Prediction of surface topography in precision hard machining based on modelling of the generation mechanisms resulting from a variable feed rate

The paper presents an original contribution to the prediction of surface topography produced by precision hard turning operations using CBN cutting tools and the variable feed rate of 0.025–0.075 mm/rev. The differences between theoretical and real surface roughness parameters Rz and Sz are quantified in terms of springback effect, additional smoothing of irregularities and side flow effect. The primary experimental study includes measurements of 2D and 3D surface roughness parameters using contact profilometer. Correspondingly, cutting forces were measured using a piezoelectric dynamometer, and based on this data, specific corresponding values of ploughing energy and friction coefficient were determined. It was found that the measured value of maximum height of the surface Sz differs from the theoretical value mainly due to elastic recovery of the machined surface and the smoothing effect at the lower feeds and the elastic recovery and the side flow effect at the higher feeds employed. An empirical model for the prediction of the Sz value in function of the feed rate is derived. The prediction accuracy can be improved by advanced numerical modelling of surface generation mechanisms and associated distortions.


Introduction
Hard machining has been established as a leading machining technology for various machine components made of highstrength steels with surface finish comparable to grinding effects [1]. Predominantly, scientific and engineering issues of hard turning do not address sufficiently surface topography and surface assessment is narrowed to the Ra or Rz roughness parameters [1,2]. In addition, the distortion of surface topography in finish hard machining has not been investigated satisfactorily. It should be noted that surface generation mechanisms developed for machining of materials of low and medium hardness cannot be applied for hard materials. This is due to the fact that strong adhesive interaction between the chip and the tool material characteristic for severe plastic deformation does not occur in hard machining [1]. The molecularmechanical theory of friction was applied for predicting the transition from interfacial sliding to micro-cutting when turning AISI 1045 steel with P10 carbide cutting tools [3]. Instead, intensive ploughing action of the cutting edge and resulting elastic recovery of the hard-machined layer are predominantly observed when removing material using CBN cutting tools with high negative rake angles. It is evident to incorporate the specific influence of the mechanics of the chip formation and the tool wear in the formation of the machined surface and subsurface layer. It was revealed [4] that in ultraprecision hard turning, the generation of surface roughness is limited by such factors as tool cutting edge defects, cutting vibration and elastic and plastic deformation of the workpiece material. Plastic deformation becomes an important component when the feed decreases down to 0.02 mm/rev (20 μm/rev) (percentage about 90%). Moreover [5], the relation between measured and theoretical values of P-V (Rz) parameter for turning and facing of aluminium with cemented carbide and PCD tools suggests that the Rz values down to 0.02 times the insert edge radii are possible. Modelling of material side flow for CBN tools in hard turning of AISI 52100 bearing steel with variable feed rate (f = 0.1, 0.2 and 0.4 mm/rev) and nose radius (r ε = 0.4, 1.2, 1.6 and 3.4 mm) indicated that it increases from about 1 μm up to about 12 μm for the tool nose radius r ε = 0.4 and 3.4 mm, respectively [6]. The side flow effect was also investigated along with 3D surface topography for high-speed hard turning of a bearing steel of 60 ± 2HRC with the cutting speed of 100-300 m/min and feed rate of 0.05 and 0.1 mm/rev [7]. It is reasoned that 3D roughness parameters better characterize the deterioration of the machined surface due to flank wear and side flow is intensified at higher speeds. Similarly, the springback effect (elastic recovery) was predicted under variable cutting speed of 10-450 m/min using FEM for aluminium and titanium [8]. For instance, it increases from about 3 up to 10 μm for Ti6Al4Valloy when cutting speed increases from 10 to 100 m/min. A high coherence between Ra parameter and the springback was found.
The prediction of the surface roughness produced by machining operations is based on the machining theory, experimental investigations, designed experiments, artificial intelligence (AI) and multi-scale analysis [9,10]. Geometrical, physical, empirical and simulation models are the mostly used ones [2,9]. In empirical models, the most important variables are the feed rate, the tool corner radius, the depth of cut and the cutting speed in that order. The practical value of empirical models is substantially limited to the set of variable factors selected by manufacturers and, as a result, they used simple theoretical formulas for both Ra and Rz roughness parameters and compare them with measured values [2]. The main drawback of such approach was that all associated effects leading to the distortion of surface topography (for instance Brammertz's model represented by Eq. 8) are considered individually. The author's analysis suggests that they occur together and their intensity depends on the cutting parameters and the real geometry of cutting tools used. The main advantage of this study is that several factors such as ploughing action of the cutting edge, elastic recovery, smoothing effect of the increased irregularities and unremoved material on the surface were considered comprehensively in precision hard turning (PHT) with variable feed of 0.025-0.075(0.1) mm/rev using chamfered CBN tools. The selection of the feed rate lower than 0.1 mm/rev results from the fact that its increase above 0.1 mm/rev causes a rapid increase of the Ra roughness parameter [11].
2 Measurements and computations of process data

