**1. Introduction**

Cutting of materials by abrasive water jets has been studied for several decades. The pioneer scientists dealing with this topic were Hashish [1,2] and Zeng and Kim [3,4]. Later, some further investigations occurred aimed at the machining process, e.g., by Kovacevic and Yong [5,6]. The current state of research of abrasive water jet technology shows that one of the important problems is the quantification and modeling of the influence of technological parameters on surface quality parameters, particularly on wear-resistant steels. Evaluation of cutting quantity and quality was continuously studied by various groups [7–10]. Sutowska et al. [11] studied the influence of cutting parameters on the kerf quality in detail. Some of the recent experiments were performed on HardoxTM 400, 450, and 500 steel plates by Filip, Vasiloni, and Mihail [12,13].

Evaluation of the cutting quality is related to the quality of the cut walls. The typical characteristics of the walls are roughness and waviness. The most common characteristics used for the evaluation of the surface roughness were measured and analyzed. These characteristics are *Ra*, the mean arithmetic deviation of the profile, and *Rz*, the height of the profile unevenness. These two quantities can be measured by contact profilometers or by non-contact profilometers [14–16]. Nevertheless, the values depend not only on cut material or depth in the kerf but also on abrasive material quality and grain size [17]. Former research works aimed at the problem of abrasive material changes in the mixing process show, however, that the problem is not easy to solve [18,19] because not only the average mean size *ao* plays the decisive role in changes to new one *an* but also the amount and type of the original damage of abrasive grains. The influence of the abrasive material and its granularity is constant for one selected material sort. This is the most common case in all commercial firms. Therefore, the influence of abrasive material can be considered as disturbance quantity identical in all experiments. The surface waviness has much higher values than roughness, generally in the order of millimeters. The quality of this part of the cut walls is incredibly low, therefore, beyond the interest of this paper.

The Hlavác group has presented another approach to the determination of the cutting wall quality ˇ than the use of the *Ra* and *Rz* values, proposing a direct relationship between the declination angle (measured between the tangent to the striation curve in the definite depth h and the impinging jet axis) and respective cutting wall quality. The angle is calculated either for a certain depth in material or some assigned traverse speed from the presented model [17]. Nevertheless, angle values are incredibly low in quality cutting, and thus even relatively small imperfections in measurements bring quite large uncertainty in quality results. Therefore, this method is better for evaluating the part of the cut walls with predominant waviness.

Although there is a constantly growing set of developed solutions to the problem, including methodologies and evaluations of experiments valid for specific measurement conditions, the current solutions still do not cover several variations. The microscopic models describing the mechanism of material cutting were prepared [18] as well as the macroscopic model of cutting front behavior [19,20]. An interesting multi-parametric phenomenological description of the cutting process has also been presented by the Ostrava group [21–23]. The group of TU Kosice researchers entered this research area as a part of systematic studies of the operational states of manufacturing processes using progressive technologies [14,15,24] and influence of the process parameters on the surface quality [25,26] a few years ago.

The recent research is focused on complementing existing models and preparing some new ones that would be simple enough to be applicable in industrial conditions to help predict and control the production quality. The most important results are presented in this paper. They can be used for the preparation of the regression models describing surface quality relationship to the cutting factors, water pressure, traverse speed, and abrasive mass flow rate.

#### **2. Experimental Section**

#### *2.1. Characteristics of the Samples*

All samples were cut from HardoxTM 500 abrasion resistant plates with a nominal hardness of 500 HBW developed for applications with high demands on abrasion resistance. Material properties were obtained by a combination of quenching and tempering performed by manufacturer SSAB Oxelösund AB, Sweden. Sheet thicknesses of 6, 10, 15, and 40 mm were used for the individual sets of experiments with the following characteristics [27]:

*Hardness* (Brinell hardness, HBW according to EN ISO 6506-1, on a milled surface 0.5–2 mm below plate surface per heat and 40 tons): 486 (6 mm)–497 (40 mm).

*Impact Properties* (longitudinal Charpy-V; typical impact energy for 20 mm plate thickness at temperature −40 ◦C): 30 J.

