Flow curves up to high strains considering load reversal and damage
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Abstract
The new idea is to produce specimens by forward rod extrusion, where in the core of the extrudate a deviatoric tensionloading is present, which is superposed by an adjustable hydrostatic pressure. Various damage levels are hence possible in the extrudate. Conducting tensile and upsetting tests with the prestrained specimens both the influence of a load reversal as well as the material weakening through ductile damage on the resulting flow curve is explored. Not only can the results be utilized to identify flow curves of materials up to high strains (ε > 1.7), but also to get new insights into the plastic material behaviour, which can be used for generating or adapting new damage models as well as kinematic hardening models under cold forging conditions. The proposed method was first assessed by means of analytical and numerical methods and then validated experimentally, by the example of the typical cold forging steel 16MnCrS5.
Keywords
Flow curve Forward rod extrusion Damage Bauschinger effectIntroduction
The flow curve is the relation between the flow stress and the true plastic strain. In the field of metal forming, flow curves are necessary for the prediction of forming forces, tool deflection, material flow as well as the resulting product properties of the produced parts. According to Bridgeman [1] the formability of metals increases significantly under hydrostatic pressure. Since in cold forging, the hydrostatic pressure is usually large, a high variety of complex part geometries can be produced. In order to predict the plastic material behaviour under cold forging conditions by means of numerical analyses with sufficient accuracy, the flow curve needs to be defined up to the large strains that actually occur during the forming process.
For bulk materials, the most common experimental methods used to determine flow curves are tensile tests, upsetting tests and torsion tests. In the case of tensile tests the maximum plastic strain is given by the onset of necking, when the true strain ε is equal to the corresponding hardening exponent n of the workpiece material. After necking the material plastically deforms further but in an inhomogeneous manner, which makes the evaluation of the flow stress difficult. Several authors have tried to take into account the plastic flow in the postnecking regime. Bridgman [2] and Mirone [3] have achieved this by means of analytical descriptions of the postnecking geometry, while other authors like Kajberg and Lindkvist [4] and Kim et al. [5] focused on the use of inverse methods based on tracking and modelling the evolution of the displacement field during deformation of the tensile specimens. Latter publication gives a good overview of methods to characterize the strain hardening behaviour in the postnecking regime. In upsetting tests, a cylindrical specimen is compressed between two parallel dies. The limit for flow curve evaluation is given by the occurrence of barrelling, caused by the friction between the dies and the specimen. In the torsion test a cylindrical bar or pipe is twisted. The main difficulty here, is to calculate representative shear stress and shear strain from experimental data, due to the statically indeterminate nature of the process.
All of the beforementioned methods for flow curve evaluation are either strictly limited by the maximum amount of plastic strain or include uncertainties regarding the actual stress and strain distributions within the specimen. Reicherter [19], Siebel [20] and Sachs [21] have taken into account the frictionrelated stresses in upsetting tests by means of analytical models based on the mechanics of the process. While a certain improvement of the flow curves is possible by mathematical methods, additional uncertainties arise, regarding the underlying friction models and the unknown specimen geometry due to barrelling. Pöhlandt et al. [22] have proposed a method to calculate a critical surface distance in torsion tests, to improve the estimation of the shear strain.
Summaries of experimental procedures for flow curve evaluation in bulk forming are given, e.g. by Krause [23], Nebe and Stenger [24], Pöhlandt [25] and Doege et al. [26].
So far, only continuous experimental procedures for flow curve evaluation were presented. Another approach for the evaluation of flow curves for high strains is the conduction of intermittent procedures. Herein, usually the goal is to produce specimens with a large known prestrain. According to Sevillano et al. [27] it should be stressed that in the utilization of intermittent procedures, the influence of load path changes should be given special care, as changes in the stress state or strain rate may have a significant influence on the workhardening. The author gives a comprehensive literature review of intermittent procedures for flow curve evaluation. Among these, