Control concept for the regulation of the surface layer properties using consecutive cuts in cryogenic hard turning of AISI 52100

Abstract

In this study, a control concept for the simultaneous adjustment of the workpiece residual stresses and the surface roughness when cryogenic hard turning AISI 52100 is presented. The turning process is comprised by two consecutive cuts in which the process parameters vary, focusing on roughing and finishing; i.e. first a high, then a low depth of cut. On the first cut, the surface residual stress is adjusted via the cutting speed. On the second cut, the surface roughness is minimized by adjusting the feed rate while maintaining high compressive residual stresses. The surface residual stresses are determined by XRD and hole drilling measurements. For the estimation of the residual stresses the passive force is used. Based on these results, a control concept is presented which makes several assumptions that are partly validated by experiments.

1 Introduction and state of the art

AISI 52100 is commonly used as a roller bearing steel and can be classified as a difficult-to-cut material. Among other materials it is the subject of current and past investigations regarding the surface layer [1]. High forces and temperatures occur during the manufacturing process because of its high hardness [2], which can have a negative impact on the surface layer [3]. Cryogenics like CO₂, LN₂ or sub-zero metal-working fluids (MWF) can be used to reduce the temperatures and tool wear, thus improving the tool life [4]. Cryogenic cooling can improve the surface layer properties by inhibiting temperature induced phase changes from martensite to austenite [5]. Preferred in most cases are high compressive residual stresses in the surface layer, because they act crack-closing thus are favorable for a longer fatigue life [6]. In general, it has been reported that the cutting parameters of the finishing step have a major influence on the surface layer properties and also the in-depth residual stress profiles [7]. Residual stresses in the surface layer of AISI 52100 can be induced and shifted to a deeper level in the workpiece with chamfered tools and increasing cutting speed [8]. The same has been found for increasing width of flank wear land [9]. Umbrello et al. proposed an FE-model for the prediction of machining induced residual stresses and white layers with good agreement to the experimental results [10]. Investigations showed that the depth of cut (0.5–1.0 mm) had no significant influence on the residual stresses [11], whereas another study came to the conclusion that depth of cut has a major influence [12]. Therefore, more studies are needed to examine this topic.

As another part of the surface layer, white etching layers can form during the machining of hardened steels [13]. The cause of their formation is investigated in [14]. It was found that the effect of the residual stresses has more impact on fatigue life than the white layers [15].

Soft sensors combine at least one physical measurement with expert knowledge or model. Meurer et al. [16] have developed a soft sensor to estimate the white layer occurrence and thickness in hard turning of AISI 4140 with good results for higher cutting speeds and higher undeformed chip thicknesses. They used the cutting force to estimate the evolving temperature and correlated it with a finite-element-model-simulation of the chip formation. Then their model was validated using micrographs of the etched microstructure of the surface layer.

An approach for an in-process measurement presented a scattered light sensor, which was integrated in the turning process to measure the surface roughness [17]. The pneumatic function allows an easier setup of the scattered light sensor and protects it against damage from chips. Also, the measuring point is cleaned by the compressed air. By combining the scattered light data with the gathered knowledge about the process and its parameters, it is possible to reconstruct the surface profile within limits. The calculation of the Fourier coefficients needed for this and the determination of the surface profile was published in [18].

Following this concept, we presented a multiple linear regression model which connects cutting parameters with process forces, temperatures and surface layer properties [19]. They can be adjusted to result in increased residual compressive stresses ${\sigma }_{RS}$ and to decrease the austenitic volume content to favor the fatigue lifetime. The austenitic phase is not preferrable in the surface layer of roller bearing steel since high hardness and wear resistance is needed [5]. In the study, it proved cutting speed and feed rate as the main influencing factors. The morphology in the surface layer was characterized by X-ray diffraction and scanning electron microscopy, revealing that the impact of the initial heat treatment was less significant compared to the influence of cryogenic hard turning. Regarding the distribution of phase components and residual stresses, the affected zone of cryogenic hard turning was determined to a depth of 80 µm beneath the surface. However, a significant influence of various turning parameters could be constricted to the first 20 µm [20]. In addition to effectively control surface roughness, it is crucial to determine the microstructural changes within the affected zone resulting from cryogenic hard turning [21].

As of now, the authors of this paper are not aware of any other implementation approaches for soft sensors in the hard turning of difficult-to-cut materials. However, in [7] only one turning step was examined. But usually there are at least two steps required resembling roughing and finishing. In the first step as much material as possible is removed while minimizing negative influences regarding the thermal and mechanical load imposed to the workpiece. In the second step the surface roughness is minimized using a much smaller depth of cut.

