reduced by increasing JT since the number of abrasive particles that are imposed on the workpiece gets reduced by increasing JT.
The semi-empirical modeling is validated by evaluating it to Ra's experimental data and regression model. The variations for each observation number for semi-empirical, observed, and regression model Ra values are shown in Figure 6. The results from the semiempirical modeling indicates a strong level of correlation between experimental and predicted values. According to Figure 6, semi-empirical models are more precise than regression models. The difference between the observed and predicted values is less than 5%.
The trial no. 22 is chosen because it provides a lower Ra value than the other experiments. Figure 7a-c illustrate the surface quality of the top, middle, and bottom sections. At a higher JT (15 mm/min), the machined surface appears with more microvoids, rough cut, edge pile up, and crater, whereas at a lower JT (5 mm/min), the machined surface appears with a full cut and smooth surface, minimal plowing, less void, and less shearing as shown in Figure 8a-c. It is possible to achieve minimum roughness even at low JT because abrasive particles have high kinetic energy and may thus penetrate. Better surface finish is obtained at trial no. 22, with SOD = 3 mm, JT = 5 mm/min and P =240 MPa.   At a higher JT (15 mm/min), the machined surface appears with more microvoids, rough cut, edge pile up, and crater, whereas at a lower JT (5 mm/min), the machined surface appears with a full cut and smooth surface, minimal plowing, less void, and less shearing as shown in Figure 8a-c. It is possible to achieve minimum roughness even at low JT because abrasive particles have high kinetic energy and may thus penetrate. Better surface finish is obtained at trial no. 22, with SOD = 3 mm, JT = 5 mm/min and P = 240 MPa.

Experimental and Modelling of Ktw, Kbw, and Ka
The models of kerf top width (Ktw), and kerf bottom width (Kbw) are determined in this investigation. Ka is calculated using the performance measure of Kbw width. The π1 term for Ktw and Kbw are generated by semi-empirical modelling. Ktw can be expressed in the formula provided in Equation (20).  The remaining terms , , , , will stay constant from the semi-empirical modeling of MRR, and the formula for the Ktw width produced by replacing all the attained terms is given in Equation (22).
where, is the proportionality constant in Equation (22) The values of the power indices (a, b, c, d, and e) must be determined. Therefore, an experiment is carried out, and the coefficients and power indices of Equation (18) have been calculated using the nonlinear estimation, respectively.
The Kbw must be determined to calculate the Ka model. The performance parameter Ka depends on both Ktw and Kbw, which necessitates a semi-empirical model usage for Kbw. The experiment was carried out to determine the values of Ktw and Kbw. The width of Kbw is directly proportional to the width of the Ktw. As a result, the term of Ktw corresponds to the term of Kbw as provided in Equation (23).

