Rotary compression in tool cavity—a new ductile fracture calibration test

Ductile fracture is one of the most common failure modes in hot metal forming. It can be predicted by means of so-called damage functions that describe the relation between stress, deformation and fracture initiation. A practical use of these functions requires the knowledge of the critical damage value of the material that is determined by calibration tests based on compression, tension and torsion. For the prediction to be correct, one must ensure that the modelled and real stresses are in agreement. Previous studies did not offer any effective test for determining critical values of damage under changing load conditions that occur in cross and skew rolling processes, among others. To compensate for this knowledge gap, researchers at the Lublin University of Technology have developed a new test consisting in rotary compression of a test-piece in a cavity between the tools, which is described in this paper. In the proposed test, a cylindrical test-piece is rolled over a cavity (impression) created by grooves on two mating tools. The cavity height is smaller than the test-piece diameter. At the critical value of the forming length, the state of stress induced thereby in the test-piece axis causes fracture. Knowing the critical forming length, it is possible to determine the critical value of damage by numerical modelling. The practical application of the proposed test is illustrated through the case of C45 grade steel subjected to forming in the temperature range 950–1150 °C. The analysis makes use of the normalized Cockcroft-Latham (NCL) criterion of ductile fracture.


Introduction
One of the predominant failure modes in metal forming is ductile fracture which is associated with nucleation, growth and coalescence of voids. In the late 1960s, McClintock [1] and Rice and Tracey [2] demonstrated that ductile fracture is accelerated with an increase in stress triaxiality. According to the results obtained by Johnson and Cook [3], the effect of stress triaxiality on ductile fracture is more significant than that of temperature and strain rate. Several years later, Wierzbicki and Xue [4] showed that besides stress triaxiality, ductile fracture is also affected by the third invariant of the stress deviator.
Over the last several years, researchers have developed a number of fracture models for determining the moment of fracture initiation. These models can be classified as follows [5][6][7][8]: & phenomenological criteria, & continuum damage mechanics, & void nucleation, growth and coalescence, & porosity Among the above models, phenomenological criteria are considered the easiest to apply in practice. For this reason, they are widely used in commercial software dedicated to analysis of forming processes. The above models are based on the assumption that ductile fracture is associated with an energy change caused by the accumulation of plastic strains leading to fracture, which can be described by the so-called damage criterion. A generalized form of this criterion is expressed as: where ε f is the critical plastic strain at fracture, Φ(σ) is the function describing the effect of stress on void nucleation, growth and coalescence and C is the critical value of damage function.
There are many ductile failure criteria which differ with respect to the function Φ(σ). A comparison of these criteria can be found in the specialist literature e.g. in works by Li et al. [9] and Kraisnik et al. [10]. For the present analysis, the highly popular ductile fracture criterion developed by Cockcroft and Latham [11] and modified by Oh et al. [12] was selected, which is expressed as: where σ i is the equivalent stress and MPa; σ 1 is the maximum principal stress, MPa. The above criterion is often called in the literature as normalized Cockcroft-Latham (NCL) and this name is used throughout this paper. The practical application of the ductile fracture criterion requires the knowledge of critical damage value, which is determined by means of so-called calibration tests. Currently employed tests are based on compression, tension and torsion, and they involve the use of specimens with shapes amplifying fracture propagation. In sheet forming, calibration tests are predominantly performed by tensile testing of sheet specimens such as dog-bone specimens, notched specimens, flat-grooved specimens and shear specimens [13][14][15][16]. The critical damage values are then validated with experimental tests using e.g. the Nakajima test [17][18][19] or the Erichsen cupping test [20].
