Compressive Strength Impact on Cut Depth of Granite During Abrasive Water Jet Machining

Abstract

Background: Compared to the conventional method of machining granite, abrasive water jet machining (AWJM) offers several benefits, including flexible cutting mechanisms and machine efficiency, among other possible advantages. The high-speed particles carried by water remove the materials, preventing heat damage and maintaining the granite’s structure. Methods: Three types of granite with different compressive strengths are investigated in terms of the effects of pump pressure (P), traverse speed (T), and abrasive mass flow (A) on the cutting depth. Results: The results of the study demonstrated that the coarse-grained granite negatively affected the penetration depth, while the fine-grained granite produced a higher cutting depth. The value of an optimal depth of penetration was also generated; for example, the optimum depth obtained for Black Galaxy Granite, M1 (32.27 mm), was achieved at P = 300 MPa, T = 100 mm/min, and A = 180.59 g/min. Conclusions: In terms of processing parameters, the maximum penetration depth can be achieved in granite with a higher compressive strength.

1. Introduction

There are several advantages to using abrasive water jet machining (AWJM) compared with traditional granite material machining methods, including cutting mechanisms and machining efficiency. These materials are removed by high-speed particles carried by water, so thermal damage will not affect the structure of the granite. This process can also be applied to various materials and thicknesses, making it versatile and cost-effective for small and large production scales in various industrial sectors [1]. Soon after its inception, AWJM was introduced for stone, concrete, and pavement cutting by the marble industry. In Italy, the development of dimension stone shaping is highly advanced, as marble is essentially concentrated in one region of the country. AWJM technology quickly became the standard for cutting in almost all stone fabricating plants, minimizing the amount of polishing and shaping required. Additionally, the emergence of conventional production processes and AWJM maximized the many possible shapes and products [1,2].

Literature Review

Studies have indicated that the performance of AWJM is influenced by hydraulic, cutting, mixing, and abrasive parameters. The measures of process quality include kerf width, kerf taper angle [3], striation marks on the machined surface, and other outputs. The major influence of AWJ machining is water jet pressure, as the kinetic energy of the AWJ depends on the water pressure level. To erode the rock, abrasive water jet machining for granite utilizes high-pressure jets containing abrasive particles [2,4]. The latest method, using premixed abrasives, indicates that higher jet pressures enhance erosion effects; namely, harder premixed abrasives produce greater cutting force and improve efficiency [4]. Another innovative technique involves testing the mechanisms of high-temperature hard rock; in this case, the high-pressure jets demonstrate three deformation stages—compressive deformation, deformation release, and stable deformation—with significant thermal effects enhancing cutting efficiency [5].

An additional study using pressurized pulsed water jets (PPWJs) indicated that the fragmentation modes of granite and sandstone are unrelated: sandstone has conical erosion craters, while granite has spoon-shaped erosion craters. The specific energy for rock breaking with pulsed water jets decreased in comparison to CWJ [6]. Nozzle arrangements can improve rock fragmentation in deep holes; for example, using straight cone and fan-type designs [7]. AWJM is influenced by many variables, including traverse speed [8]. Studies have shown that slower traverse speeds tend to be better for cutting; more material is removed with better cutting rates [9]. Furthermore, this association with pump pressure plays a vital role in the optimization of AWJM processes and improving cutting efficiency [10,11].

The impacts of abrasive mass flow and high pressure on granite cutting have been investigated in [12]. The authors discovered that the abrasive mass flow rate was precisely related to the pressure of the water jet. At low traverse rates and higher water pressure, the kerf taper angle and striation decreased.

