Keywords

1 Introduction

The production of components by flow forming holds a remarkable importance in a wide range of industrial fields due to its manufacturing flexibility, its resource optimization, and the possibility of producing components with not only a desired geometry but also specific mechanical properties [1]. The use of austenitic stainless steel as a raw material in combination with advanced manufacturing techniques makes the production of high-performance products possible. During the metal-forming of metastable austenitic steel, plastic deformation not only changes the geometrical characteristics but also the microstructure. In particular, a strain-induced phase transformation from metastable austenite to α’-martensite may occur; this modifies the magnetic and mechanical properties [2, 3].

Modeling in metal forming processes plays a crucial role in the evolution, optimization, and description of the process limits. Analytic and simulative modeling approaches include methods like slab, slip-line field, upper bound, finite difference, and finite elements [4, 5]. Some investigations focus on material behavior and the formability of sheet metal [6, 7]. Other studies have described models of phase transformation in metal-forming processes [8,9,10]. Regarding the modeling of α’-martensite phase transformation during plastic deformation, some general mathematical models can be found in the literature [3, 11, 12]; among these, the most representative is the Olson-Cohen model [13].

The primary goal of the overall research project is to develop a property-controlled flow-forming process. The implementation of online closed-loop control within metal forming processes enables the application-oriented and efficient production of high-quality components [14]. The control systems require the development of novel soft sensors to monitor the evolution of the properties during the plastic deformation. By definition, a soft sensor allows for the transformation of measured secondary variables into more useful process knowledge by means of a model [15]. A soft sensor integrates a hardware part (which, in this case, is a micromagnetic sensor device) and a software part (or soft sensor model) to transform the measured signals into numerical values of the variable(s) of interest. In this work, a soft sensor model for detecting α’-martensite evolution from measurable micromagnetic properties in combination with process parameters was developed. To achieve this, optimization by means of a non-linear regression of experimental data was carried out. This kind of optimization has been widely used in many branches of science in such applications as economics, operations research, network analysis, and the optimal design of mechanical or electrical systems [16]. In a material-behavior model during metal forming, this represents a novel approach.

2 Materials and Methods

2.1 Specimen Production by Means of Reverse Flow Forming

The specimens were produced using a PLB 400 single roller spinning machine that was manufactured by Leifeld Metal Spinning GmbH (Ahlen, Germany). The raw material for the specimens consisted of seamless tubes that were made of metastable AISI 304L austenitic steel (X2CrNi18-9, 1.4307) with an outer diameter of 80 mm. The specimens were produced at room temperature with a constant rotational speed of n = 30 rpm. The flow-forming operation was performed using a single roller arrangement in order to reduce the wall thickness of the tubes (Fig. 1a). To produce specimens that combined different geometrical features and properties, the following process parameters were used: infeed depths (the positions of the roller tool on the y-axis) of 1, 2, and 3 mm, and feed rates (the speeds of the roller tool in the axial direction [x-axis]) of 0.1, 0.2, 0.3, 0.4, and 0.5 mm/s (Fig. 1b). To ensure the reliability of the results, each flow-forming experiment was repeated three times.

Fig. 1
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Setup for production of specimens: a before reverse flow forming; b after reverse flow forming

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Figure 2a illustrates the dimensions of the produced specimens. The initial wall thickness (wIC) was reduced according to the three different infeed depths that were used to produce the resulting wall-thickness reduction (Δw). The final appearance of each of the specimens is shown in Fig. 2b.

Fig. 2
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Workpiece geometry: a dimensions; b final appearance

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Starting from the initial condition (IC), which corresponded to the metastable austenite, the forming process generated the illustrated forming zone (FZ). For each specimen, the forming zone corresponded to a plastic deformation state with a defined amount of strain-induced α’-martensite depending on the process parameters.

2.2 Procurement of Experimental Data

The experimental data that were used in the modeling of the soft sensor were obtained by means of characterizing the evolution of geometrical features and properties like wall-thickness reduction (Δw), α’-martensite amount, roughness, and the micromagnetic parameters of the workpieces. For this investigation, a total of 45 specimens were produced: three infeed depths × five feed rates × three repetitions.

Figure 3 illustrates the points that were measured to obtain the experimental data. Each specimen was measured around four equally spaced axes (A–D). Along each axis, six points were recorded for the forming zone (FZ) and one for the initial condition (IC). In summary, 28 measurements were carried out on each specimen, which adds up to a total of 1260 measurements for each characterization technique.

