1 Introduction

In recent times, the hardened steels are experiencing great demand for the fabrication and manufacturing of automotive and aircraft parts along with the machine tool parts and assemblies because of their exceptional strength, high thermal stability, and wear resistance (Ref 1). When hardened, AISI 440C stainless steel is one of the most rigid steels available for various engineering uses due to its high-alloy composition and excellent wear and corrosion resistance. Applications such as valve components, roller and ball bearings and races, knives, cutlery, gauge blocks, dies and molds, surgical tools, pumps, turbines, wear-resistant textile components, and compressor components are commonplace in the oil and gas, marine, food, and medical industries (Ref 2,3,4). Its high hardness, strength, rapid work hardening, and chemical affinity with cutting tool materials make conventional machining methods challenging (Ref 5). Tool wear, surface roughness, increased cutting forces, and elevated temperatures are some issues that can arise during conventional machining (Ref 6). In addition, the high work hardening rate causes the tool to form a built-up edge at the cutting edge (Ref 7). Because ultra-hard tool materials like polycrystalline cubic boron nitride (PCBN) are needed to machine hardened martensitic stainless steel. Due to this the machining cost rises dramatically (Ref 8). Also, Grzesik (Ref 9) highlighted the significant wear of PCBN tools while machining the hard materials. Shukla (Ref 10) highlighted that the AWJM process is one of the viable options for cutting hard materials like high-strength steel titanium (Ti) alloys and ceramics. Therefore, abrasive water jet machining is a reasonable and economical option for machining hardened AISI 440c stainless steel.

Abrasive water jet machining is a cutting-edge machining method that can theoretically machine any number of materials, regardless of how amenable they are to conventional machining techniques (i.e., in the case of traditional machining). It is one of the bulk material removal process (Ref 11). It is a hybrid process that utilizes water and abrasive-jet machining techniques. Fiber-reinforced composites, ceramics, and ceramic composites are some of the materials that benefit from this machining method. It can be machined in various ways, generates a low machining force, offers excellent adaptability, requires little time to set up, and produces almost no thermal distortion (Ref 12). Because it can remove much material at a low cutting temperature, the AWJM process is also commonly used for machining complex (nickel-based) superalloys (Ref 13).

Figure 1 shows the general topography of the machined surface as produced by AWJM process. Three distinct areas on the AWJ machined surface are clearly seen. There are two primary causes of the initial localized damaged region at the top kerf wall cut surface, as shown in Fig. 1. The first involves an increase in the size of the abrasive water jet immediately prior to impingement, and the second involves a change in the jet's energy as a function of radius. Here, the impingement angle of the abrasive particles is much greater and more forceful than required for the remaining cutting depth. The cutting region, located in the middle, is second in terms of surface finish. The cutting region lies beneath the first area of damage and can be identified by its distinctive wave-like profiles. The abrasive particle size requirement delineates this space. Surface characteristics in the rough cutting region are controlled by cutting parameters that affect the jet's kinetic energy.

Fig. 1
figure 1

Machining features in AWJM machined surface (Ref 14). Reprinted from Wear, Vol 210, Material Removal in Abrasive Water jet Machining of Metals Surface Integrity and Texture, D. Arola and M. Ramulu, Pages 50–58, Copyright 1997, with permission from Elsevier

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2 Literature Review

In the AWJM process, material removal produces the topography, as was previously mentioned. Erosion is the primary mechanism for the removal of material. Erosion in AWJM occurs due to the persistent impact of abrasive particles on a small, targeted area subjected to a liquid medium moving at a very high speed. The primary mechanism by which material is removed during the AWJM process is the force created by the abrasive particles striking the work material at high velocity, resulting in erosion of the material in that area. Erosion is the type of wear that typically occurs in machining due to the continuously targeted impingement of an abrasive water jet at high pressure and velocity over work material. In order to remove material during AWJ machining, the impact of the abrasive particles is crucial. From a more macro perspective, abrasive particles abrade the surface of the workpiece by a combination of cutting, fatigue, melting, and brittle fracture (Ref 15). Two main types of micro-cutting, namely cutting and deformation/plowing deformation erosive wear mechanism (Ref 16), account for most of the material removal in AWJM process. Figure 2 shows that sharp-edged, angular particles undergo cutting deformation, while spherical abrasive particles undergo significant plowing deformation (Ref 14, 17). Combining cutting and deformation wear is the primary mechanism behind ductile erosion. When abrasive particles impinge on the target work material at a shallow angle of attack, a smooth cutting zone is created (Ref 18). High angles of attack cause deformation wear, which results in a relatively rough surface marked by striations (Ref 19, 20). By contrast, brittle erosion is characterized by the removal of material via crack propagation and chipping due to contact stresses brought on by the impact of abrasive particles (Ref 21). The mechanism of material removal in the target materials is affected by the impact angle of the abrasive water jet (Ref 22). The highly effective ductile shearing action in the target materials was influenced by the oblique jet impact angle, which limits the fracture traces and leads to the incomplete removal of chips on the machined surface (Ref 23, 24).