Stand-off distance is the distance between the target material and the nozzle. Usually, a satisfactory distance in millimeters is maintained, as the influence of SOD is significant. One study confirmed that the kerf width grows with the stand-off distance [13]. Karakurt et al. [14] studied the taper angle of abrasive water jet cuts for many granite compositions, finding that traverse rate and stand-off distance are important controlling parameters for Baltic brown and Aksaray Yaylak granite and Bergama gray granite. An increased traverse rate resulted in a corresponding increase in the kerf angle of the granite materials. Hlaváč et al. [15] investigated the quality of cut surfaces by analyzing the declination angle between the tangent and the striation characteristics generated using abrasive water jet (AWJ) cutting on rock materials, specifically in the kerf wall cut surface. The declination angle of the cut surfaces was utilized to determine the optimal traverse rates for cutting a specific material thickness while ensuring the required quality. Furthermore, Karakurt et al. [16] examined the role of abrasive mesh, particularly sizes #80 through #120. A greater penetration depth was achieved with an #80 garnet mesh size. The cut depth decreased as the mesh size of the abrasives increased. The abrasive particles act through kinetic energy, which is induced by their mass. As a result, large abrasive particles can penetrate deeper, and the target material is quickly cut.

Hwang et al. [17] enhanced the material removal rate for granite and concrete using mixed abrasives. To understand and compare the pros and cons between garnet and steel shot in terms of their cutting capabilities and costs, they used mixed abrasives at different garnet–steel shot mass ratios. The authors also evaluated the experimental cutting characteristics and costs, finding that the best balance between the two was 50:50. Their study also provided guidance for industrial applications.

Furthermore, optimizing AWJM parameters plays an important role in this process. Researchers have undertaken numerous attempts to model and optimize abrasive water jet machining through various methodologies. These methods include genetic algorithms, simulated annealing, artificial bee colony algorithms, response surface methodology, teaching–learning-based optimization, the multi-objective Jaya algorithm, the Taguchi method, artificial neural networks, fuzzy logic, the biogeography algorithm, and sequential approximation optimization. Perec et al. [18] compared the results of algorithms such as ALO, GWO, and MFO to optimize the cutting thickness of marble. Pan et al. [19] studied granite abrasive water jet machining and obtained the best cutting parameters with a jet pressure of 40 MPa, a stand-off distance of 2 mm, a transverse velocity of 2 mm/s, and a jet inclination of 10°, improving the efficiency of the rock-cutting operation.

In this work, three kinds of granite with different compressive strengths are investigated regarding the effects of pump pressure, traverse speed, and abrasive mass flow on the cutting depth. These parameters are studied using the response surface method (RSM). Furthermore, to determine the best experimental results, Particle Swarm Optimization (PSO) is used.

2. Materials and Methods

2.1. Materials Description

Three types of granite were used in the experiment, demonstrating different levels of compressive strength.

Table 1. Major properties of the specimens.

Symbol	Material Type	Water Absorption	Compressive Strength	Density	Flexural Strength
M1	Black Galaxy Granite	0.08–0.12 by weight%	200.0–203.3 MPa	2830 kg/m3	19.2–20.2 MPa
M2	G602 Granite	0.37 by weight%	169.2 MPa	2620 kg/m3	15.5 MPa
M3	Cats Eye Granite	0.15–97 by weight%	100.0–117.0 MPa	2650 kg/m3	11.0–15.0 MPa

presents the main properties of the specimens. Granite is an igneous rock formed from a combination of deep primary and secondary magma (intrusive plutonic rocks). Macroscopically, they are dense stones and are typically challenging to work with mechanically. They have various tonalities (white, gray, blue, pink, or red), as shown in Figure 1, and a granular texture varying from small (below hundreds of microns) to coarse (up to several tens of centimeters) [20].

Granite is typically characterized by silica (SiO₂) content exceeding 40% wt., classifying it as ultrabasic rock. Common types of granite typically contain over 60% of their weight in silica, as well as Al₂O₃, Fe₂O₃, MgO, Na₂O, and K₂O. Chemical analysis can provide an estimate of the mineralogical composition of rocks [20].

Eighty-mesh garnet was used as an abrasive material in the experiment; it has been confirmed that AWJ cutting operations are influenced by the abrasive type, density, form, and hardness of the abrasive media [21].

Table 2. Garnet—quick details.