This paper follows our previous mentioned studies within the same framework, elucidating a novel concept for regulating residual stresses and surface roughness as surface layer properties in a cryogenic hard turning process of AISI 52100. In two consecutive cuts a scattered light sensor provides in-process information and is part of a soft sensor together with the expert knowledge of a previous study [19]. The concept aims to adjust surface layer properties by changing feed rate and cutting speed. First experiments were carried out including in-situ (process forces and temperatures) and ex-situ (residual stresses and surface roughness) characterizations to examine the assumptions of the concept.

In the following, the elaborated concept is presented in Sect. 2 and the required regression models in Sect. 3. Subsequently, the series of experiments carried out is described in Sect. 4 and the investigation of the assumptions and regression models required by the concept in Sect. 5. Table 1 provides a list of the important variables.

Table 1 List of symbols

2 Control concept

2.1 Conceptual idea and motivation

The machining is divided in two steps based on roughing and finishing. The objective is to reproduce specimens with the target residual stress ${\sigma }_{RS}^{T}$ and roughness ${P}_{c}^{T}$ (“mean height of profile elements” according to [22]). Therefore, a model-based controller is elaborated that sets ${v}_{c}^{1.c}$, f^2.c and ${a}_{p}^{2.c}$, where the indices 1.c and 2.c indicate first and second cut. During the first cut, soft sensors carry out in-situ measurements and supply the required measurement variables for a controlled correction in the second cut. The concept tries to set ${v}_{c}^{1.c}$ and f^2.c as high as possible to minimize production time while meeting the requirements for compressive residual stress and roughness. Due to the reliable setting of the target parameters, f^2.c is higher compared to an uncontrolled process, as it requires less certainty in the selection.

2.2 Preliminary assumptions

The following assumptions (A1 – A5) form the basis for the control concept. They are partly derived from earlier studies. At this point they are merely postulated and not validated. In this work, they are investigated for cryogenic hard turning of AISI 52100 under specific parameter ranges derived from [19]. The assumptions are as follows and the investigation is described in Sect. 5.4:

• A1: Decreasing v_c increases the compressive ${\sigma }_{RS}$ in axial direction (derived from [19]).

• A2: With a sufficiently low a_p in the second cut, ${\sigma }_{RS}^{1.c}$ can be further influenced.

• A3: F_P carries information about the induced ${\sigma }_{RS}$.

• A4: Only f^2.c influences the roughness after the second cut.

• A5: Scattered light data can be used to approximate the roughness profile (derived from [18]).

2.3 Controlled two-step process flow

The proposed concept is described by a two-step process flow as illustrated in Fig. 1. Besides the parameters that can be adjusted by the controller ${v}_{c}^{1.c}$, f^2.c and ${a}_{p}^{2.c}$, other parameters are fixed for each cut and cannot be changed. This simplification to fixed parameters reduces the parameter space for the required regression models and makes their parameter identification more robust.

Fig. 1

Scheme of the controlled two-step process flow (the dashed lines indicate the scope of future studies, as explained in Sect. 2.4)

The process flow can be explained by the following steps: the user selects the residual stress in the axial direction ${\upsigma }_{RS}^{T}$ and roughness ${P}_{c}^{T}$ as target variables and passes them to the controller. Based on three regression models (M₁ to predict ${\sigma }_{RS}^{1.c}$, M₂ to predict ${\sigma }_{RS}^{A}$ and M₃ to predict ${P}_{c}^{A}$), the controller plans the whole machining process with the aim to set the target variables accurately and to reduce the production time. For this purpose, ${v}_{c}^{1.c}$ and f^2.c are always selected as large as possible while meeting the targets.

After ${v}_{c}^{1.c}$ has been set based on this planning and to create a desired ${\sigma }_{RS}^{1.c}$ (cf. A1), the first cut is made while soft sensors are used: The soft sensor “residual stress” uses F_P and calculates ${\sigma }_{RS}^{1.c}$ based on M₁ (cf. A3). The soft sensor “roughness” uses the data measured by the scattered light sensor and calculates an approximated roughness profile after the first cut (cf. A5). The approximated roughness profile is transferred to the soft sensor “process condition”. This knows the machining log of previous cuts and also receives real-time data from the two soft sensors mentioned above. With this information, it can determine the tool wear for a correction in the second cut or identify specimens that deviate significantly from the model and mark them as outliers.