Fig. 3
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Points measured for acquisition of experimental data

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The conducted characterization techniques for acquiring the experimental data are described as follows:

Wall-thickness Reduction, Average Roughness, and α’-Martensite Measurements. The wall-thickness reduction (Δw) was measured offline on the produced specimens by using a D4R50 mechanical gage (Kroeplin GmbH, Schluechtern, Germany). The arithmetic mean roughness (Ra) of the formed specimens was measured with a MarSurf M300 tactile device (Mahr GmbH, Goettingen, Germany).

The α’-martensite content measurements were used as reference values in order to characterize the phase transformation during the plastic deformation of the material. The data was recorded by means of an FMP30 Feritscope (Helmut Fischer GmbH, Sindelfingen, Germany). This device works under the magneto-inductive principle to measure the ferritic phase content, which is correlated with the strain-induced α’-martensite (according to [17]). These measurements could not be used for closed-loop property control, as the online acquisition and real-time transmission of the data to a control system was not possible. As a result, the use of micromagnetic testing was a convenient alternative as a soft sensor for property control.

Micromagnetic Measurements. By means of the magnetic Barkhausen noise (MBN) analysis, different material features such as lattice defects, dislocation density, phase changes, grain sizes, and residual stresses could be detected. In previous publications, the suitability of MBN testing has been demonstrated for detecting the amounts of strain-induced α’-martensite phases—especially by means of the maximum amplitudes of the MBN profiles (Mmax) [18,19,20]. MBN testing has been used by many authors for detecting ferromagnetic α’-martensite phase amounts [21,22,23]. In this work, the micromagnetic measurements were performed using the 3MA-II system (Fraunhofer IZFP, Saarbruecken, Germany), with the SN18201sensor in contact with the outer surfaces of the specimens. The measurements were carried out by emulating the real conditions for the online measurements. This meant protecting the sensor head against wear with two layers of Kapton bands and the appropriate testing parameters for the online application. The micromagnetic parameters for the measurements were as follows: a 200 Hz magnetization frequency, an 80 A/cm magnetization amplitude, a 15° magnetic phase offset, a 20 dB gain, and without bandpass filters on. This allowed us to perform an integral measurement of the entire wall thickness.

3 Results and Discussion

3.1 Wall Thickness, Roughness, α’-Martensite Content, and MBN Signals

As mentioned in Sect. 2.2, four different characterization methods were used on the 45 specimens in order to obtain the experimental data for the soft sensor model. A summary of the quantitative results of the measurements is presented in Table 1.

Table 1 Quantitative results for characterizations of wall-thickness reduction (Δw), α’-martensite content (α’), and maximum amplitude of MBN (Mmax) and their respective standard deviations for flow formed workpieces
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3.2 Correlation Between α’-Martensite, Mmax, and Feed Rate

Figure 4a illustrates the evolution of the α’-martensite content with respect to the wall-thickness reduction (Δw). Keeping constant production parameters like the feed rate and varying infeed depth, more α’-martensite was present on the microstructure of the flow formed workpieces due to the increased plastic deformation with higher Δw. The lower feed rates of the roller tool on the x-axis during the flow forming process (see Fig. 1) entailed the higher local deformation of the areas that were in contact with the roller tool; this led to increased α’-martensite formation. In this way, the strain-induced deformation promoted the transformation of the austenite into α’-martensite. These aspects have been discussed in previous publications (e.g., [18, 19]).

Fig. 4
figure 4

a Correlation between α’-martensite and wall-thickness reduction for flow formed specimens at different feed rates; b influence of feed rate on average roughness of specimens

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Evaluating the roughness on produced specimens is crucial for applying non-destructive testing by means of MBN measurements. As has been demonstrated in earlier papers, higher roughness values increase the numbers of detected MBN signals [24, 25]. Since roughness cannot be measured online during the production process, their effects can be considered by means of other variables. The feed rate of the roller tool during the flow forming of the specimens has a great influence on the roughness of flow-formed components. From our numerical results (Table 1, Fig. 4b), higher feed rates increased the average roughness (Ra). At lower feed rates, the distance between the microgrooves that were produced during the flow-forming process was smaller; this entailed lower roughness values. For this reason, the feed rate was chosen as an important production parameter for the soft sensor modeling.