Specification	Details
Place of Origin	Jiangsu, China (Mainland)
Material	Almandine garnet
Color	Red
Bulk Density	1.96–2.15 g/cm3
Brand Name	Jin Hong
Hardness	7.5–8.0 Mohs Scale
Abrasive Grain	80 MESH

provides details about the abrasive material.

2.2. Experimental Setup

The experiment was performed on the KMT international pumping system. The cutting table illustrated in Figure 2A has a work area of 2000 mm × 3000 mm. Figure 2D illustrates a streamlined SL-V 50 HP-type pumping system that can operate at up to 300 MPa.

The water jet was controlled by a CNC unit with the help of NC Studio CAM, as shown in Figure 2C. The machine has an accuracy and repeatability of ±0.1 mm. The lateral speed ranges from 0 to 15 m/min. Figure 2E illustrates the orifice nozzle, with a diameter of 0.25 mm, and Figure 2F shows the focusing tube, with a diameter of 0.79 mm.

In the experiment, pre-dimensioned granite specimens were used with a trapezoidal shape with the following parameters: 60 mm thickness, 250 mm length, and 60 mm width. To achieve 60 mm thickness, epoxy-based adhesive was used; this type of adhesive is strong enough to bond granite effectively [22,23]. Once the surfaces had been prepared and the required quantity of epoxy had been applied to one of them for joining, clamps were used to hold the stone pieces together. This ensures effective contact when the epoxy cures.

3. Machining Factors of the Experiment

The experimental model was developed using the response surface method (RSM). The RSM led to the selection of the Box–Behnken technique. It is a 2^k factorial design with k representing the factors. The three factors of these designs are their vertices, centers, and star points, resulting in 15 combinations [24].

The input factors selected in this work were pressure (P), abrasive mass flow (A), and traverse speed (T). The process parameters (factors) were established based on the literature, prior experiences, and preliminary experiments, and their levels were subsequently determined. Each parameter was tested at three levels, as shown in

Table 3. Process variables and their levels.

Factor Symbol	Variables	Levels
		Level 1	Level 1	Level 3
P	Pressure [MPa]	200	250	300
A	Abrasive mass flow rate [g/min]	150	200	250
T	Traverse speed [mm/min]	100	140	180

. The responses considered for assessment are the cut depths for three types of material (M1, M2, and M3), as explained in Table 3. The plan followed the procedures defined by the RSM, as shown in

Table 4. Design matrix and results.

Process Variables				Response
Run	Pressure [MPa]	Abrasive Mass Flow Rate [g/min]	Traverse Speed [mm/min]	Cut Depth
				[mm]
				M1	M2	M3
1	200	150	140	24.41	23.66	19.81
2	300	150	140	28.68	27.32	22.47
3	200	250	140	22.94	21.88	21.46
4	300	250	140	29.61	28.33	23.77
5	200	200	100	27.15	26.76	25.9
6	300	200	100	32.14	28.86	26.46
7	200	200	180	17.9	17.17	17.01
8	300	200	180	26.16	25.88	24.02
9	250	150	100	29.78	27.31	25.88
10	250	250	100	30.12	28.44	26.56
11	250	150	180	16.99	16.66	16.36
12	250	250	180	24.85	23.13	19.42
13	250	200	140	26.94	25.01	21.46
14	250	200	140	26.81	25	21.32
15	250	200	140	26.77	25.89	21.44

.

The other parameters were kept constant during the tests, including the nominal jet impact angle (90°), orifice diameter (0.25 mm), mixing tube or nozzle diameter (0.79 mm), and stand-off distance (2 mm). The cutting depth of each experimental case was measured three times, and the average results were derived to examine the cutting performance.

Different methods are available for finding the cut depth. In general, the accuracy of the measurement may be affected by using ordinary instruments, like steel rules and height gauges, to measure the depth of penetration [25,26,27,28,29]. In the present study, a profile projector was used to measure the length of the cut (L) on the sloped surface, and the exact depth of penetration was calculated. Three decimal places can help determine the value of the cut’s length. Common measuring tools, including height gauges and steel rules, cannot reach this degree of precision [25,26].