The chemical composition of the material is presented in Table 1.


**Table 1.** Chemical composition of the HardoxTM 500 plate samples [27].

To study the dependencies of parameters, a 3-level full 3-factor experiment was designed with a total number of combinations of technological parameter values of 27 (Table 2). These combinations were applied to 4 di fferent sample thicknesses (6, 10, 15, and 40 mm). It follows that 9 samples with 3 cut surfaces were cut from each sheet thickness. Therefore, the shape with the plan view of an equilateral triangle was chosen as the most suitable sample shape. Transverse speeds *v* was used for sample thicknesses of 10 and 15 mm; *v*<sup>+</sup> are increased speeds for 6 mm samples because the speeds *v* for 6 mm sheet metal would leave minimal roughness and at the same time almost identically rough-cut surfaces. Traverse speeds *v*<sup>−</sup> were used for 40 mm sheet metal, as *v* would not be enough to cut the plate, so decreased speeds were chosen.


**Table 2.** Combinations of technological parameter values for sets of experiments.

#### *2.2. Characteristics of the AWJ System and Procedure*

The experiments were performed on the AWJ system comprising technological (cutting) head PaserIIITM, X-Y table WJ1020-1Z-EKO with X-Y Computer Numerical Controlled (CNC) system and pump Flow HSQ 5X (see Figure 1) with combination of the following parameters:


**Figure 1.** Experimental AWJ system (on the **left**) and pump Flow HSQ 5X (on the **right**).

These combinations represent 135 single cuts. Therefore, 45 triangle-shaped samples were cut, each side being cut with a different traverse speed *v* (Figure 2). All samples' cut surfaces were chemically treated with a passivation bath immediately after the end of the experiments—a solution of 5 g of sodium nitrite per 1 liter of water. The samples were immersed for 2–3 s in a solution at a temperature of about 60 ± 5 ◦C. Immediately after application of the solution, drying with hot air and storage in a dry environment followed. Surfaces treated in this way will resist corrosion for sufficient time to perform measurements.

**Figure 2.** The cutting program's screen and detail of the HardoxTM steel plates cutting (on the top); below samples prepared from (thicknesses from left to right 6, 6, 10, 15 and 40 mm).

The samples with thickness 6 mm were cut two times, ones with the same traverse speed as thicknesses 10 and 15 mm, ones with higher traverse speeds. Thickness 40 mm was not possible to cut using the same traverse speeds as other thicknesses. Therefore, the lower ones were utilized.

#### *2.3. Roughness Measurement of Cut Surfaces*

The roughness parameters *Ra* and *Rz* were measured in the middle height of the sample, i.e., at half the cut material's thickness. The roughness parameters *Ra4* and *Rz4* were measured on the cut surfaces of samples of all examined thicknesses (6, 10, 15, 40 mm and 6 mm<sup>+</sup>) at a distance of 4 mm from the upper cutting edge (from the surface of the sheet where the jet enters the material) using the Mitutoyo Surftest SJ-301 roughness tester. Repeated control measurements were performed for the reliability of all measured sets of values. The control measurements' total errors for the roughness *Ra*, *Rz*, *Ra4,* and *Rz4* are in the range <3.06; 5.09> percent.

The Dixon test of extreme values is applied to selected sets of measured values in which some values di ffer significantly from the other values of the set. Based on the comparison of the calculated value and the tabular critical value of the test criterion, it can be stated with 95% probability that Rz and *Ra*'s assessed values are not extreme values and can therefore remain in the sets of measured values for evaluation.

Results of *Ra* and *Rz* measurements performed on the middle line of the thickness of samples cut from 6, 10, and 15 mm thick plates using the identical jet parameters mentioned above are summarized in Table 3. The additional results were measured on samples prepared from a 40 mm thick plate applying lower traverse speeds. They are presented in Table 4.


**Table 3.** Values of roughness *Ra* and *Rz* measured on cut surfaces of samples.


**Table 4.** Measured values of declination angle θ and roughness characteristics *Ra* and *Rz* on the cut walls at samples with a thickness of 40 mm.