Measurements of cutting forces and specific energies
Measurements of three components of the resultant cutting force (F c , F f and F p ) were performed in the xyz coordinate system as shown in Fig. 1a. These three cutting forces were transformed into the l mn coordinate system ( Fig. 1 b) using two transformation matrixes given by Eq. 1 [12] in order to determine the friction coefficient for the rake face. The geometrical details of the machined layer generated by means of rounded nose inserts is shown in Fig. 1a.
According to Fig. 1, the l mn system is obtained by rotating the xyz system by the inclination angle λ s around the x-axis and by the normal rake angle γ n around the y-axis. The product of relevant transformation matrixes [TM] x and [TM] y is defined by Eq. 1.
Hence, the transformation of force components from the xyz coordinate system to the l mn system using the matrix [TM] is given by Eq. 2.
By multiplying symmetrical 3 × 3 matrix (Eq. 1) and column matrix in Eq. 2, one obtains three equations for determining F l , F m and F n forces in terms of the measured F x , F y and F z forces. They are as follows: Specific cutting k c and ploughing k p pressures are calculated based on the equivalent cutting edge of the length l k and the mean uncut thickness (UCT) h m shown in Fig. 1 a. Hence: A c ð4aÞ where the cross-sectional area of cut A c = h m × l k .
Values of the specific cutting e c and ploughing e p energies are determined by the transformation of units from N/mm 2 (10 6 × N/m 2 ) to 10 6 × N × m/m 2 × m = MJ/m 3 . Finally, the unit of the specific cutting energy was GJ/m 3 (10 3 × MJ/m 3 ).

Measurements of surface roughness parameters
Surface topographies generated by CBN tools were measured by means of the stylus contact method using a TOPO-01P contact profilometer and three non-contact methods: confocal, white-light scanning interferometer and optical with focus variation using versatile S-lynx 3D profiler by Sensofar Metrology, because the contact profilometer is not equipped with the function of precise contour shape measurement. The approximation of the cutting edge radius was performed automatically using a set of reference circles.
3D roughness parameters were determined according to ISO 25178 standard, and surface topographies were visualized using a Digital Surf, Mountains® Map package. They include S and V standardized ISO texture parameters [14]. The definitions, practical interpretations and measurements of 3D surface roughness parameters termed "areal surface texture" are presented by Leach [15]. The choice of the measurement (profiling) technique (stylus profilometer versus laser profilometer or atomic force microscope) depends on the scale of the surface roughness, i.e. from micro-to nano-scale [10,14]. In this investigation, height parameters were analyzed in comparison to their computed values with a different scale of distortion shown in Fig. 4. In particular, a special searching algorithm which allows the selection of minimum and maximum values of Rz (Rt) parameter from 2, 3 or 10 surface profiles was implemented in order to consider the scattering of the measured data (see Fig. 5).

Computations of friction coefficient and elastic recovery
The friction coefficient for the rake face-chip contact can be determined as the ratio of the F n (Eq. 3.3) and the F m (Eq. 3.2) forces as follows: where F γ is the friction force and F γN is the normal force on the rake face (Fig. 1b).
The elastic recovery of the machined surface corresponding to friction at the tool-chip interface defined by Eq. 5 is determined using the following equation [16,17]: where r n is the measured cutting edge radius (Fig. 3b) and μ γ is the friction coefficient on the rake face. Equation (6) was derived based on the theory of elasticity applied to the case when an indenter of r n radius loads an elastic half-space with defined friction coefficient [17,18]. For this reason, the cutting edge radius was measured precisely ( Fig. 3a) to determine accurate values of the elastic recovery (see Appendix Table 1). Figure 2 shows that elastic recovery of the machined surface changes practically linearly with both the ploughing energy and friction coefficient μ γ . It should be noted that PHT with the minimum feed of 0.025 mm/rev corresponds with the highest value of the friction coefficient equal to 2.5 and the specific ploughing energy of 43.1 GJ/m 3 . In comparison, the corresponding value of the specific cutting energy in this case is equal to 17.6 GJ/m 3 which is in accordance with metal-cutting data [13,19].