Fig. 2
figure 2

Erosion process of ductile materials by abrasive particles (Ref 25). Reprinted from Wear, Vol 265, Finite Element Model of Erosive Wear on Ductile and Brittle Materials, Y.F. Wang and Z.G. Yang, Pages 871–878, Copyright 2008, with permission from Elsevier

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Several studies have examined how varying AWJM process parameters affect surface roughness, kerf quality, and metal removal rate in metals and alloys (MRR). One group of researchers, Babu and Chetty (Ref 26), looked at how using garnet abrasives with a single mesh size affected kerf geometry and surface roughness in AWJM of aluminum. In terms of top and bottom width and kerf taper, it was determined that single mesh size abrasives are distinct from multi-mesh size abrasives. Hascalik et al. (Ref 27) examined how increasing the traverse speed in AWJM of Ti-6Al-4V alloy affected the kerf taper ratio and surface roughness. It has been reported that the nozzle traverse rate plays a major role in determining the surface morphology and the kerf widths. It was also found that as the traverse speed was increased, the kerf taper ratio and surface roughness followed increasing pattern. To determine the effect of AWJM process parameters on the mean kerf width and surface roughness of TRIP 700 CR-FH and TRIP 800 HR-FH steel sheets, Kechagias et al. (Ref 28) conducted an experimental investigation. Process parameters affecting surface roughness in abrasive water jet machining of aluminum were investigated by Selvan et al. (Ref 29). The study factored in the traverse speed of the nozzle, the jet-water pressure, the stand-off height, and the abrasive mass flow rate. According to the authors, water pressure significantly affects surface roughness. Surface roughness decreases, nozzle traverse speed increases, and abrasive particle mass flow rate rises in response to water pressure increases. To get a good surface, the authors suggested using a slow traverse speed, which increases machining time but decreases productivity. Jain et al. (Ref 30) investigated the work material's kerf properties using a slurry made of abrasives and water. It was also looked into that how the temperature of the slurry affected the kerf taper angle of mild steel A36 that was machined using the AWJM method. An investigation revealed that the temperature of the slurry, which in turn is a function of its viscosity, played a significant role in determining the kerf taper angle. For this reason, the kerf taper narrows when the slurry temperature drops. Researchers Babu and Muthukrishnan (Ref 31) investigated the impact of AWJM parameters on the surface roughness of brass-360 and found that the parameters could be optimized to reduce the roughness. Overall, the roughness of a surface is found to be most affected by the pump pressure, with the abrasive flow rate coming in second. Trivedi et al. (Ref 32) studied the effect of water pressure, nozzle traverse rate, and stand-off height on the roughness of austenite steel (AISI 316L) surfaces. The results indicate that higher water pressure is responsible for enhanced surface quality. The surface roughness (Ra) varies directly as a function of the nozzle traverse rate, making it the most important parameter. When cutting EN AW-6060 aluminum alloy sheets with an abrasive water jet machining process, Jankovic et al. (Ref 33) studied the impact of process parameters on cut quality characteristics like perpendicularity of cut and surface roughness. They discovered that an important parameter of the surface morphology is the jet's feed rate (nozzle traverse speed). The influence of cutting speed on surface roughness in AWJM of 316L stainless steel was studied by Löschner et al. (Ref 34). As cutting or traverse speed decreases, surface roughness improves, according to the authors. As cutting speed is slowed, machining marks spread over a smaller area. In their investigation into the relationship between abrasive water jet machining parameters and the quality of mild steel cuts, Dumbhare et al. (Ref 35) found that traverse speed is the most important factor in determining surface roughness and kerf taper angle, followed by stand-off distance and abrasive flow rate. Researchers found that the stand-off distance or stand-off height and abrasive flow rate were the most influential factors in determining the surface roughness and kerf taper angle, with traverse speed coming in a close second. Kulisz et al. (Ref 36) studied the influence of process parameters on the surface roughness of Si-enriched Al alloys, specifically AlSi10Mg and AlSi21CuNi. They found that the surface roughness worsened with increasing jet feed rate but showed little correlation to the abrasive flow rate. Liao et al. (Ref 13) proposed a dual-process AWJM process in which two abrasive water jet functions were used. Subsequent processes aim to modify the surface of the workpiece, enhancing its structural stability, while the primary process involves material removal. Kerf taper properties of aluminum alloy 6061-T6 were investigated by Wang et al. (Ref 37), who looked at how factors like cutting speed, material thickness, water pressure, and abrasive flow rate affected the cutting edge. It can be concluded that the kerf taper has a major impact on the processing accuracy of the AWJM process; however, the kerf taper can be reduced or eliminated by slowing the cutting speed, though this is less efficient and not applicable when the material thickness is great. The manufactured Al/Si7 composite was machined by Garikipati et al. (Ref 38) using the AWJM technique. Material removal rate and surface roughness (Ra) are investigated as a function of jet pressure (JP), stand of distance (SOD), and traverse rate (TS). It is discovered that SOD and TS have a major impact on MRR, while JP and SOD have a major impact on Ra. The surface roughness of titanium alloy was predicted using an ANN-based model developed by Ting et al. (Ref 39). They found that the ANN-based predictive model outperformed the regression model in predicting titanium alloy surface roughness.