This method of determining penetration depths provides an accurate value compared with other methods. Figure 3 shows the geometry of the trapezoidal specimen used to measure the length of cuts. A light beam source was thrown into cuts generated by an abrasive water jet. The cut shadow was projected on the display screen of a profile projector, and the length of each cut was measured from the start to the end of the cut [25]. The real penetration depth is determined using the following formula: depth of cut = L tan θ (where L is the length of the cut at the bottom of the specimen, as shown in Figure 2A, and θ is the angle of the trapezoidal sample). In this research, we consider only two decimal places for the length of cuts projected on the display of the profile projector.

Figure 3A illustrates the geometry of trapezoidal specimens mounted on the glass slide of the profile projector. A ray of light was passed through the cut, as shown in Figure 3B. The shadow of each cut was cast on the display screen of the profile projector, as shown in Figure 3C, and subsequently, the length of each cut was obtained by measuring the length between the beginning and the end of the cut with the aid of a digital projector measuring unit. Generally speaking, a shadow reflects the full length of a cut.

The SEM model FEI Quanta FEG 450 was used (Thermo Fisher Scientific, Brno, Czech Republic). The samples were mounted on aluminum stubs with double-sided sticky disks of conductive carbon and then coated with 5 nm gold using a sputter coater with the help of a Quorum Q 150R.

4. Response Surface Methodology

The effect of AWJM parameters on cut depth was investigated experimentally using the response surface methodology. The relationship between the response and independent factors can be represented in the RSM using a polynomial regression model, fitting the response function to the experimental findings using the following equation.

$$\mathrm{y}={\mathrm{B}}_1+{\mathrm{B}}_2{\mathrm{X}}_1+{\mathrm{B}}_3{\mathrm{X}}_2+{\mathrm{B}}_4{\mathrm{X}}_1{\mathrm{X}}_2+{\mathrm{B}}_5{\mathrm{X}}_1^2+{\mathrm{B}}_6{\mathrm{X}}_2^2$$

(1)

In this context, y represents the dependent variable (response), X_i denotes the value of the control parameter, and B_i signifies the regression coefficients. The models for cut depths were developed based on the experimental values gathered, and a mathematical relationship was formulated. An empirical model of the influence of the pressure (P), abrasive mass flow rate (A), and traverse speed (T) was developed. Table 4 presents the results of experimental studies. The cut depth ranged from 16.99 mm to 32.14 mm for Black Galaxy Granite (M1), 16.66 mm to 28.86 mm for G602 Granite (M2), and 16.36 mm to 26.56 mm for Cats Eye Granite (M3). Analysis of variance (ANOVA) was employed to assess the significance of each independent variable in the response function. The ANOVA test was performed at a significance level of 5% (α = 0.05). The corresponding F-value followed a continuous probability distribution. A p-value of less than 0.05 for each factor indicated that the model factor was significant, while values greater than 0.05 suggested that the model factor was non-significant.

The ANOVA results for cutting depth across the three specified materials are detailed in

Table 5. ANOVA for cut depth—Black Galaxy Granite (M1).

Source			DF	Adj. SS	Adj. MS	F	p-Value
Model			9	241.325	26.814	11.94	0.007
Linear			3	219.007	73.002	32.50	0.001
Pressure			1	73.145	73.145	32.56	0.002
Abrasive mass flow rate			1	7.334	7.334	3.26	0.131
Traverse speed			1	138.528	138.528	61.66	0.001
Square			3	4.068	1.356	0.60	0.640
Pressure [MPa] × Pressure			1	0.001	0.001	0.00	0.987
Abrasive mass flow rate × Abrasive mass flow rate			1	0.640	0.640	0.28	0.616
Traverse speed × Traverse speed			1	3.610	3.610	1.61	0.261
Two-Way Interaction			3	18.251	6.084	2.71	0.155
Pressure × Abrasive mass flow rate			1	1.440	1.440	0.64	0.460
Pressure × Traverse speed			1	2.673	2.673	1.19	0.325
Abrasive mass flow rate × Traverse speed			1	14.138	14.138	6.29	0.054
Error			5	11.232	2.246
Total			14	252.558
S = 1.49882	R2 = 95.55%	R2 (adj) = 87.55%

,

Table 6. ANOVA for cut depth—G602 Granite (M2).