Experimental and Modelling of K tw , K bw , and Ka
The models of kerf top width (K tw ), and kerf bottom width (K bw ) are determined in this investigation. Ka is calculated using the performance measure of K bw width. The π1 term for K tw and K bw are generated by semi-empirical modelling. K tw can be expressed in the formula provided in Equation (20).
The power indices are calculated similarly to the MRR and Ra model formulations. The dimensionless coefficients a, b, c, and d have values of 0, 1, 0, and 0 respectively. Thus, by replacing the acquired power indices values for the K tw width model, the derived π 1 term is provided in Equation (21). The remaining terms π 2 , π 3 , π 4 , π 5 , π 6 will stay constant from the semi-empirical modeling of MRR, and the formula for the K tw width produced by replacing all the attained terms is given in Equation (22).
where, γ is the proportionality constant in Equation (22) The values of the power indices (a, b, c, d, and e) must be determined. Therefore, an experiment is carried out, and the coefficients and power indices of Equation (18) have been calculated using the nonlinear estimation, respectively.
The K bw must be determined to calculate the Ka model. The performance parameter Ka depends on both K tw and K bw , which necessitates a semi-empirical model usage for K bw . The experiment was carried out to determine the values of K tw and K bw . The width of K bw is directly proportional to the width of the K tw . As a result, the π 1 term of K tw corresponds to the π 1 term of K bw as provided in Equation (23).
δ is the proportionality constant in Equation (24). The power indices values, (δ, a, b, c, d, and e) must be determined. Therefore, an experiment is carried out, and the coefficients and power indices of Equation (24) have been calculated using the nonlinear estimation. Thus, the Ka performance parameter is predicted using the K tw and K bw models. Equation (25) can be used to calculate Ka. Table 6 shows that the interaction of SOD, JT, P, and the second-order interaction of JT and P has a physical significance, with SOD having the highest contribution of 50.78% and the second-order interaction of P having a contribution of 2.92%. It was found that the SOD-JT, P, and JT interaction did not affect the Ka.  Figure 9 shows the contour plot of the machining parameters on Ka, which increased linearly with SOD. Ka initially decreased from 0.727 to 0.720 while the P varied. Then, Ka increased exponentially to 0.746 for a further increase in P. There was a significant relationship between SOD, P, JT, and Ka. While the JT increased, the Ka decreased, as shown in Figure 8a. When the JT increased from 5 mm/min to 10 mm/min, the Ka increased from 0.752 • to 0.722 • ; when the JT increased to 15 mm/min, the taper angle was reduced to 0.719 • . The variance in Ka was due to the hardness in the material. The SOD and JT should be kept to a minimum. The Ka decreased as the hydraulic P increased. This is because kinetic energy is enlarged in the velocity transfer of abrasives, resulting in a diminished K bw . The kinetic energy and cutting performance of the particles were enhanced with increased P. As P increased from 240 MPa to 260 MPa, the top kerf increased due to the increase in Ka. At the base of the workpiece, the waterjet kinetic energy decreased. It could not remove the material efficiently and therefore created a narrow K bw , resulting in a decrease in Ka. A similar trend was seen in armor steel cutting. Ka increased with an increase in SOD. At high speeds, the jet will deviate. The jet increased as it exited the nozzle. The workpiece surface is now exposed to the bottom of the waterjet after the SOD is raised. As the water jet diverged, it lost uniformity, which diminished the cutting area seen in Ka immediately after that. Similarly, increasing JT tended to increase Ka. When the JT value is low, the nozzle allows for a high volume of abrasive particles to strike the workpiece. As a result, a larger volume of material is removed, resulting in a higher Ktw. At the same time, the waterjet thoroughly penetrates the workpiece, maintaining kinetic energy and efficiently cutting the bottom material. The kerf width variation reduces the taper angle. The correct taper was obtained at SOD (1 mm), JT (10 mm/min), and P (220 MPa). The reverse taper was obtained at SOD (3 mm), JT (5 mm/min), and P (240 MPa), as shown in Figure 10. The Ka semi-empirical model was verified by comparing the experimental data and regression model. Figure 11 shows the variation for each observation number in the semiempirical modeling, experimental, and regression models with Ka values. The findings of the semi-empirical model showed a good connection between the experimental and predicted values. The semi-empirical models outperformed the regression models. The observed and anticipated values varied by less than 5%. As the water jet diverged, it lost uniformity, which diminished the cutting area seen in Ka immediately after that. Similarly, increasing JT tended to increase Ka. When the JT value is low, the nozzle allows for a high volume of abrasive particles to strike the workpiece. As a result, a larger volume of material is removed, resulting in a higher K tw . At the same time, the waterjet thoroughly penetrates the workpiece, maintaining kinetic energy and efficiently cutting the bottom material. The kerf width variation reduces the taper angle. The correct taper was obtained at SOD (1 mm), JT (10 mm/min), and P (220 MPa). The reverse taper was obtained at SOD (3 mm), JT (5 mm/min), and P (240 MPa), as shown in Figure 10. As the water jet diverged, it lost uniformity, which diminished the cutting area in Ka immediately after that. Similarly, increasing JT tended to increase Ka. When th value is low, the nozzle allows for a high volume of abrasive particles to strike the w piece. As a result, a larger volume of material is removed, resulting in a higher Ktw. A same time, the waterjet thoroughly penetrates the workpiece, maintaining kinetic en and efficiently cutting the bottom material. The kerf width variation reduces the t angle. The correct taper was obtained at SOD (1 mm), JT (10 mm/min), and P (220 M The reverse taper was obtained at SOD (3 mm), JT (5 mm/min), and P (240 MPa), as sh in Figure 10. The Ka semi-empirical model was verified by comparing the experimental data regression model. Figure 11 shows the variation for each observation number in the s empirical modeling, experimental, and regression models with Ka values. The findin the semi-empirical model showed a good connection between the experimental and dicted values. The semi-empirical models outperformed the regression models. The served and anticipated values varied by less than 5%. The Ka semi-empirical model was verified by comparing the experimental data and regression model. Figure 11 shows the variation for each observation number in the semiempirical modeling, experimental, and regression models with Ka values. The findings of the semi-empirical model showed a good connection between the experimental and predicted values. The semi-empirical models outperformed the regression models. The observed and anticipated values varied by less than 5%.