Before the second cut, the controller knows the required ${\sigma }_{RS}^{1.c}$, based on its planning, and, at the same time, the actual ${\sigma }_{RS}^{1.c}$, provided by the soft sensor. With this information, it calculates a correction to adjust ${a}_{p}^{2.c}$ (cf. A2). Furthermore, based on the model M₃ it calculates the required f^2.c to achieve the target roughness (cf. A4).

If a variation of ${a}_{p}^{2.c}$ is excluded, corrections can be made with f^2.c. Since in this case both targets cannot be set accurately by changing only one parameter, f^2.c is chosen so that ${\upsigma }_{RS}^{A}\ge {\upsigma }_{RS}^{T}$ and ${P}_{z}^{A}\le {P}_{z}^{T}$. One target is set accurately while the other is set better than the target value. However, this is only possible in those parameter regions where reducing f^2.c leads to an improvement of both targets.

2.4 Specification for the scope of this study

The introduced control concept takes the variation of three parameters into account: ${v}_{c}^{1.c}$, f^2.c and ${a}_{p}^{2.c}$. Since, on the one hand, taking all three parameter variations into account would require a large experimental effort and, on the other hand, a statement about the functionality of the concept can be made solely with ${v}_{c}^{1.c}$ and f^2.c, the following specification is chosen for the framework of this study: ${a}_{p}^{2.c}=const$. The investigation regarding the adjustment of ${a}_{p}^{2.c}$ is carried out by analyzing the residual stress in-depth.

The influence of tool wear is the scope of a separate study and will be investigated in the future. In a defined series of experiments, cutting tool replacements were carried out at an early stage to keep the influence of tool wear negligible. For this reason the soft sensor “process condition” is not part of this study as it primarily addresses tool wear.

For the realization of M₁, M₂, and M₃, regression models have to be derived and an identification has to be carried out, as will be described in the next section. After the identification a validation with new data has to be performed. The validation is not part of the present work, since the main objective is to establish the concept and to determine the control parameters while mapping an industry-related process and proving the assumptions A1–A5 for plausibility.

The exact operating principle of the soft sensor "roughness" was described in detail in [18]. Briefly summarized, the soft sensor works in such a way that initial Fourier coefficients, which represent an ideal profile depending on the manufacturing parameters, are optimized using sequential quadratic programming so that the scattered light distribution calculated with this corresponds to the scattered light measurement data.

3 Regression models

3.1 Model approach on residual stress

In the following it is assumed that the final residual stress ${\sigma }_{RS}^{A}$ can be modelled by adding the surface residual stresses ${\sigma }_{RS}^{1.c}$ and ${\sigma }_{RS}^{2.c}$:

$${\sigma }_{RS}^{A}= {\sigma }_{RS}^{1.c}+{\sigma }_{RS}^{2.c}$$

(1)

To describe the machining process, non-linear approach functions ${f}_{1}\left(*\right)$ and ${f}_{2}\left(*\right)$ are to be used for ${\sigma }_{RS}^{1.c}$ and ${\sigma }_{RS}^{2.c}$. Because ${v}_{c}^{1.c}$ is varied in the first cut, ${f}_{1}\left({v}_{c}^{1.c}\right)$ applies. Because ${f}^{2.c}$ is varied in the second cut and additionally the residual stress built up in the second cut is assumed to be dependent on the first cut, ${f}_{2}\left({v}_{c}^{1.c},{f}^{2.c}\right)$ applies. Furthermore in line with Assumption A3, an influence of the passive force is expected and taken into account by adding the linear correction term ${k}_{F}\cdot ({F}_{P}^{1.c}-\overline{{F }_{P}^{1.c}}$), where ${F}_{P}^{1.c}$ denotes the passive force, $\overline{{F }_{P}^{1.c}}$ the averaged passive force from comparable experiments and k_F a constant to be identified. It follows:

$${\upsigma }_{RS}^{1.c}={f}_{1}\left({v}_{c}^{1.c}\right)+{k}_{F}\cdot \left({F}_{P}^{1.c}-\overline{{F }_{P}^{1.c}}\right).$$

(2)

$${\upsigma }_{RS}^{2.c}={f}_{2}\left({v}_{c}^{1.c},{f}^{2.c}\right)$$

(3)

It can be advantageous to combine the functions ${f}_{1}\left({v}_{c}^{1.c}\right)$ and ${f}_{2}\left({v}_{c}^{1.c},{f}^{2.c}\right)$ within one approach function ${f}_{12}\left({v}_{c}^{1.c},{f}^{2.c}\right)$ with fewer parameters to be identified and to carry out a separation only after identification. This has the additional advantage that the samples only have to be evaluated after the second cut, which reduces the experimental effort and allows more specimens to be manufactured. This approach is presented below. For ${\upsigma }_{RS}^{A}$ then follows:

$${\upsigma }_{RS}^{{\text{A}}}={f}_{12}\left({v}_{c}^{1.c},{f}^{2.c}\right)+{k}_{F}\cdot ({F}_{P}^{1.c}-\overline{{F }_{P}^{1.c})}$$

(4)

For ${f}_{12}\left({v}_{c}^{1.c},{f}^{2.c}\right)$ an arctangent function rotated around the vertical axis according to Fig. 2 is chosen. Because of its shape, an arctangent is suitable to model the behavior from one input to one output variable with continuously increasing or decreasing and, if necessary, saturation effects. By adding a second dimension and rotating the arctangent around the vertical axis, the dependence of a second input variable can be taken into account. The more the function is rotated in the direction of one input variable, the stronger the effect of that input and the weaker the effect of the other input. The function is defined by the Eqs. (5), (6) and (7), where k_min, k_max, k_g, k_o and k_c denote identification constants. The effect of the identification constants is shown in Fig. 2:

Fig. 2

Effect of the identification constants on ${f}_{12}\left(*\right)$. The displayed parameter combination is: k_min = 100, k_max = 800, k_g = 10, k_o = 0.5, k_c = 1. On the left side the shape of the arc tangent approach function is shown by showing it from the side, on the right side the same function is shown from the top

$${f}_{12}\left({\text{q}}\right)=\frac{({k}_{max}+{k}_{min})}{2}+\frac{\left({k}_{max}-{k}_{min}\right)}{\pi }\cdot {\text{arctan}}({k}_{g}\cdot (q-{k}_{o}))$$

(5)

$$q={q}_{f}\left({f}^{2.c}\right)+{{\text{k}}}_{c}\cdot {q}_{v}({v}_{c}^{1.c})$$

(6)

$${q}_{v}\left({v}_{c}^{1.c}\right)=\frac{{v}_{c}^{1.c}-{v}_{c,min}}{{v}_{c,max}-{v}_{c,min}},{q}_{f}\left({f}^{2.c}\right)=\frac{{f}^{2.c}-{f}_{min}}{{f}_{max}-{f}_{min}}$$

(7)

The separation of ${f}_{12}$ into ${f}_{1}$ and ${f}_{2}$ is not explicit. This results in two further identification constants k_p and k_b. ${f}_{12}= {f}_{1}+ {f}_{2}$ is valid with the following equations (with $\pm 2$ equal to $+2$ for ${{\text{arg}}}_{vc}+{{\text{arg}}}_{f} \ge 0$ and equal to $-2$ otherwise):

$${f}_{1}\left({v}_{c}^{1.c}\right)=\frac{{k}_{b}\cdot ({k}_{max}+{k}_{min})}{2}+\frac{\left({k}_{max}-{k}_{min}\right)}{\pi }\cdot {\text{arctan}}({{\text{arg}}}_{vc})$$

(8)

$${f}_{2}\left({v}_{c}^{1.c},{f}^{2.c}\right)=\frac{(1-{k}_{b}\pm 2)\cdot \left({k}_{max}+{k}_{min}\right)}{2}+\frac{\left({k}_{max}-{k}_{min}\right)}{\pi }\cdot {\text{arctan}}({{\text{arg}}}_{f})$$

(9)

$${{\text{arg}}}_{vc}={k}_{g}{k}_{c}{q}_{v}\left({v}_{c}^{1.c}\right)-{k}_{g}{k}_{o}{k}_{p}$$

(10)

$${{\text{arg}}}_{f}=\frac{{k}_{g}{q}_{f}\left({f}^{2.c}\right)-{k}_{o}+{k}_{p}{k}_{o}}{{[k}_{p}{k}_{o}-{k}_{c}{q}_{v}\left({v}_{c}^{1.c}\right)]\bullet [{k}_{o}-{q}_{f}\left({f}^{2.c}\right)-{k}_{c}{q}_{v}\left({v}_{c}^{1.c}\right)]{k}_{g}^{2}+1}$$

(11)

From the comparison of Eqs. (5) and (8) it can be seen that ${f}_{1}$ is based on a similar function shape as ${f}_{12}$ shown in Fig. 2 on the left. Due to the two additional identification constants k_p and k_b, this curve can be shifted vertically or horizontally and, depending on k_c, is stretched or compressed compared to Eq. (5).