Source			DF	Adj. SS	Adj. MS	F	p-Value
Model			9	187.406	20.823	9.66	0.011
Linear			3	162.282	54.094	25.10	0.002
Pressure			1	54.706	54.706	25.38	0.004
Abrasive mass flow rate			1	5.831	5.831	2.71	0.161
Traverse speed			1	101.745	101.745	47.20	0.001
Square			3	5.126	1.709	0.79	0.548
Pressure [MPa] × Pressure			1	0.562	0.562	0.26	0.631
Abrasive mass flow rate × Abrasive mass flow rate			1	0.569	0.569	0.26	0.629
Traverse speed × Traverse speed			1	3.860	3.860	1.79	0.238
Two-Way Interaction			3	19.998	6.666	3.09	0.128
Pressure × Abrasive mass flow rate			1	1.946	1.946	0.90	0.386
Pressure × Traverse speed			1	10.923	10.923	5.07	0.074
Abrasive mass flow rate × Traverse speed			1	7.129	7.129	3.31	0.129
Error			5	10.255	2.155
Total			14	198.184
S = 1.46813	R2 = 94.56%	R2 (adj) = 84.77%

and

Table 7. ANOVA for cut depth—Cats Eye Granite (M3).

Source			DF	Adj. SS	Adj. MS	F	p-Value
Model			9	142.676	15.8529	17.67	0.003
Linear			3	123.181	41.0603	45.77	0.000
Pressure			1	19.656	19.6564	21.91	0.005
Abrasive mass flow rate			1	5.595	5.5945	6.24	0.055
Traverse speed]			1	97.930	97.9300	109.16	0.000
Square			3	7.648	2.5492	2.84	0.145
Pressure [MPa] × Pressure			1	2.870	2.8702	3.20	0.134
Abrasive mass flow rate × Abrasive mass flow rate			1	0.623	0.6232	0.69	0.443
Traverse speed × Traverse speed			1	4.142	4.1422	4.62	0.084
Two-Way Interaction			3	11.847	3.9491	4.40	0.072
Pressure × Abrasive mass flow rate			1	0.031	0.0306	0.03	0.861
Pressure × Traverse speed			1	10.401	10.4006	11.59	0.019
Abrasive mass flow rate × Traverse speed			1	1.416	1.4161	1.58	0.264
Error			5	4.486	0.8971
Total			14	147.161
S = 0.947163	R2 = 96.95%	R2 (adj) = 91.47%

. Table 5 and Table 6 present the ANOVA results for M1 and M2, respectively. The findings reveal that traverse speed (T) has the most significant impact on the cutting depth, with pressure (P) being the second most influential variable. Furthermore, the ANOVA results for M3 in Table 7 indicate that traverse speed (T) has the most significant impact on cutting depth, with pressure (P) being the second most influential variable, followed by the interaction between pressure (P) and traverse speed (T).

We calculated the coefficient of determination, the R-squared, and the adjusted R-squared, representing the percentage of variance explained by the model. When the values of the R-squared and adjusted R-squared approach unity, and the difference between the two is smaller than 0.2, this indicates that the established model is adequate in representing the process, and the regression equation would fit more accurately with the research results. The following polynomial functions for each of the three materials describe the cut depth.