In the metal forming of solids, ductile fracture calibration tests are more difficult to perform. It is known that ductile fracture criteria can predict fracture initiation effectively if the modelled and real stresses are in agreement. As a result, compression, tensile and torsional tests are performed on testpieces of different shapes in order to vary stresses. Compression tests under negative stress triaxialities are performed on cylindrical, ring, flanged and tapered test-pieces [21,22]. To reduce the impact of friction, Aleksandrov et al. [23] recommend making a recess on the test-piece end and filling it with lubricant. It must be emphasized that if compression tests are performed in hot working conditions, it is difficult to determine fracture initiation. This results from strong thermal radiation of the material, which renders it significantly difficult to capture the initiation and propagation of ductile fracture with a high-speed camera [24]. This problem does not occur in tensile tests where the occurrence of a crack is tantamount to workpiece rupture. Test-pieces used in tensile tests can be smooth round bars or notched round bars [9]. The stress triaxiality obtained in these tests exceeds 0.3 and depends on the notch radius. The least frequently used torsional tests are performed in a two-dimensional state of stress where the stress triaxiality is equal to zero. Test-pieces can be solid [6] or hollow [5]. Sometimes, to change the existing state of stress, additional loads acting on the test-piece in the axial direction are applied [25].
Given the difficulty with determining the critical damage value in hot metalworking conditions, no wonder the number of publications devoted to this problem is scarce. This can particularly be observed with respect to cross and skew rolling processes in which variable stresses occurring in the workpiece axis lead to fracture, a phenomenon which is known as the Mannesmann effect. As shown by Yang et al. [26], in these processes, the material is prone to standard ductile fracture that can be predicted using the fracture criteria described with Eq. (1).
Regarding the use of phenomenological models of ductile fracture in the analysis of cross and skew rolling processes, it must be noted that the first work on this problem was published by Piedrahita et al. in 2005 [27]. These authors used Forge 2005 to determine the effect of basic parameters of cross wedge rolling (CWR) on the damage value calculated by the NCL criterion. In 2009, Wang et al. published the first study showing the distribution of damage on the workpiece surface in the CWR process [28]. The numerical simulation was performed based on the NCL criterion, using Deform 3D. The same criterion was also employed by Silva et al. [29], who used Forge 2008 to investigate the effect of forming angle, temperature and velocity on crack formation in CWR. The propagation of fracture was modelled by the kill element technique. The critical damage value was however determined by tensile testing, which led to differences between the numerical and experimental results. In 2014, Pater and Tofil [30] presented the results of a numerical analysis investigating the tube rolling process in the Diescher mill. Among others, the results showed the NCL ductile fracture in the axial section of the workpiece. Also in 2014, Liu et al. [31] employed the Oh criterion to analyse the relationship between the number of tool passes and the final damage value in CWR. They found that when the process is performed in two stages, it is beneficial to use the highest cross-sectional reduction ratio (approx. 75%) in the first stage. Hai and Hue [32] analysed the CWR process for a railway sleeper screw using the Johnson-Cook fracture criterion. The results of the simulation demonstrated that fracture is caused by low-cycle fatigue of the material. Another solution was proposed by Novella et al. [33], who modified the Oyane-Sato fracture criterion to include the effect of temperature and strain rate. Implemented as a subroutine to Forge 2011, this modified criterion proved to be effective in modelling fracture in the CWR of an AA6082-T6 alloy part formed at 510°C. The NCL criterion was also used to compare two cases of helical rolling of balls analysed using Simufact.Forming in [34]. The results demonstrated that the highest damage value occurs in the loci where the balls get separated. The NCL criterion of ductile fracture was also investigated with respect to skew rolling performed in a numerically controlled three-roll mill [35]. Using Simufact.Forming, the authors analysed the rolling process for producing a crankshaft preform. Bulzak et al. [36] determined the distribution of the NCL damage value in ball studs produced by CWR using two tools. The analysis was performed with the use of Deform 3D v.11. Skripalenko et al. [37] used both the same software and ductile fracture criterion to investigate the effect of feed angle on the skew rolling of bars in a two-roll mill. The CWR process for a stepped shaft described in [38] was simulated using Forge NxT1.1. The authors of the study arbitrarily assumed that the loss of material cohesion occurs when the NCL damage is equal to 2.5. As a result, they were able to model the process of cutting end scrap material with end-face cavities. The same approach was adopted in the cross rolling of balls [39] and helical rolling of balls [40]. This time the critical damage value was equal to 3. The analysis of CWR of a drive shaft [41] made use of the critical NCL criterion to predict ductile fracture in C45 steel during tensile testing performed in the temperature range of 900-1200°C. A comparison of the numerical and experimental results demonstrates that the use of the critical damage values determined by tensile tests in cross rolling processes is incorrect and leads to erroneous conclusions. Therefore, it is necessary to develop a new test for determining critical damage values under varying stress conditions, feasible to perform in hot metalworking conditions. This study proposes a test which consists in performing rotary compression in a cavity between the tools.