In order to enhance the cut quality characteristics like surface roughness, kerf properties, depth of cut, many researchers have looked into the influence of critical process parameters of the AWJM process for different metals and alloys. Unfortunately, there is a lack of data regarding hardened AISI 440C stainless steel. It could be because annealed standard grade 440c stainless steel is very amenable to the typical machining process. However, heat-treated AISI 440C stainless steel results in tough alloy steel. As highlighted by various researchers in their research work, machining it using conventional methods proves to be very difficult. The machined surface quality is often degraded, or the machining cost is increased if conventional machining techniques are used, such as rapid tool wear, high surface roughness, larger cutting forces, and high temperature.

In light of this, it is essential to study the results of unconventional machining on hardened AISI 440C stainless steel. As the conventional machining methods have limitations while machining this material, the novelty of the present work can be realized in the term that there is a lack of any information in the literature regarding the applicability and suitability of advanced machining methods like AWJM to the hardened AISI 440C stainless steel. The literature also needs more information on the machining and applicability of the AWJM process on high-thickness hard material to cut material. Surface roughness and kerf taper on the material are analyzed to determine how effective process parameters are on them.

3 Material and Methods

With the AWJM method, the work material can be of any size or shape. In the present investigation, AISI 440C stainless steel, 16 mm in thickness, and three process parameters (jet-water pressure, nozzle traverse rate, and stand-off height) were chosen for the experimental work. The material undergoes a hardening and tempering process at high temperatures. The material is machined using a computer-controlled flying-arm abrasive water jet (AWJ) machine (Model: Jet Cut 30B of M/s An Innovative International Ltd., Ahmadabad). The experiments use garnet abrasives with a constant abrasive mass flow rate (380 g/min) and a mesh size of #80 throughout. Figure 3 shows some general shape of garnet abrasive used in machining.

Fig. 3
figure 3

Some general shapes of the garnet abrasives (Ref 24)

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The diameter of the nozzle is 0.30 mm, and the impingement angle is 90 degrees. In Fig. 4, the AWJM process being used to machine hardened AISI 440c stainless steel. Analysis of variance is used for data interpretation after experiments are conducted (ANOVA). Using regression analysis, a model is built to predict surface roughness and kerf taper concerning critical parameters. In addition, the process parameters are fine-tuned to ensure the smoothest possible surface and the smallest possible kerf taper.

Fig. 4
figure 4

Machining of hardened AISI 440C stainless steel using AWJM

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In this investigation, sample of AISI 440C stainless steel that measures 210 mm by 54 mm by 16 mm was used for experiments. Tables 1 and 2 detail the chemistry of AISI 440C stainless steel and the annealed mechanical properties of this material.

Table 1 Mechanical properties of AISI 440C stainless steel in annealed condition
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Table 2 Chemical composition of AISI 440C stainless steel
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Initially, AISI 440C stainless steel material was in the annealed condition with a hardness of 22 HRC. The heat treatment process is carried out to obtain the highest hardness, resulting in wear resistance and dimensional stability. After heat treatment, it can sustain the hardness, toughness, strength, and resistance ability (i.e., corrosion, wear) for a long time. Hardening is done on the AISI 440C stainless steel to improve its mechanical properties. The material is heated up to 960 degrees Celsius and held at that temperature for 30 min to get a uniform temperature throughout the section. After that, the material is removed from the furnace and quenched in warm oil. Generally, oil quenching is carried out for thicker sections.