Regression Equations:

$$\begin{array}{l}\mathrm{Depth}\mathrm{of}\mathrm{cut}\mathrm{for}\mathrm{M}1=56.0-0.042\mathrm{P}-0.106\mathrm{A}-0.221\mathrm{T}-0.000006\mathrm{P}\times \mathrm{P}-0.000166\mathrm{A}\times \mathrm{A}-0.000618\mathrm{T}\times \mathrm{T}+\\ 0.000240\mathrm{P}\times \mathrm{A}+0.000409\mathrm{P}\times \mathrm{T}+0.000940\mathrm{A}\times \mathrm{T}\end{array}$$

(2)

$$\begin{array}{l}\mathrm{Depth}\mathrm{of}\mathrm{cut}\mathrm{for}\mathrm{M}2=73.8-0.197\mathrm{P}-0.083\mathrm{A}-0.250\mathrm{T}+0.000156\mathrm{P}\times \mathrm{P}-0.000157\mathrm{A}\times \mathrm{A}-0.000639\mathrm{T}\times \mathrm{T}+\\ 0.000279\mathrm{P}\times \mathrm{A}+0.000826\mathrm{P}\times \mathrm{T}+0.000668\mathrm{A}\times \mathrm{T}\end{array}$$

(3)

$$\begin{array}{l}\mathrm{Depth}\mathrm{of}\mathrm{cut}\mathrm{for}\mathrm{M}3=85.7-0.251\mathrm{P}+0.0496\mathrm{A}-0.534\mathrm{T}+0.000353\mathrm{P}*\mathrm{P}-0.000164\mathrm{A}\times \mathrm{A}+0.000662\mathrm{T}\times \mathrm{T}-\\ 0.000035\mathrm{P}\times \mathrm{A}+0.000806\mathrm{P}\times \mathrm{T}+0.000298\mathrm{A}\times \mathrm{T}\end{array}$$

(4)

Analyses of the cutting depth indicated that the values of the R-squared for the three materials were 95.55%, 94.56%, and 96.95%, respectively. The regression models successfully illustrated the relationship between the independent variables and the cutting depth.

5. Results and Discussion

The following results were obtained by focusing on how the compressive strengths of different granite types affect the cut depth in abrasive water jet machining with respect to independent factors. Overall, the compressive strength of granite is the most important parameter in reflecting the bearing capacity of rock mass, and its variability can be attributed to various reasons [30]. However, as the average grain size of minerals that make up rocks becomes smaller, their strength increases [31]. Weaknesses like microfractures, grain boundaries, mineral cleavages, and twinning planes are more likely to occur when the size of the grains decreases, as the microstructure of a rock influences its strength characteristics [32,33]. Accordingly, compressive strength is usually higher for smaller grains.

The grain size was measured for each granite sample using SEM, and the results confirm that the three materials have different grain sizes, as shown in Figure 4. Even though the products were labeled by the suppliers, the grain sizes vary, with the lowest value for Black Galaxy Granite and the highest value for Cats Eye Granite. Normally, a lower cut depth can be obtained for granite with lower compressive strength. Our results demonstrated that coarse-grained granite had a negative effect on the depth of penetration [34], while fine-grained granite produced a higher cutting depth [35]. In other words, the highest depth of penetration relating to process parameters was achieved with granite with smaller grains and higher compressive strength [11]. When the size of the grains decreases, the weaknesses may control the direction in which rock failure occurs, and they may favor deeper depths [11,34,35]. These findings suggest that the union of more individual grains and grain boundaries may have an active effect on the cut depth [11,12,13,14,15,16].

Limited to the effects of three parameters, the following graphs relate pressure to traverse speed, and the holding values for abrasive mass flow and traverse speed are 200 g/min and 140 mm/min, respectively. The findings indicate that, for all three materials, pressure increases with depth. High pressure may help raise traverse speed [36].

The effect of pressure and traverse speed is shown in Figure 5; the jet stays for a longer time, so the depth of cut increases [37,38]. At the same time, Figure 5 shows the reduced cutting depth at a higher traverse speed. Accordingly, the cutting depth is greatly affected by the feed rate. It is essential to maintain a consistent speed during the operation to avoid rough edges. Traverse speed is measured in millimeters per minute [35].