Computations of heights and smoothing rate of irregularities
The theoretical values of roughness height were determined using both the classical circle models (Eq. 7), expressing its correlation with the feed and the corner radius and more advanced elliptical model (known as the Brammertz's formula) which additionally considers the minimum UCT (h min in Eq. 8) and in consequence a small unremoved area of the rough surface.  Fig. 1 Dimensioning of uncut chip (a) and resolution of resultant cutting force in l mn coordinate system (b) [12,13] Rzt ¼ In this study, the smoothing effect resulting from the additional cut of irregularities during subsequent revolutions of the workpiece after their elastic recovery was taken into consideration. The appropriate formulas are as follows [19,20]: Equation 9a represents the case when the individual irregularity is re-generated after smoothing and Eq. 9b expresses the smoothing effect occurring in the second revolution of the workpiece. The second case takes place when the feed f ≤ ffiffiffiffiffiffiffiffiffiffiffi ffi r ε h min p (for r ε =0.8 mm and h min = 0.8 μm, f ≤ 0.08 mm/ rev). Equations 9a and 9b were derived by adding the first correction terms to Brammertz's formula given by Eq. 8 [20].

Experimental details 3.1 Workpiece material and cutting tool
The workpiece material was a 41Cr4 (AISI 5140 equivalent) alloy steel with Rockwell's hardness of 57 ± 1 HRC and initial Sa roughness of about 0.4 μm. In order to determine an accurate value of the minimum UCT, the cutting edge radius was determined using 3D image obtained on a focus variation microscope as shown in Fig. 3. It varies slightly in the range of 8-10 μm and, in consequence, computations for the elastic recovery (Eq. 6) were carried out assuming its lower and upper values of 8 and 10 μm. In Fig. 3b, a small land b α ≈ 6 μm can be noticed, so the smoothing effect can be expected as well. According to the well-established relation between the minimum UCT and the cutting edge radius, it was assumed that the ratio h min = 0.1r n [3,16,19].

Hard turning conditions
Turning operations were performed on a CNC turning center, Okuma Genos L200E-M with an installed three-component Kistler dynamometer (model 9129A) and consumed energyrecording system. The resultant cutting force was resolved into three components-F c , F f and F p . The measured signals were processed with a sampling rate of f = 1 kHz and a lowpass filter with a cut-off frequency of f c = 300 Hz.
Initial finish hard turning (FHT) was performed with v c = 150 m/min, f = 0.1 mm/rev, a p = 0.15 mm and subsequent precision hard turning (PHT) operations with the same cutting speed but variable feed rate of 0.025, 0.035, 0.050, 0.060 and 0.075 mm/rev, respectively. CBN TNGA 160408 S01030 chamfered inserts with electro-erosion (ER) honed cutting edges. Typically, the cutting edge is prepared by sinking it into a counterface [21]. This special preparation technology allows to produce cutting edges with the minimum radius of about 5 μm. The cutting edge radius of r n = 8-10 μm and chamfer width b γ ≈100 μm were measured (see Fig. 3b). The cutting tool angles in the tool-in-hand system were the following: κ r = 91°, λ s = −6°, γ nc = −30°, γ n = −6°.

Factors influencing surface roughness in PHM
As pointed out in Section 1, surface generation in precision hard turning (PHT) is influenced by several distortion effects depending on the feed rate applied. They include such effects as elastic recovery caused by intensive ploughing action of the cutting edge, smoothing effect of the irregularities in subsequent revolutions and plastic side flow due to the lateral flow of the thermally softened material.
Values of measured and computed surface roughness parameters presented graphically in Fig. 4 are specified in Appendix Table 2.
In this study, all surface distortion effects are related to the maximum surface height Sz (surface topography) and corresponding 2D parameter Rz (surface profile). According to Fig. 4, the elastic recovery modelled by Eq. 6 (curves #3a and 3b) was established to be mostly predominant at the minimum feeds of 0.025 and 0.035 mm/rev. This corresponds well with the Brammertz's model represented by curves #2a and 2b. It should be noted that springback effect seems to be important also when PHT with the feed rate higher than 0.06 mm/rev. The smoothing effect occurs simultaneously with the previous one and exists also at a higher feed up to about 0.05 mm/rev. When the feed exceeds 0.05 mm/rev, the Brammertz's model (Eq. 7) fits the measured Sz values better than the elastic recovery because it decreases as shown in Fig. 2. Finally, at the feeds between 0.075 and 0.1 mm/rev, surface profiles become more regular with a visible plastic flow effect at the highest feed (Fig. 6c, d). This observation is in a strong agreement with previous reports by Kishawy et al. [6]. All these influences were documented quantitatively by the introduction of appropriate analytical models.
In particular, in PHT the minimum value of Rz parameter is predicted with sufficient accuracy using theoretical formula,  Fig. 3 Stereometric image of the cutting edge (a) and its dimensioning (b) at magnification ×200 but its maximum value needs the smoothing and springback effects to be considered (curves #4a and 4b). In contrast, the Sz value is closer to the Brammertz's formula or more accurately to the smoothing effect at the minimum feeds applied. Figure 5 shows the graphical method for fitting the predicted values of the maximum roughness height to the measured values of Sz parameter (curve #2b). In the first step, the model of the function Sz = f ( f ) (curve #2a) was determined using polynomial model in the form of Sz ¼ −0:3023 þ 54:545 f −394:97f ð10Þ and using this model, the values of cutting edge radii were computed for all the values of feed rate (0.025, 0.035, 0.050, 0.060 and 0.075 mm/rev) and constant tool nose radius of 800 μm. In case of empirical Eq. 10, the R-squared is equal to R 2 = 0.9718 and the residual sum of squares is equal to 0.0122. It was found that the relevant values of the cutting edge radius which satisfy Eq. 10 are equal to 6, 14.3, 19 and 13 μm, respectively. On the other hand, the next possibility is to change the tool nose radius, i.e. decrease it for lower feeds and increase it for higher feeds.
The measured values and values of the maximum height of the surface Sz predicted by Eq. 10 are presented separately in Fig.6. Figure 7 shows a series of zoomed surface topographies recorded by means of a confocal 3D profilometer showing characteristic distortion modes obtained at feed rates of 0.025, 0.050 and 0.075 mm/rev and additionally, a reference surface topography with regular feed marks generated at the feed of 0.1 mm/rev. Representative surface textures obtained in PHT operations are labelled by the measured values of 3D roughness parameters. The measured values of Sa and Sz parameters range from 0.07 to 0.21 μm and 0.2 to 1.6 μm, respectively.