Principle of the rotary compression test
The term "rotary compression" refers to the type of loading in rotary forming processes such as rotary forging, rotary piercing (Mannesmann process), cross rolling and skew rolling. This type of loading can be induced using two tools (flat, concave or convex) which compress the billet, causing it to rotate at the same time. Figure 1 shows the schematic design of a rotary compression process that is performed using two flat tools, each tool having a longitudinal tapered cavity. The lower tool is stationary while the upper tool executes plane motion along the cavity. A test-piece of the dimensions d 0 × b is put inside the lower tool cavity; the test-piece diameter is bigger than the spacing between the cavity bottoms (impression height) which is set equal to 2h. In its front-end part, the moving tool is notched at an angle γ to ensure that the test-piece is radially compressed and thus cannot be pushed out of the cavity. Along with being deformed the testpiece is rotated by action of the friction forces, which causes it to roll in the tool cavity over a forming length s. The role of the cavity side walls is to prevent axial flow (elongation) of the material. During the forming process, the radial stresses in the test-piece axis oscillate from compressive (the minimum values occur vertically) to tensile (the maximum values are observed in a horizontal direction, reflecting the sliding motion of the upper tool) [42]. The stresses change twice per one revolution of the test-piece. Once the specimen has completed the limit number of revolutions, a fracture is formed in its axial zone. In the proposed test, the main task is to determine the forming length of the lower tool (forming length s) at which cracks begin to occur. Figure 2 shows the test stand for cross rolling used in experimental tests, available at the Lublin University of Technology. The rolling mill has flat tools with a maximum length of 1000 mm. The speed of the hydraulically-driven moving (upper) tool was maintained constant at 300 mm/s. Calibration tests were performed on specimens of C45 steel, each test-piece having a diameter of 40 mm and a length of 20 mm. The cavity height was set equal to 2h = 38 mm. The forming surface of the upper tool was cut at an angle γ of 3.5°. The type of material was selected on purpose, as C45 steel is widely used in machine design and has thoroughly been investigated with respect to workability, which is of vital importance in complex metal forming processes such as cross wedge rolling. Test-pieces were preheated in an electrical chamber furnace to the following temperatures T 950°C, 1000°C, 1050°C, 1100°C and 1150°C. After preheating, the test-pieces were put inside the lower tool cavity, its dimensions corresponding to those given in Fig. 1. Set into motion, the upper tool gripped the test-piece and rolled it over the lower tool. An example of the test is shown in Fig. 3. When the forming length s was higher than its critical value, a crack would occur in the axial region, as shown in Fig. 3. In such cases, the forming length s was reduced, and the test was repeated until no crack occurred on the test-piece flank (Fig. 4). After that, the test was repeated two more times, each time at the same parameters; if still no crack could be observed, the applied forming length was considered to be critical for the given billet temperature. It should be mentioned that results of the destructive tests demonstrate that fracture is initiated practically over the entire length of the test-piece axis (Fig. 5). The knowledge of this fact significantly facilitates realization of the proposed test, because then it is enough only to conduct a macroscopic examination of the test-piece flank to determine whether fracture has occurred or not.

Experimental tests
The tests also involved measuring the temperature on the testpiece surface. Figure 6 shows the temperature of the test-piece preheated to 1000°C and rotary compressed over the forming length s = 230 mm. The results demonstrate that despite the relatively long forming time, the temperature on the specimen flank, in the axial zone of the test-piece, remains close to the initial temperature of the billet. The layers of the material located on the circumference of the test-piece undergo cooling due to contact with the cold tools. This particular distribution of temperature is desired for the test because the temperature at the locus of fracture remains relatively constant.