The tempering process is carried out immediately after the hardening process to obtain optimum corrosion resistance and maximum hardness. Grade AISI 440C stainless steel is heated to 150 degrees Celsius and held at that temperature for 1.5 h. After that, it cooled in the air. Tempering above 400 degrees Celsius should be avoided due to the material's reduction in corrosion and impact resistance properties. The tempering of this material in the range of 590 degrees Celsius to 675 degrees Celsius results in lower hardness and high impact resistance. The heating, cooling, temperature, and soaking time for the materials vary because of factors like workpiece size, quenching medium, workpiece transfer facilities, and furnace type employed. Testing revealed a hardness of 58 HRC in the material after entire heat treatment cycle.

Various process parameters impact AWJM's cutting performance. Jet-water pressure (JWP), nozzle traverse rate (NTR), and stand-off height (SOH) are the three primary process parameters considered across the five levels in this investigation. The values for these variables are determined by critically analyzing the available literature, the machine's setup range, and the results of trial experiments.

The pressure of the water used in a water jet is a crucial factor in the accuracy and efficiency of AWJ machining. The water's pressure affects the AWJ's kinetic energy. When no material is removed, the pressure is below the threshold range. Similarly, the adequate cutting pressure is equal to the critical pressure range. If the machining process continues past this point, it is useless. Both the depth of penetration and the rate of material removal when using a water jet are directly related to the pressure of the jet. It affects how the water and particles in the abrasive water jet are distributed. MPa, bar, or PSI are the standard units of measurement. Table 3 displays the five distinct pressure ranges that constitute the present study, which extends from 320 to 400 MPa.

Table 3 Process parameters and levels
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The quality of the AWJ process is cut, and the nozzle traverse rate determines surfaces. Determining the exposure time has a significant impact on the nozzle traverse rate during AWJ machining. Because more abrasive particles can impact the surface of the target material at a slower nozzle traverse rate, the surface quality is improved. That factor also affects the process is cutting speed. mm/min represents the velocity of the nozzle traverse. Table 3 displays the results of this study's five levels, where the sampling rates ranged from 20 mm/min to 60 mm/min.

Stand-off Height is the physical separation between the source and the nozzle. Considering the heavy impact of Stand-off height (SOH) on the kerf profile, it is typically kept at an optimal distance. Table 3 displays the results of the current investigation at five distinct levels ranging in size from 1 to 5 mm.

The abrasive mass flow rate is the total mass of the abrasive particles that pass-through a given area in a given time. While working with hardened AISI 440c, a strange phenomenon was discovered during machining. When the water jet strikes the surface of the work material, the abrasive particle in the water jet slams into it at a very high speed. This collision of abrasives with the hardened surface of work material, causes an instantaneous spark. An electron striking the surface and producing a spark is visually reminiscent of electro-discharge machining. When multiple abrasives strike the workpiece's surface at once, a shower of sparks erupts. To the untrained eye, it could be mistaken for a well-established flame, as if the material is being cut by the gas-cutting process under water.

Additionally, the provision of varying abrasive mass flow rates may not be available in most abrasive water jet machines. As a result, in this study, abrasive mass flow rate is kept constant to its steady-state rate of 380 g/min. In AWJM, machining performance is represented by response parameters like kerf property and surface roughness. In this study, both the average surface roughness (Ra) and the kerf taper are taken.

The experiments are created using the central composite design (CCD) response surface methodology (RSM) after finalizing the process parameters. The CCD is a workable design method used in continuous experimentation because it requires fewer design points but still provides enough data to test for lack of fit. Based on the CCD, the design expert software proposes 20 experiments for the current investigation: 6 centered runs, 6 axial runs, and 8 factorial runs. Figure 5 is a picture of the machined components. CCD results in slightly more no. of experiments as compared to the box Behnken design (BBD) of the experiment as used by Sisodia and Kumar (Ref 40) and Sisodia and Kumar (Ref 41). It requires 46 experiments for five factors and three levels. However, if compared with a complete factorial design for five factors and five levels, then, by definition of full factorial design, the number of experiments obtained is 125. Although every possible combination of process parameters is taken in the complete factor design of experiments, performing 125 experiments is quite tedious. Also, the amount of material required and experimental information processing is more in the case of 55 full factorial designs, which will significantly contribute to the research cost. However, sufficient and relevant information could be obtained from the CCD. Therefore, for the present work, CCD is selected for the present research. After machining, a surface roughness tester is used after machining to evaluate the typical Ra value (Model—Mitutoyo Surftest SJ-310). This measuring instrument defines the sampling length as the filter's cut-off length (λc) to separate roughness and waviness. For example, using a cut-off length of 0.8 mm and 5 sampling lengths, parameters will be estimated on each segment (Ra1, Ra2,., Ra5), and the parameter value will be given as the mean of these estimated values. The Ra is the average peak-to-valley height of the machined surface.