Results measured on samples 6 mm thick, cut at higher traverse speeds than samples presented in Table 3, were used to broaden the confidence interval of a regression derivation of the relationships useful for analyses, simulations, and control of the surface quality.

Results of surface roughness characteristics *Ra* and *Rz* presented in Table 3 indicate supposed relationships—increasing quality for lower traverse speeds, higher pressures in pump, and higher abrasive mass flow rates. Similar conclusions can be drawn from the results presented in Table 4. Nevertheless, the relationship between roughness and declination angle values can be derived from values in Table 4. Subsequently, the results can be compared with the model presented by Hlavác [ ˇ 17].

Summarizing all combinations of factors, it is possible to obtain functions describing speed-dependent roughness for each doublet *ma* and *p*. However, it is necessary to measure the values in a certain selected identical depth on the cut wall for all samples (to compare the values). The depth equal to 4 mm was selected for presentation in this paper (values are marked *Ra4* and *Rz4*). The typical series of roughness values for all used traverse speeds is presented in Table 5. The selected abrasive mass flow rate is typical for applied nozzle diameter, focusing tube characteristics, abrasive material type, grain size, and pump pressure. Traverse speeds were completed from all experimental sets.


**Table 5.** Typical series of roughness values *Ra4*, *Rz4* (abrasive mass flow rate 220 g/min and pressure 380 MPa are typical technological parameters used for cutting).

The graph of relation between traverse speed and surface roughness parameters is presented in Figure 3 (for values presented in Table 5).

**Figure 3.** Roughness dependence on traverse speed with linear regression (graph).

The summary regression models for factors *x*1 (*ma*), *x*2 (*p*) and *x*3 (*v*) can be written as

$$Ra = 7.905 - 0.012x\_1 - 0.007x\_2 + 0.011x\_3 \tag{1}$$

$$Rz = 39.103 - 0.049x\_1 - 0.027x\_2 + 0.046x\_3 \tag{2}$$

Values calculated from these models were compared with further experimental results, and the comparison is presented in Table 6. The modeled surface roughness values for the respective combinations of technological parameter values were subsequently experimentally verified. The result of the verification confirmed the correctness of the verified mathematical models and the subsequently performed calculation. The deviation between the values of *Ra*, *Rz* obtained from the simulation, and the values from the subsequent verification experiment ranges from −5.4 to +5.6% for *Ra* and in the range of −4.9 to +0.3% for *Rz*. From that, it is evident that the model is simple but works e ffectively.


#### *2.4. Measurements of the Angle of Declination of the Jet*

The declination angle θ of the abrasive water jet for 40 mm thick samples was measured at 5 locations on each cut surface on series of successive measurements distinct approximately 5 mm from the previous measurement in the cutting direction according to Figure 4. The resulting values of the jet deflection on the cut surfaces were obtained by the arithmetic mean of the measured repeated values on the individual cut surfaces (θ1–θ5).

**Figure 4.** Locations of zones for measuring roughness parameters *Ra, Rz*, and *Ra4, Rz4* (**upper left**); measurement of roughness *Ra, Rz*, and *Ra4, Rz4* on cut surfaces of samples (**upper right**); declination angle on the cut surface (**bottom**).

A Vogel-Germany Universal Winkelmesser device with measuring range distribution 4 × 90◦ and scale resolution 5 was used to measure jet declination on cut surfaces of 40 mm thick samples. Presented results also make it possible to prepare the regression equations describing the relations between declination angle value and roughness parameters. The equations obtained from regression by the processing of all measured values for both the *Ra* and *Rz* characteristics of roughness for factor *x*4 (θ) are as follows

$$Ra = 0.2195x\_4 - 0.2239\tag{3}$$

$$Rz = 0.4442x\_4 + 12.25\tag{4}$$

The obtained model reveals linear behavior within the tested range, as shown in Figure 5. Testing of this model accuracy on other materials is just in the stage of preparation for future work.

**Figure 5.** Roughness dependence on traverse speed with linear regression (typical graph).