The loading force acting on the moving (upper) tool was measured on the test stand. Figure 7 shows the loading force variation in rotary compression performed at T = 1000°C and s = 230 mm. An analysis of the loading force variation reveals that the force reaches the highest value at the beginning of the forming process when the test-piece undergoes considerable ovalization. After that, the loading force decreases and remains relatively stable until the end of rotary compression. It is worth emphasizing the fact that the loading force is relatively small in the rotary compression test.
where σ F is the flow stress, MPa; ε is the effective strain, −; ε is the strain rate, s −1 ; T is the temperature,°C. The GMT model used in the analysis was taken from the Simufact.Forming material database containing material data imported from MatILDa, which is a registered trademark of GMT (Gesellschaft für metallurgische Technologie-und Softwareentwicklung mbH, Berlin). Friction on the test-piece-tool contact surface was modelled as constant friction described with the following relation: where τ is the shear stress on contact surface, MPa; m is the friction factor (set equal to m = 0.8), −; k is the yield stress at pure shear (k ¼ σ F = ffiffi ffi 3 p Þ, MPa. The analysis took account of thermal phenomena occurring in the forming process. They depended on the following parameters: test-piece temperature (950°C, 1000°C, 1050°C, 1100°C, 1150°C), tool temperature (50°C), ambient temperature (20°C), material-tool heat transfer coefficient (10,000 W/m 2 K) and material-environment heat transfer coefficient (200 W/m 2 K).
The material was modelled with the use of hexahedral finite elements. The size of a single element was set equal to 1 mm. In total, more than 26,000 finite elements were used to model the test-piece. An automatic remeshing tool was used to reconstruct the mesh at every jump of the strain rate by 0.4. The boundary conditions were defined by the shape of the tools that were modelled as ideally rigid bodies. The implicit finite element solver was used in the calculations. Figure 9 shows the rotary compression process performed at T = 1000°C and s = 230 mm. It can be observed that during the forming process, the test-piece assumes an oval shape characteristic of cross and skew rolling processes. This shape must be maintained in order to induce in the test-piece axis the type of stress that will trigger the Mannesmann effect. To this end, it is necessary to prevent elongation of the test-piece, which can be ensured via action of the side walls of the tool cavity (impression). The circumferential metal flow in the outside layer is accompanied by the displacement of the material in the central region of the test-piece toward its centre, which leads to the formation of funnels on the test-piece flanks. As a result of such metal flow, the test-piece thickness in its axis is reduced (the bigger the reduction, the longer the  Obtained effective strains of the test-piece preheated to 1000°C after compression over the forming length s = 230 mm are shown in Fig. 10. The plot reveals that the highest strains occur on the test-piece flanks i.e. in the region where the test-piece is in contact with the working planes of the cavity. In contrast, the strains in the test-piece axis are almost homogenous, which can be considered positive from the point of view of the present analysis. A slight increase in the strains can only be observed on the test-piece end-face. Figure 11 shows the damage function as calculated by the NCL criterion expressed with Eq. (2). The plot confirms that in the analysed test, fracture will occur in the test-piece axis. The damage function is the highest in the centre of the testpiece, and decreases the closer it is to its end-face.
Interesting observations can be made with respect to temperature distribution in the rotary compressed test-piece. An analysis of the temperature shown in Fig. 12 reveals that due to contact with the tools, the outside layers of the material undergo cooling, and their temperature drops by 100°C or even more. At the same time, the temperature in the test-piece centre increases, which can be associated with the change of deformation work into heat. The numerical test-piece temperatures agree very well with the experimental findings (Fig. 6). This clearly proves that the temperature slightly increases in the axial zone of the test-piece during rotary compression. Figure 7 shows the FEM results of the loading force obtained for the analysed case of rotary compression (performed at T = 1000°C and s = 230 mm); the figure also shows the experimental loading force. The experimental and numerical loading forces show good qualitative agreement. For the purpose of a quantitative comparison, the loading force results    were used to determine deformation work for the upper tool velocity set equal to v = 300 mm/s. The experimental deformation work was 4.77 kJ, while the numerically calculated deformation work was 4.30 kJ. Hence, the difference between the results is 9.8%. It must be stressed that the experiments involved measuring the loading force on the tool that equalized the deformation force (simulated) and the force associated with overcoming motion resistances generated on the slide guides (not included in the simulation).