Based on the above equations, the model M₁ calculates ${\sigma }_{RS}^{1.c}$ according to Eqs. (2), (8) and (10) and the model M₂ calculates ${\sigma }_{RS}^{A}$ according to Eq. (4), (5) and (6). Both Models M₁ and M₂ represent a data-driven approach and are examined in Sect. 5.5 based on measurements. For both ${\sigma }_{RS}^{1.c}$ and ${\sigma }_{RS}^{2.c}$, the residual stresses at the surface are used for identification.

3.2 Model approach on roughness

According to the literature [23] the theoretical total height R_th can be calculated with the following formula:

$${\text{R}}_{\text{th}}\text{=}\frac{{\text{f}}^{2}}{{8}{\text{r}}_{\upvarepsilon }}\text{+}\frac{{\text{h}}_{\text{min}}}{2}\left(\text{1} + \frac{{\text{r}}_{\upvarepsilon }{\text{h}}_{\text{min}}}{{\text{f}}^{2}}\right)\text{in mm}$$

(12)

The formula of Brammertz is mainly depending on f. If $f\le \mathrm{0,1}$ mm/rev the right term becomes especially important. Since no investigations have been conducted to determine the minimum chip thickness h_min in this case, it will be estimated in Sect. 5.5.

4 Experimental setup

The specimens consist of AISI 52100 roller bearing steel and were austenitized at 850 °C for 120 min, then quenched in oil and heat treated for 1000 min at 180 °C tempering temperature. The retained austenite content after the heat treatment was 6 vol%. The length of the machined surface was 25 mm and the specimen diameter 15 mm.

The machining was carried out on the CNC lathe Boehringer NG200¹ equipped with a Siemens¹ Sinumerik 840d control system. A setup providing cryogenic cooling during these experiments was implemented into the lathe allowing to regulate the mass flow of the CO₂. The nozzle had an outlet diameter of 1.2 mm and was 20 mm away from the contact zone facing the workpiece and tool at the rake face.

The arrangement of the cooling nozzle and the compressed air is depicted in Fig. 3. The compressed air is needed to ensure that the cold specimen is not covered in CO₂ snow or condensate from the ambient air humidity, what would hinder the measurement of the scattered light sensor.

Fig. 3

Experimental setup in the lathe and details of the cutting process

For the force measurement a three-component dynamometer type 9129AA from Kistler¹ was used. The temperatures were measured with a total of four thermocouples of type K with Ø = 0.5 mm each. Three thermocouples were inserted in the back of the specimens into eroded holes with different depths of 40, 47.5 and 55 mm (see Fig. 3). These holes lie on a circle with a diameter of 13.5 mm equidistant to each other. The fourth thermocouple was inserted in an eroded hole in the back of the indexable insert with 0.5 mm distance to the cutting edge.

A modified self-developed scattered light sensor from Optosurf¹ was used to measure the surface in-process. A pneumatic unit was built in front of the actual sensor. In this setup, it was only used to clean the measuring point. To maximize cleaning performance, a cleaning pressure of 8 bar was used with a 2 mm diameter nozzle. The scattered light sensor measured the gradients of the surface in the range of ± 8° in a measuring range of 0.9 mm. The measuring frequency of 2 kHz ensured a 50% overlap of the measurements.

To conduct residual stress analysis, X-ray diffraction (XRD) measurements were performed on the specimen’s surface, employing a Cu-Kα₁ radiation source with a focused beam size of 2 × 2 mm². The scanning conditions included a voltage of 40 kV and a current of 40 mA, with a scanning rate of 0.004°/s in the 2θ range from 80.32° to 84.28°, specifically targeting the (211)—bcc lattice plane. The determination of residual stress was accomplished by utilizing the sin²ψ method with an interval of 0.1 for sin²ψ values ranging from 0 to 0.5. The peak positions corresponding to all measured inclination angles ψ, reflections, and samples were analyzed and fitted using the X’Pert¹ Stress software (Malvern Panalytical, the Netherlands). The reference lattice parameter d₀ was derived from [24], utilizing the calculation method based on an isotropic material model. Considering its significance in practical applications, this study places its focus on measuring and modeling the residual stress along the axial direction. Thus, all the following residual stresses indicate stresses along the axial direction.