The penetration of particles into the granite material and the associated kinetic energy are influenced by the abrasive mass flow rate. This indicates that higher abrasive mass flow rates can enhance cutting performance [34]. However, it is important to recognize that an excessively high abrasive flow rate may cause the abrasive particles to lose some of their kinetic energy [28]. It is essential to closely monitor any increases in the abrasive mass flow rate. Moreover, there is an advantage in increasing the abrasive mass flow rate, which reduces the cut depth, as seen in Figure 6; however, the cut depth and abrasive mass flow rate generally maintain a directly proportional relationship.

6. Optimization

Finding an optimal parameter for each material is one of the most important steps in AWJM. Thus, the established method will be useful in determining the relationships between the input parameters and AWJM responses [18]. In this work, the Particle Swarm Optimization (PSO) algorithm was applied as a computational tool to determine ideal values for the input variables, thus producing the maximum depth for a given material. PSO was inspired by group behaviors observed among fish and birds. Within this framework, a collection of potential solutions, referred to as particles, moves around the search space. Based on its own historical best position, as well as the best position recognized by the entire swarm, every particle adjusts its location and velocity to obtain the best possible results. This iterative procedure continues until either the ideal solution is found or a sufficient number of iterations have been accomplished, whichever comes first [39,40].

The initial three independent variables were defined: T (traverse speed), A (abrasive mass flow rate), and P (pressure). These factors are the design or operational features that affect the ideal depth of a material. Data for these variables and the corresponding depth were obtained from the experimental results. A “fitting curve” methodology was then exploited to obtain a mathematical equation that relates the variables in terms of depth. These procedures were adopted for the three materials investigated, M1, M2, and M3. A quadratic equation was obtained for the materials as follows:

$${\mathrm{y}=\mathrm{a}}_1{\mathrm{X}}_1^2+{\mathrm{a}}_2{\mathrm{X}}_1{\mathrm{X}}_2+{\mathrm{a}}_3{\mathrm{X}}_1{\mathrm{X}}_3+{\mathrm{a}}_4{\mathrm{X}}_1+{\mathrm{a}}_5{\mathrm{X}}_2^2+{\mathrm{a}}_6{\mathrm{X}}_2{\mathrm{X}}_3+{\mathrm{a}}_7{\mathrm{X}}_2+{\mathrm{a}}_8{\mathrm{X}}_3^2+{\mathrm{a}}_9{\mathrm{X}}_3+{\mathrm{a}}_{10}$$

After obtaining these equations, the PSO algorithm was used again to determine the optimal values for the variables T, A, and P that achieve the highest possible depth value, as well as the highest curve fitting coefficient, as shown in

Table 8. Curve fitting coefficients.

Coefficient	M1	M2	M3
a1	−0.00062	−0.00064	0.000662
a2	0.00094	0.0006675	0.0002975
a3	0.0004088	0.000826	0.00080625
a4	−0.2211875	−0.25028125	−0.5338854
a5	−0.0001665	−0.000157	−0.000164
a6	0.00024	0.000279	−3.5 × 10−5
a7	−0.10585	−0.083325	0.049558
a8	−5.5 × 10−6	0.000156	0.000353
a9	−0.042	−0.19717	−0.25086
a10	55.9659	73.795	85.712

for each material. A software model was developed using MATLAB version 19 to implement the optimization process. An objective function was constructed for each material based on the extracted regression equation. The PSO algorithm parameters were then adjusted to ensure a balance between exploration and exploitation in the search space, namely, the number of particles, the number of iterations, the variable limits, and the cognitive and social coefficients.

The system was programmed flexibly, allowing the process to be automatically repeated for each material. The depth equation was entered into the PSO model, and the optimal results are displayed in a table containing the best depth value and the best combination of the three variables.

At the end of this phase, the study objective was achieved: to develop a computational framework based on a hybrid (combination) methodology between mathematical modeling and the PSO algorithm to obtain the best depth for a specific material based on three independent variables, with the ability to generalize the framework to more than one material.