Considering the flank temperature and loading force results, one can state that the developed numerical model is a good representation of the real process. Therefore, it can be claimed that the numerical model can be used to determine the critical damage by the rotary compression in tool cavity test.

Results and discussion
The experimental tests led to determination of the critical forming length at which fracture takes place. Obtained results are plotted in Fig. 13 and demonstrate that the critical forming length s increases with increasing the temperature. In addition to this, the experimental results demonstrate that at lower forming temperatures cracks propagate rapidly, while in the case of test-pieces preheated to high temperatures, the crack propagation rate is slower.
It should be stressed that the accuracy of the proposed calibration test strongly depends on the accuracy of calculating the critical forming length s. Although the cracking is initiated in the centre of the test-piece, it rapidly propagates into its lateral surfaces. This results from the small thickness of the test-piece, which is additionally reduced due to the formation of cavities on the lateral surfaces. This pattern of crack propagation occurs during a linear displacement of the test-piece over a few millimetres length. Thanks to the research method employed in this study, according to which the fracture initiation was associated with the propagation of cracking into the lateral surface of the test-piece, it was considerably easier to conduct the experimental tests. To increase the accuracy of determining the critical damage, the experimental forming length should be increased only to a very small extent. This, however, will entail more experimental tests and higher research costs.
Knowing the critical forming length s, it was possible to numerically model the rotary compression in cavity test. Calculations were made separately for every tested test-piece temperature, assuming that in every tested case the forming Fig. 11 Damage distribution (as measured by NCL criterion) in a test-piece preheated to T = 1000°C and rotary compressed over a forming length of 230 mm Fig. 10 Effective strain distribution in a test-piece preheated to T = 1000°C and rotary compressed over a forming length 230 mm process is realized over the critical length s. To give an example, for the test-piece preheated to T = 1000°C the critical forming length is s = 230 mm (Fig. 13).
For the purpose of quantitative assessment, 11 virtual sensors were located every 2 mm in the test-piece axis to measure individual parameters. Figure 14 shows the effective strain variations calculated with the sensors, the location of which is also marked in this plot. An analysis of the effective strains demonstrates that they increase proportionally in the time range from 0.2 to 1.6 s. In the range 0-0.2 s, the upper tool moves freely, without any contact with the test-piece. On the other hand, for the time t > 1.6 s, the strains do not change, which results from that fact that the test-piece is no longer in contact with its rotation-inducing upper tool. The effective strains in the test-piece axis range 1.1-1.2 approx., achieving higher values at the test-piece surface rather than in its axis.
Interesting observations can be made analysing variations in the principal stress recorded by the virtual sensors for the specific time of rotary compression i.e. t = (0.2-1.6 s), which are plotted in Figs. 15, 16, and 17. The plots reveal the presence of a complex stress state in the test-piece axisbiaxial tension coupled with uniaxial compression inside the testpiece. It can be observed that the further it is from the testpiece end-face, the higher the tensile stresses and the lower the compressive stresses become. At the same time, pure torsion occurs on the end-face in the test-piece axis, which is proved by the fact that the intermediate principal stress determined with sensors 1 and 11 is equal to zero.