The depth-resolved hole drilling (HD) method was used to supplement the XRD measurements. The optical PRISM system from Stresstech¹ used the electronic speckle pattern interferometry (ESPI) to measure σ_RS. The ESPI method uses laser light to interfere an illumination beam with an object beam to produce a speckle pattern. A phase shift contains information about the deformation of the pixels [25]. With this deformation data a mathematical model based on Fourier analysis calculates residual stresses regardless of rigid-body motions or temperature changes [26]. For the reduction of measurement noise, a regularization factor of 0.01 according to the ASTM E837-13a standard [27] was applied. As drilling tool a 0.8 mm diameter end mill with the specification “474P080.029” from Zecha¹ with a feed rate of 0.05 mm/s, an illumination angle of 53° and a camera angle of 26° were used.

The surface profiles were measured with the stylus instrument Hommel¹ T8000 with a traverse length of 24 mm, a 5 µm stylus, a speed of 0.2 mm/s and a point spacing of 0.2 µm. Before each measurement, an alignment was made by the instrument to compensate for any tilt of the samples and a 2.5 µm Gaussian S-filter was used.

5 Experiments, evaluation and identification

5.1 Experimental plan

The motivation of the experiments was to validate the initial assumptions as well as the identification of the regression models. 20 specimens in total have been manufactured, the first 16 each in two consecutive cuts, plus four additional specimens in one cut to investigate ${\sigma }_{RS}^{1.c}$ in a depth of 30 µm, see Fig. 4.

Fig. 4

Scheme of the experimental plan: same colors mark the same specimen. The run ID consists of the specimen number followed by the number of the cut

For the first cut ${v}_{c}^{1.c}$ ranged from 25 to 100 m/min to adjust ${\sigma }_{RS}^{1.c}$ and f^1.c was kept constant at 0.15 mm/rev. Whereas f^2.c was varied from 0.025 to 0.15 mm/rev to minimize surface roughness after the second cut and simultaneously further increase compressive ${\sigma }_{RS}$. Here ${v}_{c}^{2.c}$ was kept constant at 50 m/min. a_p of the first cut was 0.15 mm and for the second cut 0.03 mm. For specimens 1, 2, 5, 9, 13 and 17 new tools were used (“+” in Fig. 4).

5.2 Evaluation of the measurement data

The cutting forces as well as the temperature in the workpiece and tool were recorded with the start of the manufacturing process. The data acquisition started via NC code about one second before the machining and ended with the tool exiting the workpiece. The analysis of the data was done in MatLab¹. 10% at the beginning and at the end was neglected for the analysis eliminating singularities of the infeed and the outfeed of the cutting edge according to Fig. 5. This window was also transferred to the analysis of the temperature. Then the median of the force was built and the arithmetic mean of the temperatures to condensate the data to a scalar value. The values of the three thermocouples in the workpiece were averaged.

Fig. 5

Evaluation of the process forces (left) and temperatures (right)

The roughness evaluation is carried out with the measurement data of the stylus instrument T8000 on primary profiles and the parameter P_c according to [22]. For the evaluation, the hills are selected in such a way that they have a distance equal to f^2.c to each other, the dales are always located in between.

5.3 Preliminary considerations

In Fig. 6 process temperatures and passive force of all specimens mentioned in Fig. 5 are shown. Passive forces increase almost linear in every block of four except for specimens 1–4. Both temperatures show a gradual decrease and it seems to have an inverse correlation with the passive force. The passive force of specimen 1 is much higher than all of the other specimens. The authors do not attribute this to the occurrence of built-up-edges, which were neither observed in any of the experiments nor in previous studies within this framework. Also, it must be noted, that specimens 1, 5, 9, 13 and 17 had the same cutting parameters and exhibit much lower passive forces. Therefore, specimen 1 is declared as an outlier. Specimen 4 shows high process temperatures in the second cut compared to specimens 1–3. Also, compared to the temperature differences within the other blocks specimen 4 can be declared as an outlier. Thus, they will be neglected in the models.

Fig. 6

Tool and workpiece temperatures with passive force for the first and second cut; the four additional specimens are listed in the first cut (specimen no. 17–20)

In accordance with previous research [21], the affected residual stress zone can be proven up to a depth of 80 µm after the first cut (see Fig. 7, right). The maximum compressive ${\sigma }_{RS}$ were identified either directly on the surface layer or at a depth of approximately 20 µm. In the latter scenario, a distinct drop in compressive ${\sigma }_{RS}$ is present, attributable to localized overheating resulting from the rough cutting parameters. From about 30 µm in depth to the surface, ${\sigma }_{RS}$ distributions of various cutting parameters converge and gradually subside, ultimately reaching a residual stress-free state.