Figure 5 illustrates the correlation between the α’-martensite content and the MBN measurements (specifically, Mmax); the data is plotted for feed rates between 0.1 and 0.5 mm/s. The results showed a good correlation for feed rates between 0.1 and 0.3 mm/s; however, the separation between the measured data was not clear enough for feed rates of 0.3 through 0.5 mm/s. In Fig. 6, a 3D plot of the experimental data allows one to interpret the results in a clearer way.

Fig. 5
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Correlation between α’-martensite and maximum MBN amplitude Mmax for flow-formed specimens at different feed rates

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Fig. 6
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Three-dimensional plot of correlations among α’-martensite, maximum MBN amplitude Mmax, and feed rate for flow-formed specimens

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3.3 Non-linear Regression by Means of Data-Fitting Optimization

Definition of Ansatz Function. The Ansatz function must be defined as an initial assumption for solving the problem. In this case, the starting equation must describe the behavior of the quantity of interest (namely, the α’-martensite) with respect to secondary measurable quantities like maximum MBN amplitude (Mmax) and process parameters like the feed rate. As previously discussed, the selected feed rate for the production of specimens had a remarkable influence on the roughness of our specimens; this also had an impact on the measured Mmax signals.

In Fig. 7a, a non-linear regression of the experimental data of the α’-martensite with respect to Mmax was carried out using the Simple Fit tool from OriginPro (2021b). A good fitting of the points was obtained by means of an exponential function. Similarly, a potential function performed the best description of the behavior of the α’-martensite with the feed rate. The product of both functions was expected to generate the combined effect of the variables. The final Ansatz function is expressed in Eq. 1.

Fig. 7
figure 7

Non-linear regression based on experimental data for α’-martensite content depending on a maximum MBN amplitude Mmax and b feed rate f

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$$\mathrm{\alpha }{\prime}={\upbeta }_{1} \left(1 - {{\upbeta }_{2}}^{{{\text{M}}}_{{\text{max}}}}\right) \left( {{\text{f}}}^{{\upbeta }_{3}}\right)$$
(1)

The problem is now to find the numerical values of the βn coefficients that better fit the Ansatz equation to the experimental data.

Derivatives of Ansatz Function. To calculate the numerical values of the βn coefficients in Eq. 1, it is necessary to compute the partial derivatives of the Ansatz function with respect to each coefficient. The partial derivatives are detailed in Eqs. 2 through 4.

$$\partial \alpha /\partial {\upbeta }_{1}=\left(1 - {{\upbeta }_{2}}^{{{\text{M}}}_{{\text{max}}}}\right) \left( {{\text{f}}}^{{\upbeta }_{3}}\right)$$
(2)
$$\partial \mathrm{\alpha }/\partial {\upbeta }_{2}=-{\upbeta }_{1} \left({M}_{{\text{max}}}\right) \left({{\upbeta }_{2}}^{{{\text{M}}}_{{\text{max}}}-1}\right) \left( {{\text{f}}}^{{\upbeta }_{3}}\right)$$
(3)
$$\partial \alpha /\partial {\upbeta }_{3}={\upbeta }_{1} \left(1 - {{\upbeta }_{2}}^{{{\text{M}}}_{{\text{max}}}}\right) \left( {{\text{f}}}^{{\upbeta }_{3}}\right) \left(ln\left({\text{f}}\right)\right)$$
(4)

Optimization by Means of Objective Function. The definition of an objective function or cost function is crucial for solving the optimization problem. The problem is formulated in terms of a real function of several real variables. The goal of the optimization is to find a set of arguments that provide a minimal function value [16]. For this optimization, the convergence is defined by means of Newton’s method, where f is the objective function, and \(\widehat{{{\varvec{\upalpha}}}^{\boldsymbol{^{\prime}}}}\) is the minimizer. In the implementation, the objective function (Eq. 5) and its derivatives (Eq. 6) are used.

$${f}_{n+1}={f}_{n}+1/2 {\left({\mathrm{\alpha }^{\prime}}_{n}-\widehat{{\mathrm{\alpha }}^{\prime}} \right)}^{2}$$
(5)
$$d{f}_{n+1}=d{f}_{n}+ \left({\mathrm{\alpha }^{\prime}}_{n}-\widehat{{\mathrm{\alpha }}^{\prime}} \right) d{\mathrm{\alpha }^{\prime}}_{n}$$
(6)

MATLAB Implementation. The implementation in MATLAB for the solution of the problem involves the inclusion and plotting of the experimental data, the formulation of the Ansatz function (and its derivatives), and the convergence function. To compute the unknown βn coefficients, the initial values were obtained by means of the non-linear regressions that were discussed previously in this section (Fig. 7). The fmincon solver was used to perform the optimization and find the quantitative values of the coefficients.