As mentioned in the introduction, ductile fracture depends on the stress invariants; it was therefore considered justified to show variation in the stress triaxiality η and the Lode parameter Θ at individual measurement points. The abovementioned parameters were determined using the following equations: where p, q, r denote the stress invariants that are calculated using the equations: where σ 1 , σ 2 , σ 3 are the principal stresses, σ m is the mean stress and σ i is the effective stress. Figure 18 illustrates the stress triaxiality whichdepending on the sensor location-ranges from − 0.08 to 0.46. The stress triaxiality is higher in the test-piece centre, where its values are close to those obtained by uniaxial tensile testing. An analysis of the Lode parameter variations recorded with individual sensors (Fig. 19) shows that for the principal forming zone, this parameter ranges (− 0.4; 0.1). Such values of this parameter point to the presence of strong shear stresses in the test-piece axis. Summing up this part of the analysis, it can be stated that the stresses in the axis of the test-piece subjected to the rotary compression in tool cavity test differ from those occurring in compression, tensile and torsional tests that were previously used for determining critical values of the damage function. Figure 20 shows the numerically simulated variations in temperature at individual measurement points. The variations demonstrate that the temperature increases during the forming process up to even 20°C, depending on the sensor location. This increase is caused by the change of deformation work into heat. Due to the fact that the test-piece cool-down   resulting from its transfer from the furnace to the test stand was omitted in the numerical simulation, the critical damage function was ultimately determined based on the assumption that the temperature in the test-piece axis is equal to the assumed billet temperature.
The increase of the damage function in the test-piece axis (Fig. 21) is similar to the behaviour pattern obtained for the effective strains (Fig. 14). Nevertheless, one can observe bigger differences between the damage function values recorded with the sensors located in the test-piece centre and those positioned on its ends, which results from more intensive tension in the test-piece centre.
To determine the critical damage of the material in a given temperature, the damage function values recorded with individual sensors were averaged. In this way, the plot shown in Fig. 22 was obtained. The data given in this plot clearly indicate that an increase in the billet temperature leads to an increase in the critical value C which-for the tested temperatures-ranges from 0.922 to 3.913.
A flowchart explaining the procedure for determining the critical damage in the rotary compression test is shown in Fig. 23. The determination of the critical damage function (for a given material and a forming temperature T) requires the assumption of the input forming length applied in the rotary compression test. If the fracture occurs in the test-piece, the forming length must be gradually decreased and the calibration test must be repeated until the cracking no longer occurs. If the fracture does not occur in the test-piece during the first test run, the forming length must be increased until the occurrence of cracking. Having the critical forming length s established in this way, it is now necessary to model the rotary compression test numerically in order to determine the mean damage value in the test-piece axis. This value will be equal to the required critical damage value C.
The critical values C calculated by the rotary compression test are much higher and more dependent on the temperature than the critical values determined in the tensile test of notched bars. In accordance with the experimental results presented in [41], the critical NCL damage determined in the tensile test is C = 0.756 ± 0.125 (for C45 steel specimens subjected to tensile testing in the temperature range of 900-1200°C). By applying the critical damage values established in the rotary compression test to predict fracture in the CWR case described in [46], one would ensure agreement between the numerical and experimental results.
To facilitate the practical application of obtained results, the critical damage C of C45 steel is described with the following equation dependent on the material temperature T: which is suitable for the temperature range of 950-1150°C.

Conclusions
The following conclusions have been drawn from the study: & in the proposed rotary compression in tool cavity test, material damage (fracture) occurs in the test-piece axis as a result of the so-called Mannesmann effect; & despite a relatively long duration of the test (for the forming velocity v = 300 mm/s), the temperature in the test-piece axis is similar to that of the billet; & the state of stress in the test-piece axis is characterized by the presence of maximum and intermediate principal tensile stresses as well as minimum principal compressive stresses, which causes intense shear of the material; & to determine the critical damage value by means of the proposed rotary compression test, one must know the forming length value at which fracture initiates. In the present analysis, it was the highest forming length value for a given temperature at which no fracture was identified in three repeated tests; & the critical damage value determined by the rotary compression in tool cavity test strongly depends on the temperature of the material; increasing this temperature leads to an increase in the limit damage value; & in the hot forming of C45 steel, it is recommended using Eq. (10) that describes the critical damage based on the Cockcroft-Latham ductile fracture criterion. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.  Fig. 23 Flowchart describing the procedure for determining the critical damage value by rotary compression in tool cavity