Fig. 7

Surface residual stress after the second cut (left); residual stress profile of the additional specimens after the first cut (right)

5.4 Validation of the assumptions

A1 “Decreasing v_c increases the compressive ${\sigma }_{RS}$ in axial direction.”: This assumption is derived from [19] where a model was presented predicting the ${\sigma }_{RS}$ with an R² = 0.75 and containing a linear regression between ${\sigma }_{RS}$ and v_c. It is important to note, that this only is valid for the surface residual stress. As in Fig. 7 (right) depicted, the results of the XRD measurements show a similar tendency. In contrast, the hole drilling (HD) experiments show that the in-depth profiles differ from the values at the surface, in fact the correlation between v_c and ${\sigma }_{RS}$ seems to be inversed. At a depth of 30 µm there are still compressive residual stresses, which are important for the second cut. A steadily rising trend in ${\sigma }_{RS}^{1.c}$ from there on can be seen. Despite the similar residual stress value at 30 µm, it is worth noting that the residual stress states can be strongly influenced by the local microstructure. A previous study utilizing GIXRD (grazing incidence X-Ray diffraction) revealed significant microstructural distinctions, e.g. grain size, phase component, residual stresses within the initial 20 µm beneath the surface [20]. Consequently, the influence of various cutting speeds during the first cut is not negligible.

A2 “With a sufficiently low a_p in the second cut, ${\sigma }_{RS}^{1.c}$ can be further influenced.”: Regardless of ${\sigma }_{RS}^{1.c}$, an influence of f^2.c can be seen in Fig. 7(left). A decreasing f^2.c led to a significant increase of compressive ${\sigma }_{RS}$. Comparing this to the maximum of the additional specimens ${\sigma }_{RS}^{1.c}=-450 {\text{MPa}}$, it is obvious that ${\sigma }_{RS}^{2.c}$ can be further influenced by the residual stresses emerging from the second cut. The thesis is that the residual stresses from the first cut can superimpose with the residual stress from the second cut.

Assuming this is true, then the compressive ${\sigma }_{RS}$ of specimens produced with the same cutting parameters as in the second cut should be lower. This needs to be validated in the future. Although in the previous model [19] a_p had no influence on ${\sigma }_{RS}$, here a_p is 20 µm smaller and outside of this model. Thus, an influence on ${\sigma }_{RS}$ induced in the first cutting step cannot be excluded.

A3 “F_P carries information about the induced ${\sigma }_{RS}$.”: Similar as reported in the literature [28], a lower v_c as well as higher f led to higher F_P. However, since ${\sigma }_{RS}^{1.c}$ could be influenced by many other factors, such as temperature, a direct correlation between F_P and ${\sigma }_{RS}$ could not be clearly identified within this study.

A4 “Only f^2.c influences the roughness after the second cut.”: There is no influence from any parameter of the first cut to the roughness after the second cut. Otherwise in Fig. 8(left) would be an influence within the specimens cut with the same f.

Fig. 8

Roughness after the second cut depending on ${v}_{c}^{1.c}$ (left) and ${F}_{P}^{1.c}$ (right)

A5 “Scattered light data can be used to approximate the roughness profile.”: The reconstruction of roughness profiles with scattered light was investigated in [18]. A method is proposed to achieve an approximation of the stylus measurement data as shown in Fig. 9. With the scattered light sensor an approximated in-process measurement of the surface roughness is possible. The reconstructed data is shown in Fig. 9. A reconstruction of the sinusoidal signal shows some minor deviations of about 0.2 µm but in general it is sufficient for an approximation regarding the surface roughness depicted in Fig. 8.

Fig. 9

Sample 14: measured with T8000 and reconstructed from scattered light data

The assumptions A2 and A4 could be validated. The assumptions A1, A3 and A5 have been investigated and continue to appear plausible but require further analysis.

5.5 Model identification for regression models

In the following, for the models M₁, M₂ and M₃, a fitting into the measurement data is carried out. The models and the identification parameters are described in Sects. 3.1 and 3.2 and should be recalled here. Since the identification parameters of all models produce a predictable effect (Fig. 2), it is possible to perform the fitting manually to have more freedom in the alignment. At the same time, an automated fit can identify the parameters to minimize the deviation between measured data and model. For M₁ and M₃ the former way is chosen, for M₂ the latter.