Conclusions
The influence of the AWJM process parameters on the MRR, RA, and Ka values while machining an armor steel plate was investigated in this study. Using Buckingham's πtheorem and the regression equation, a semi-empirical model was created to estimate three key parameters: MRR, Ra, and Ka. ANOVA was used to evaluate the model's significance and performance. The experimental findings were compared with the predicted values of MRR, Ra, and Ka using the proposed semi-empirical model and regression model, and the following decisions were made:

•
The confirmation experiments showed that the residuals between the predicted and experimental values were less than 5%. The values of the semi-empirical modeling and regression models were compared with the experimental values. For MRR, Ra, and Ka, the semi-empirical model was proven to be more accurate than the regression model to predict the relationships of the variable.

•
The use of the analysis of variance approach in combination with the dimensional analysis reduced the number of insignificant variables and resulted in simpler models for MRR, Ra, and Ka.

•
The maximum MRR was obtained at SOD = 2 mm, JT = 15 mm/min, and P = 260 MPa. Improved MRR was achieved due to the greater kinetic energy of the jet produced by this combination. These findings demonstrate that a higher surface finish was attained with a higher SOD = 3 mm and lower JT = 5 mm/min. This allows for a more uniform MRR, which resulted in a lower roughness. Ra was determined to be 2.23 µm.

•
The SOD process parameter significantly impacts the response, Ka. The variation in Ka is due to the hardness of the material. The SOD and JT should be kept at the minimum. The lower Ka (0.651°) was obtained at SOD = 1 mm, JT = 10 mm/min, and P = 220 MPa.

•
The microstructural analysis of the machined surfaces revealed three regions: top, middle, and bottom. In the top region, smooth surfaces were obtained, and in the middle region, fewer voids were present, and less ploughing was identified at the bottom.

•
In future, the research may be extended on the comparison of experimental results with numerical modeling studies. The results obtained can be applied in battlefield vehicle applications. Figure 11. A comparison of Ka between the experimental, semi-empirical, and regression methods.

Conclusions
The influence of the AWJM process parameters on the MRR, RA, and Ka values while machining an armor steel plate was investigated in this study. Using Buckingham's π-theorem and the regression equation, a semi-empirical model was created to estimate three key parameters: MRR, Ra, and Ka. ANOVA was used to evaluate the model's significance and performance. The experimental findings were compared with the predicted values of MRR, Ra, and Ka using the proposed semi-empirical model and regression model, and the following decisions were made:

•
The confirmation experiments showed that the residuals between the predicted and experimental values were less than 5%. The values of the semi-empirical modeling and regression models were compared with the experimental values. For MRR, Ra, and Ka, the semi-empirical model was proven to be more accurate than the regression model to predict the relationships of the variable.

•
The use of the analysis of variance approach in combination with the dimensional analysis reduced the number of insignificant variables and resulted in simpler models for MRR, Ra, and Ka.

•
The maximum MRR was obtained at SOD = 2 mm, JT = 15 mm/min, and P = 260 MPa. Improved MRR was achieved due to the greater kinetic energy of the jet produced by this combination. These findings demonstrate that a higher surface finish was attained with a higher SOD = 3 mm and lower JT = 5 mm/min. This allows for a more uniform MRR, which resulted in a lower roughness. Ra was determined to be 2.23 µm.

•
The SOD process parameter significantly impacts the response, Ka. The variation in Ka is due to the hardness of the material. The SOD and JT should be kept at the minimum. The lower Ka (0.651 • ) was obtained at SOD = 1 mm, JT = 10 mm/min, and P = 220 MPa.

•
The microstructural analysis of the machined surfaces revealed three regions: top, middle, and bottom. In the top region, smooth surfaces were obtained, and in the middle region, fewer voids were present, and less ploughing was identified at the bottom.

•
In future, the research may be extended on the comparison of experimental results with numerical modeling studies. The results obtained can be applied in battlefield vehicle applications. Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.