Fig. 5
figure 5

Machined samples of hardened AISI 440C stainless steel for surface roughness measurement

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The surface characteristics observed at a glance are depicted in Fig. 6(a) while three distinct areas where the surface roughness is evaluated: the top, the middle, and the bottom are depicted in Fig. 6(b). Since the surface roughness varies in different areas, the average roughness across all three is considered. Table 4 displays the experimentally determined surface roughness values.

Fig. 6
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(a) Surface profile observed of entire sample at a glance. (b) Surface roughness measurement over three specified regions

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Table 4 Measured experimental values of surface roughness
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Figure 7 shows a thoroughly examined sample surfaces under an optical microscope. An abrasive water jet at high pressure and speed erodes the work material. The abrasive has left visible marks on the material being worked on. The following slash is clean through cut with uniform grooves. Jet deflection is usually to blame for this wavering. The rapid traversal speed causes the rapid acceleration into the jet. For cost-effective machining, it is necessary to maintain a high traverse speed.

Fig. 7
figure 7

Surface characteristics and features of hardened AISI 440c post machining

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Figure 8 shows the geometry of the kerf. The present investigation utilises a vision measurement system to assess the machined sample's top and bottom kerf width (Fig. 9). (Model—Sipcon SDM-TRZ 5300). To prevent measurement errors caused by the ballooning effect, we take the top kerf width and bottom width measurements at a constant height of 2 mm below the top surface and 2 mm above the bottom surface of the machined sample. For this reason, Eq. 1 is used to determine the kerf taper angle using a thickness of 12 mm. Table 5 displays the measurable top and bottom kerf widths:

$$\theta = \tan^{ - 1} \frac{{W_{{\text{t}}} - W_{{\text{d}}} }}{2t}$$
(1)

where θ Kerf taper angle (degree), \(W_{{\text{t}}}\) Kerf top width (mm), \(W_{{\text{b}}}\) Kerf bottom width (mm), h height or thickness (mm).

Fig. 8
figure 8

Representation of Kerf Taper Profile in AWJM process

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Fig. 9
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Machined Sample for kerf taper measurement

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Table 5 Measured values of kerf top width and bottom width
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4 Results and Discussion

The central composite design dictates the order in which the experiments should be carried out. After that, an analysis of variance (ANOVA) is performed to identify significant factors that affect surface roughness and kerf taper. Also, it is the most widely used and very quickly applicable technique in multiple regressions for analyzing the validity of the developed model (Ref 42). Table 6 displays the results of an ANOVA performed on the Ra value. Analysis of variance (ANOVA) is used to assess the statistical significance and impact of process parameters on surface roughness. The analysis is performed with a 95% degree of certainty (alpha value = 0.05). Those variables with a p value below the threshold for statistical significance are accepted as relevant. Surface roughness’s ANOVA data are shown in Table 6. NTR and JWP are the two most important parameters when evaluating surface roughness, with stand-off height coming in a distant third.

Table 6 ANOVA for Avg. surface roughness (Ra)
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4.1 Critical Analysis and Discussion of Obtained Results

Surface roughness is briefly affected by process parameters like jet-water pressure, nozzle traverse rate, and stand-off height. Figure 10 illustrate the impact of process parameters on surface roughness, but they are not presented in a quantitative format. Figure 10(a) shows that as the nozzle-traversal rate increases, so does the surface roughness. The Physics behind this is that when the traverse rate is high, there is less time for the jet to interact with any given material area and, thus, less time for the material to be eroded by the abrasive particles. Greater surface roughness is produced when the nozzle traverse rate is high, and the machining action does not overlap much. Thereafter, increases in jet-water pressure are mirrored in a corresponding decrease in surface roughness. The jet's kinetic energy increases with the pressure of the water, and a jet with high kinetic energy cuts materials with a smooth, even edge. Surface roughness is reduced because abrasive particles are broken down into smaller pieces when subjected to high water pressure.