We start with a reduced form of M₂, which initially excludes the correction with ${F}_{P}^{1.c}$ by setting k_F = 0. By minimizing the summed deviation of model and measured data with an automated fit, the following parameter combination is obtained: k_min = 282.3, k_max = 768.6, k_g = − 9.5, k_o = 0.49, k_c = 0.47 and k_F = 0. The resulting model has a mean deviation of 38.9 MPa and is shown in Fig. 10a). If, on the other hand, the correction with F_P is included, a slightly better fit is possible. The mean deviation of the model and the measured data then is 32.1 MPa. The identification parameters change to: k_min = 263.8, k_max = 752.0, k_g = -6.8, k_o = 0.50, k_c = 0.47 and k_F = 5.0 (using four values for $\overline{{F }_{P}^{1.c}}$ obtained by averaging the trials of the first cut with the same cutting parameters). The improvement of the model by taking F_P into account provides an argument in favor of the assumption that the passive force carries information about the produced residual stress (cf. A3).

Fig. 10

Data and model in each plot: a surface residual stress after the second cut, b surface residual stress after the first cut and c roughness after the second cut

M₁ uses the five previously identified parameters of M₂ and has also two additional parameters. The parameters k_b = 0.35 and k_p = 0.95 of M₁ are chosen by hand and result in the model shown in Fig. 10b). As described in Sect. 3.1, k_b and k_p place a span of the arctangent function, which can be shifted up or down, into the measured data. k_p is chosen to achieve a reduction at ${v}_{c}^{1.c}=100$ m/min and to position k_b in the middle of the measured data. It would also be possible to make the decrease weaker or to shift the complete function curve downwards. An upward shift seems implausible because residual stress would then have to be reduced in the second cut. It should be noted that the model M₁ can only be varied to a limited extent, as it results directly from M₂ and only two parameters can be varied. The success of the fit, as demonstrated in Fig. 10b, suggests that the approach function (cf. Eq. (5)) for ${f}_{12}$ is suitable for modelling the residual stresses for consecutive cuts. Nevertheless, this statement must be questioned critically, because on the one hand only four samples were available for the fit and on the other hand the deviation between model and measurements are still significant. In particular, the deviation of the sample at ${v}_{c}^{1.c}=75$ m/min cannot be explained by the model and fluctuations of neighboring parameters cannot be explained in principle because of the continuous course of the arc tangent function. The consideration of the passive force to model such fluctuations is not enough here and further consideration in the model would be necessary.

The roughness model M₃ according to Eq. (12) has the given parameters r_ε and f^2.c as well as the identification constant chip thickness h_min. If h_min is chosen in such a way that the deviation is compensated for the samples with feed rates of 0.1 and 0.15 mm/rev, the result is h_min = 0.001 mm. This model is shown in Fig. 10c) together with the measurement data. For small feed rates, the model can predict well. For large feed rates, there is a deviation. Further tests are necessary to exclude systematic errors caused by the cutting edge.

6 Conclusion and outlook

A control concept for a two-step cryogenic hard turning process of AISI 52100 was presented. It was designed to enable the production of specimens with defined surface residual stress and roughness by adjusting cutting speed and feed rate. Six assumptions serve as basis for the concept undermining the following models.

They mathematically describe the non-linear residual stress relationship in the cryogenic process and are translated into a soft sensor for practical application. After first experiments the stated assumptions seem to be plausible and are in line with the literature but still need final validation. Especially assumptions A3 and A4 since they contain the information about the residual stresses. Three derived regression models are the key to the presented control concept. Notably, a theory was established that the residual stress from the initial cut is not completely overwritten but superimposed with a subsequent cut with a sufficiently low depth of cut ultimately enabling a control of the residual stress.

In addition, the tests were used to identify the regression models and to determine a proper fit: The residual stress model represents the real residual stresses with an average deviation of 32.1 MPa. This suggests that the non-linear relationships can be modelled and used within a model-based controller. Ultimately it can be stated that the goal of presenting a control concept for the cryogenic hard turning of AISI 52100 was reached. Even first steps to a validation have been undertaken.

Making the model more robust, the next phase involves implementing and testing the proposed control concept into our hardware. Future investigations need to expand the assumptions and models outside of the borders of this framework (e.g. other material, cutting parameters, surface layer properties). Within the experiments of this study reliable data has been acquired for the austenitic phase content as surface layer property, but it needs further investigation.