Analysis of surface topography in terms of constitutive conditions
The minimum and maximum values of Rz (Fig. 4) determined from two and three automatically selected profiles varied between 0.22-0.95 μm and 0.28-0.97 μm, respectively. The regular distribution of feed mark characteristic for CBN turned surface with feed rate of 0.1 mm/rev is visualized in Fig. 7d. In Fig. 7d also, regular material pile-ups at the secondary cutting (trailing) edge caused by material side flow can be observed. This effect corresponds well quantitatively with FEM predictions made by Kishawy et al. [6] and Schaal et al. [7]. For sharp tools, the maximum springback in metal cutting is equal to δ s = 0.3-1.6 μm.
It should be emphasized that plastic side flow results from lateral plastic flow of the material locally heated up to 900-1000°C [22]. In addition, associated abrasive wear of the cutting tool is more likely than adhesion [23].
In this study for precision machining it ranges from 0.2 to 0.8 μm depending on the ploughing intensity (Fig. 2). On the other hand, the height of lateral flashes resulting from the material side flow which develops at  Fig. 7d. Moreover, the corresponding average nodal displacement is about δ x = 5 μm which coincides with computed data in Ref. [6]. The regular feed marks produced in hard turning operations using low and high feed rates are visualized in Fig. 8. In general, the measured distances between feed marks are practically equal to the nominal values of feed rates selected in this study.

Conclusions
1. In precision, hard turning performed with CBN cutting tools and feed rate of 0.025-0.075 mm/rev such surface distortion effects as elastic recovery (springback) caused by an intensive ploughing action of the cutting edge, smoothing effect of the generated irregularities in subsequent revolutions and plastic side flow resulting from lateral plastic flow are revealed depending on the feed rate applied. 2. The elastic recovery was established to be mostly predominant at the minimum feeds of 0.025 and 0.035 mm/rev. The smoothing effect occurring simultaneously with the elastic recovery is also extended to a higher feed of 0.050 mm/rev.
3. At the feeds of 0.075 μm and 0.1 mm/rev the minimum uncut chip thickness increases and both these phenomena weaken visibly while surface profiles (topographies) become more regular. Additionally, visible flashes of about 0.6 μm in height caused by the side flow effect appear.
4. The three effects were documented quantitatively by the introduction of appropriate analytical models. In particular, in PHT the minimum value of Rz parameter is predicted with sufficient accuracy using theoretical formula but its maximum value needs the cutting edge sharpness to be considered. In contrast, the Sz values are closer to the Brammertz's formula or more accurately to smoothing effect at the minimum feed rates applied.
5. An original graphical method for fitting the predicted and measured values of the maximum surface height by searching the correct values of the cutting edge radius is proposed. As a result, an empirical model for prediction of the Sz parameter in terms of the feed rate used is derived. Moreover, the possible distortions of the machined surface topographies are visualized in the form of isometric views. 6. Future trends can be focussed on experimental investigations in micro-scale using specially designed devices and accurate FEM-based modelling of surface generation mechanisms including associated distortion effects. It is also important to design cutting tools with precisely prepared cutting edges, keeping the recommended cutting edge radius and the tool corner radius. It is also plausible to extend this analysis to higher tool corner radiuses, i.e. 1