7. Optimization Results

The results proved that our approach could accurately control the depth of the material mathematically, optimizing it using the PSO technique. This equation also fits the first material, M1, perfectly with R² = 0.955 for the reference data, RMS = 0.8653, and adj = 0.8755. R² = 0.955 demonstrated high representation accuracy. By incorporating this equation into PSO, the best depth value was obtained; for example, M1 = 32.27 mm when T = 100, A = 180.59, and P = 300. These values reflect the combined effect of the three variables and demonstrate the interaction between them in obtaining the best possible result. Figure 7 demonstrates the relationship between the calculated and measured cut depth for M1.

For the second material, M2, the fitting equation was more complex, with a coefficient of determination of R² = 0.9456, with RMS = 0.8476 and adj = 0.8477, indicating some variability in the experimental data. However, the PSO model could identify a set of optimal values for the variables, resulting in a significant increase in the depth value, achieving a depth of M2 = 29.55 mm, the highest compared with the original measurements when T = 128.74, A = 250, and P = 300. Figure 8 demonstrates the relationship between the calculated and measured cut depth for M2.

For the third material, M3, a good-quality equation was also obtained (R² = 0.9695, with RMS = 0.5468 and adj = 0.9147); the algorithm achieved the highest recorded depth when PSO was applied to the equation. The results indicated the material’s sensitivity to these variables and demonstrated the model’s ability to handle the different physical properties of each material. M3 = 26.957 mm when T = 100, A = 220.015, and P = 200. Figure 9 demonstrates the relationship between the calculated and measured cut depth for M3.

In terms of programming, the flexible model showed good iterative performance, and it was possible to adapt it to any material with only marginal code modification. Each of the objective functions was validated separately, and we ensured the solutions were void of undesired local extremes, confirming PSO’s ability to effectively find optimal solutions.

The results show that the main goal of the study has been achieved: we have developed an efficient hybrid method suitable for industry, especially in regions needing multiple solutions based on multiple variables, considering the properties of granite. This approach is a pioneering contribution in a field where computational intelligence algorithms are used to integrate mathematical formulations with iterative optimizations to ensure accurate and reliable results.

8. Practical Implication

In technical applications, optimization results are most often expected, and this type of requirement is emphasized more than others. From the technical point of view, our findings can be applied in industries where AWJM is often used; for example, to create artistic shapes from a minimum of three types of granite with different thicknesses while minimizing the time needed to change the parameters for each type, as shown in Figure 10 [41].

Another potential practical implication is employing an inverse model to determine the AWJM settings based on the desired material type, predicting the cut depth without the need for experimental work.

9. Conclusion

We conducted an experimental study of the cut depth produced by abrasive water jet machining for three materials with different compressive strengths. We examined the influences of different machining parameters on extreme cut depths, and a clear correlation between the compressive strength of the granite and the cut depth was observed. Various models were developed to predict the penetration depth of the cut, and we made the following main conclusions:

• The highest depth of penetration relating to process parameters is achieved with granite with smaller grains and higher compressive strength.
• The PSO model identified the best set of optimal values for the three variables to achieve maximum depths for M1, M2, and M3: M1 = 32.27 mm when T = 100 mm/min, A = 180.59 g/min, and P = 300 MPa; M2 = 29.55 mm when T = 128.74 mm/min, A = 250 g/min, and P = 300 MPa; and M3 = 26.957 mm when T = 100 mm/min, A = 220.015 g/min, and P = 200 MPa.
• For all three materials, increased pressure increases the cut depth.
• High pressure helps to increase the traverse speed and reduce the cutting depth, which was achieved at higher traverse speeds for all types of granite. Accordingly, the cutting depth is greatly affected by the feed rate.
• This study had some limitations. We focused only on the impact of compressive strength on the cut depth of granite and a limited number of factors. Further research on various other key parameters is necessary; for example, additional research is needed to address the effects of process parameters on surface texture and kerf angles. Possible investigations might determine the effects of new abrasive materials on the cut depth of granite.