Fig. 10
figure 10

Effect of Jet-water pressure, nozzle traverse rate, and stand-off height on surface roughness (Ra)

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Figure 10(b) demonstrates that as the stand-off height increases, the surface roughness also rises. The physics behind this that in order to overcome the resistance, it encounters from the air, the abrasive water jet (AWJ) expands as it leaves the nozzle. This means that the kinetic energy of the AWJ decreases with increasing stand-off height because the jet flares out further. As a result, the jet's cutting ability is diminished, leading to an uneven finish. The density of AWJ decreases as the jet flares further. The surface of the cutting material is more likely to have random peaks and valleys created by the jet because of its low density.

The primary goal of this study is to determine the range of cutting conditions for abrasive water jet machining of hardened 440c stainless steel that will yield the required surface roughness. Figure 11 shows contour plots of average surface roughness for jet-water pressure, nozzle traverse rate, and stand-off height. The linear profile in the contour plot is consistent with the fitted linear model. Surface finishes on the order of 2 microns are demonstrated in Fig. 11.

Fig. 11
figure 11

Contour plot for Jet-water pressure, nozzle traverse rate, and stand-off height for avg. surface roughness (Ra)

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Table 7 displays the results of an ANOVA on the kerf taper, and Table 6 displays the analysis results at the 95% confidence level. Parameters with p values less than 0.05 in Table 7 include jet-water pressure, nozzle traverse rate, and stand-off height. JWP is the most critical parameter for the kerf taper, and NTR is the least important.

Table 7 ANOVA for kerf taper
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Figure 12 illustrates the impact of jet-water pressure, nozzle traverse rate, and stand-off height on the kerf taper. Figure 12(a) shows that as the jet-water pressure increases, the kerf taper decreases because the increased kinetic energy of the water jet causes the jet to deflect less, resulting in a cleaner cut. Increased kerf width uniformity and reduced kerf taper result from high jet-water pressure, which also increases the effective diameter of the jet at its lower part. It can also be seen that the kerf taper increases with the nozzle traverse rate. This is because a smaller fraction of the jet's energy is expended in any given area of the workpiece when the nozzle traverse rate is high. There will be less machining action overlap, and fewer abrasive particles impinging on the surface of the nozzle traverse rate are increased. Kerf taper increases because the jet cannot cut the material in the bottom region.

Fig. 12
figure 12

Effect of jet-water pressure, nozzle traverse rate and stand-off height on kerf taper

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Because the flaring out of the jet reduces the cutting ability, Fig. 12(b) shows that the kerf taper increases with an increase in stand-off height. As the jet expands, the effective diameter of the jet decreases, and with it, the density of abrasive particles on the jet's outer surface. Since the outer layer of the diverged jet cannot cut the lower part of the kerf, the effective diameter of the jet decreases, resulting in a narrower bottom width of the cut. As an abrasive density, distance, and drag are all reduced, the jet's kinetic energy decreases and the kerf tapers increases.

The primary goal of this study is to determine the range of cutting conditions for the abrasive water jet machining process of hardened 440c stainless steel that will yield the desired kerf tape angle. Figure 13 shows the kerf taper in jet-water pressure versus nozzle traverse rate and stand-off height, respectively, as contour plots. When a linear model is used to fit the kerf taper angle, the resulting linear profile of the contour plot is consistent with that model. The values of the process parameters shown in Fig. 13 allow the machined part to achieve specified minimum kerf taper for hardened AISI 440c.

Fig. 13
figure 13

Contour plot for Nozzle traverse rate, Jet-water pressure, and stand-off height for Kerf Taper angle

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5 Regression Equation Modeling

To estimate how different settings of process parameters affect the response variable, a regression analysis is conducted on the sample data. Surface roughness prediction equations (Eq 2 and 3) are developed using significant process parameters obtained from ANOVA.

$$\begin{aligned} {\text{Avg}}.\,{\text{Surface}}\;{\text{Roughness}}\;\left( {R_{{\text{a}}} } \right) = & 18.18569 \\ &-\left( {0.053978 \times {\text{Jet}}-{\text{water}}\;{\text{Pressure}}} \right) \\ & + (0.11554 \times {\text{Nozzle}}\;{\text{Traverse}}\;{\text{Rate}}) \\ & + (0.63206 \times {\text{Stand}}-{\text{off}}\;{\text{height}}) \\ \end{aligned}$$
(2)
$${\text{Avg}}.\,{\text{Surface}}\;{\text{Roughness}}\;\left( {R_{{\text{a}}} } \right) = 5.27 - 1.08 \times A + 1.16 \times B + 0.63 \times C$$
(3)

In this case, the adjusted R-squared and predicted R-squared are 0.9432 and 0.9244, respectively. The “predicted R-squared” value of 0.9244 indicates the good agreement with the "adjusted R-squared" value of 0.9432.