Surface and Equation of Soft Sensor Model. The numerical results of the unknown βn parameters that were calculated by means of the implementation in MATLAB were β1 = 37.9134, β2 = 0.6495, and β3 = -0.534. Including the numerical values of the coefficients in the Ansatz equation, the soft sensor model can be expressed by means of Eq. 7.

$$\mathrm{\alpha }^{\prime}= 37.9134 \left(1 - {0.6495}^{{{\text{M}}}_{{\text{max}}}}\right) \left( {{\text{f}}}^{-0.5309}\right)$$
(7)

Figure 8 shows a 3D plot of Eq. 7. The experimental data is plotted for a qualitative comparison to the soft sensor model.

Fig. 8
figure 8

Surface of soft sensor model resulting from data-fitting optimization by means of non-linear regression of experimental data (experimental data plotted with symbols)

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Quantitative Error Analysis of Soft Sensor Model with Respect to Experimental Data. The deviation of the α’-martensite that was calculated with the model and the measured experimental data is plotted in Fig. 9. The quantitative values of the maximal deviation between the calculated and experimental data of the α’-martensite are summarized in Table 2. The highest deviation was 13.4 α’-%; however, most of the deviation values were below 6%.

Fig. 9
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Deviation plot of α’-martensite calculated by model vs. experimental data

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Table 2 Deviation and root mean square error (RMSE) of soft sensor model with respect to experimental data for different feed rates
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The root means square error (RMSE) of the soft sensor model was calculated for the group of data on each value of the feed rate according to Eq. 8; these results are presented in Table 2. The calculated errors were below 5α’-%. It can be concluded that the α’-martensite amount that was calculated with the soft sensor model was accurate with respect to the experimental data. A new challenge for applying the soft sensor model consisted of comparing the online measurements that were recorded under real production conditions with the offline α’-martensite measurements that were taken on the produced specimens. This is the topic of the next section.

$$RMSE=\sqrt{\sum_{i=1}^{N}{\left({\mathrm{\alpha }^{\prime}}_{{\text{calculated}}}-{\mathrm{\alpha }^{\prime}}_{{\text{experimental}}}\right)}^{2}/N}$$
(8)

3.4 Online Validation of Soft Sensor Model

Since the goal of the soft sensor is to apply it in property-controlled flow-forming processes, validation under real production conditions is necessary. Figure 10 illustrates the setup for the online measurement of α’-martensite by means of the soft sensor model. For the experiments, the physical 3MA-II sensor was coupled with the soft sensor model, which was implemented in the machine control via a TCP/IP approach that was developed by the authors (see [26]). The micromagnetic soft sensor parameters were set to be equal to the characterization that was described in Sect. 2.2. The online measurements were recorded during the formation of the AISI 304L tubes with a constant infeed depth of 2 mm and feed rates of 0.1, 0.3, and 0.5 mm/s (Tests 1, 2, and 3, respectively). The data was recorded along the x-position. The flow forming process required a liquid coolant to refrigerate the workpieces and keep them at a constant temperature; this coolant had no remarkable influence on the measured signals.

The results of the online measurements of the α’-martensite amount by means of the soft sensor are plotted in Fig. 11 versus the x-position (see Fig. 10—right). The highest α’-martensite amounts were detected in Test 1, which was performed with the lowest feed rate of the roller tool. The lowest values of α’-martensite corresponded to Test 3 due to the high feed rate that was used during the production of the components. In all of the performed tests, the infeed depth of the roller tool remained constant (at 2 mm of the outer surface of the tubes). The results of the online measurements under the real production conditions agreed with the measurements that were performed on the specimens that were produced to generate the experimental data that was used to develop the soft sensor model (see Sect. 3.2).

Fig. 10
figure 10

Setup for online measurements of α’-martensite by means of soft sensor model using 3MA-II micromagnetic sensor during flow forming of workpiece with infeed depth of 2 mm and feed rates of 0.1, 0.3, and 0.5 mm/s

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Fig. 11
figure 11

Online measurements of α’-martensite by means of soft sensor model during flow forming of workpiece with infeed depth of 2 mm and feed rates of 0.1, 0.3, and 0.5 mm/s

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The produced specimens were measured offline with the FMP30 Feritscope to determine the reference values of the amounts of α’-martensite on each test. To increase the resolution of the α’-martensite measurements, the measured points along the x-position and around the A-D axes were increased with respect to the measurements that were performed in Sect. 2.2 (see Fig. 3). The results of the offline measurements (plotted in Fig. 12) corresponded well with the online measurements that used the soft sensor model.

Fig. 12
figure 12

Offline measurements of α’-martensite by means of Feritscope after flow forming of workpiece with infeed depth of 2 mm and feed rates of 0.1, 0.3, and 0.5 mm/s

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Qualitatively, the measured α’-martensite amount for each test could be well-differentiated. The results of Test 1 showed the greatest scattering in both the online and offline measurements; in both cases, the data was between 45 and 65 α’-%. However, peaks of up to 80 α’-% were recorded between 0 and 40 mm of the x-position in the online measurements. These could be the effects of surface roughness areas that were considered to be disturbances. In Test 2, the drop that was measured offline at 15 mm in the x-position was not detected in the online measurements. In this region, the lower α’-martensite amount could not be detected on the online measurements due to the large measuring spot of the micromagnetic sensor that was used in the soft sensor (which produced delays in the detected signals). The online and offline signals that were measured in Test 3 corresponded very well. Some peaks were detected in the online measurement from 70 mm in the x-position; these can be evaluated as measurement noises since they were not representative.

A quantitative analysis was carried out with the average values of the α’-martensite amount that was calculated for each test of the online and offline measurements; these results are reported in Figs. 11 and 12 and summarized in Table 3. The maximal deviation of the α’-martensite amount could be found in Test 2 due to the non-detected drop in the online measurement at 15 mm of the x-position. These deviations were acceptable within the window operation of the soft sensor since the repeatability of the measurements that were made with the Feritscope reached up to 5 α’-%.

Table 3 Average α’-martensite and deviation measured online with soft sensor model compared to offline reference measurements with Feritscope for different feed rates
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3.5 Future Perspective for Application of Soft Sensor in Closed-Loop Property Control

The proposed soft sensor can be prospectively used inside a closed-loop property control of the α’-martensite volume fraction during the flow-forming process (as shown in Fig. 13). As discussed, Mmax is measured online during the manufacturing of the specimen and transformed to the α’-martensite volume fraction by the soft sensor model. Subsequently, the control variable α’-martensite volume fraction is compared to the desired α’-martensite value by the controller. The correction signal from the controller output directly manipulates process parameters such as the feed rate and infeed depth or the tool trajectory of the flow-forming machine during the manufacture of the workpiece. Thus, the α’-martensite fraction of the manufactured forming part is set to the desired value. The wall-thickness closed-loop control [26] complements the soft sensor-based control of the α’-martensite to concurrently ensure the desired geometry of the part. Overall, the closed-loop property control is, therefore, a multi-variable control. For the design of such a control, additional numerical model-based approaches can be used in terms of control-oriented real-time simulation models (cf. [26, 27] for further information).

Fig. 13
figure 13

Soft sensor-based multi-variable control concept of closed-loop property control during flow forming of metastable austenitic stainless steels (note for simplicity, feed-dependence of soft sensor model is not drawn)

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4 Conclusions

To achieve the property-controlled production of flow formed components, a soft sensor model was proposed and validated by means of two strategies. The soft sensor aims to compute the variable of interest for the property control (namely, the α’-martensite content), which is not directly measurable within the process. Secondary measurable variables like the maximum amplitude of the magnetic Barkhausen noise were successfully correlated with the variable of interest. The influence of process parameters such as the feed rate were included in the soft sensor model, as they play a crucial role in the forming process, the phase transformation of the material, and the online measurements. The model was obtained through optimization by means of a non-linear regression of experimental data. The validation of the soft sensor model first took place with respect to the experimental data that was used on the modeling and then online under real flow forming-process conditions. The influence on the measurements of different factors like the liquid coolant was evaluated.

The developed soft sensor model requires additional measurements that combine different production parameters and strategies in order to complete the validation for use on a property-controlled production. The operation of the sensor under real conditions demands an investigation of the compensations of any disturbances of the real process. There is also an intent to look for an alternative instrumentation for detecting α’-martensite with lower sensitivities to disturbances (such as the roughness of a specimen). In this sense, sensors and devices working under eddy-current principles are promising.