Ra’s predicted and measured values agree well (Fig. 14). The fitted trendline has an R2 of 0.9521, suggesting this is the case. The percentage of error is also acceptable. The designed space can be predicted in terms of Ra using the developed model. To predict kerf characteristics, a regression model for kerf taper is developed using design expert software based on critical process parameters. Equation 4 and 5 provide the regression equations for kerf taper in terms of the critical and correlated factors, respectively.

$${\text{Kerf}}\;{\text{Taper}} = \left( {6.30551} \right){-}\left( {0.015066 \times {\text{Jet-water}}\;{\text{Pressure}}} \right) + \left( {0.00889375 \times {\text{Nozzle}}\;{\text{Traverse}}\;{\text{Rate}}} \right) + (0.24994 \times {\text{Stand-off}}\;{\text{Height}})$$
(4)
$${\text{Kerf}}\;{\text{Taper}} = \left( {1.99} \right){-}\left( {0.30 \times A} \right) + \left( {0.089 \times B} \right) + \left( {0.25 \times C} \right)$$
(5)
Fig. 14
figure 14

Scatter plot between predicted and measured avg. surface roughness (Ra)

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The 0.9063 adjusted R2 value is consistent with the 0.8844 predicted value. Observed and predicted kerf taper values are highly correlated, as shown in Fig. 15. That is what the fitted trendline's R2 of 0.9211 suggests. The developed mathematical model for kerf taper can be used to predict kerf characteristics within the designed space as the percentage error is within the allowable limit.

Fig. 15
figure 15

Scatter plot between predicted and measured kerf taper angle

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The coded equation helps identify the relative impact of the factors by comparing the factor coefficients. At the same time, the equation in terms of fundamental factors can be used to make predictions about the response for given levels of each factor. Here, the levels should be specified in the original units for each factor. This equation should not be used to determine the relative impact of each factor because the coefficients are scaled to accommodate the units of each factor, and the intercept is not at the center of the design space.

Hardened AISI 440c steel exhibits its best corrosion resistance in its hardened and tempered form. The structure comprises dendrites containing austenite, surrounded by a fine lamellar eutectic of austenite. The presence of austenite is further confirmed with the help of Fig. 18. Further striation marks made by the abrasive water jet during machining are visible on the machined sample. The mechanism of striation formation as demonstrated by Orbanic et, al. (Ref 43) is the linear difference between the centers of the non-moving and moving jet cross sections with the material surface at the jet outlet from the material is known as the trail-back “σ.” The trail-back “σ” causes the difference between the radii of the curves on the inlet surface and on the outlet surface of the curved trajectories cut by the jet when all material and jet parameters are constant (Ref 44). The same phenomenon is also observed in Fig. 16.

Fig. 16
figure 16

SEM image of machined sample (hardened 440c stainless steel)

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Figure 17 shows the SEM image of the machined sample. The striation mark created by jet entry and exit can be observed along grain boundaries. The structure comprises dendrites, containing austenite surrounded by a fine lamellar eutectic of austenite, is visible in the SEM image. Figure 18 depicts the SEM image of the presence of austenite in the hardened 440c stainless steel. It adds more clarity to the austinite presence in the structure.

Fig. 17
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SEM image of machined sample (Hardened AISI 440c stainless steel)

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Fig. 18
figure 18

SEM Image depicting austenite presence in hardened 440c stainless steel

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6 Optimization of Process Parameters

Surface roughness and kerf taper were reduced by optimizing process parameters using the desirability approach in the present study. Individual parameters and responses have their own goals and bounds. Each optimizable parameter and response have an upper and lower bound given. The parameter and response optimal values are listed in Table 8.

Table 8 Criteria for optimization
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The dimensionless measure of response, the composite desirability function based on the desirability analysis procedure, is adopted for multiple responses. Each response is transformed into a desirability function (df). This method has merits like simplicity, weight flexibility, ease of use, and focuses on individual response. The desirability scale ranges from df = 0 to 1, where df = 0 is not acceptable, whereas df = 1 is a highly desired response. The desirability of each response is determined using Eq 6 (Ref 45):

$$d_{{\text{f}}} = \left[ {\frac{{H_{i} - Y_{i} }}{{H_{i} - L_{i} }}} \right]^{{w_{i} }}$$
(6)

where Hi upper limit of the response, Li lower limit of the response, Yi respective value of the response, and wi weight of the response.

The desirability function is varied according to response using the weight field. Generally, the weights are altered within the range of 0.1 to 20. Weights less than 1 are less critical, and weights > 1 provide much importance to the desired goal. The df varies weight = 1 linearly. The optimum values of stand-off height, Jet-water pressure, and nozzle traverse rate are obtained based on statistical analysis in Design Expert software. The optimized setting of stand-off height, Jet-water pressure and nozzle traverse results in a combination of levels that fulfill the desired optimization conditions for the response. Figure 19 shows a bar chart displaying the optimal values for surface roughness and kerf taper process inputs. It is highly desirable and acceptable that the value for all factors and surface roughness is 1 and that the value for kerf taper is 0.905888, which is very close to 1. All factors and responses have a combined desirability value of 0.957022, which is very close to 1 and is satisfactory.

Fig. 19
figure 19

Bar graph of desirability

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Variations in jet-water pressure and nozzle traverse rate that are considered desirable are demonstrated in Fig. 20. Total desirability equals 0.95702. This desirability value is acceptable because it is close to 1. Figure 20 also displays the predicted values of surface roughness (2.404 m) and kerf taper angle (1.34 degrees) under various conditions of water pressure and nozzle traverse rate. Examining the surface roughness and kerf taper behavior close to an optimal condition are beneficial.

Fig. 20
figure 20

Combined desirability and predicted of avg. surface roughness and kerf taper angle at optimum values

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7 Conclusions

The abrasive water jet machining (AWJM) process is one of the fastest-growing cutting processes in manufacturing. It has replaced the traditional means of machining thick, hard, and difficult-to-cut materials. In the present study, the applicability of AWJM process has tested for thick and hard material. The experimental investigation is carried out to study the influence of process parameters of AWJM on surface roughness and kerf taper of hardened AISI 440C stainless steel. The experimental investigation has been carried out to study the influence of process parameters (i.e., jet-water pressure, nozzle traverse rate, and stand-off height) on surface roughness and kerf taper angle in AWJM of hardened AISI 440C stainless steel. Experimental design is based on the central composite design of response surface methodology. The average surface roughness and kerf geometry are measured and analyzed. Further, ANOVA is performed to determine the significant process parameters on surface roughness and kerf taper of machined hardened AISI 440C stainless steel. Based on experimental results and study nozzle traverse rate is the most significant parameter followed by jet-water pressure and stand-off height, the following conclusions are drawn for surface roughness:

Surface roughness.

  1. 1.

    decreases with an increase in jet-water pressure,

  2. 2.

    decreases with a decrease in stand-off height,

  3. 3.

    decreases with a decrease in nozzle traverse rate.

Further, Jet-water pressure is the most significant parameter followed by stand-off height and nozzle traverse rate. Following conclusions are drawn for kerf taper:

Kerf taper decreases.

  1. 1.

    with an increase in jet-water pressure,

  2. 2.

    with a decrease in nozzle traverse rate,

  3. 3.

    with a decrease in stand-off height.

For each response feature, a mathematical model has been constructed to predict the values of critical parameters (i.e., surface roughness and kerf taper). Model predictions are found to be consistent with experimental findings. Finally, kerf taper and surface roughness are minimized by adjusting AWJM's process parameters. The reduced surface roughness of the order of 2.404 µm and kerf taper of the order of 1.352° is achieved through careful optimization of the process parameters. The optimum conditions of process parameters are.

  1. 1.

    Jet-water pressure—380 MPa

  2. 2.

    Nozzle Traverse rate—30 mm/min

  3. 3.

    Stand-off Height—2 mm

8 Scope of Future Work

Even though this work examines how AWJM process parameters affect the average surface roughness and kerf taper of hardened AISI 440C stainless steel, it could be further extended and improved by including the following:

  1. 1.

    One way to expand on the present experimental work is to vary the input process parameters (such as the abrasive mass flow rate) and observe how that impacts the output parameters.

  2. 2.

    The other response characteristics of AWJM of hardened AISI 440C stainless steel should be investigated, including MRR and the possible depth of cut.

  3. 3.

    Consider various values and levels of process parameters to reduce the striation effect and the kerf taper.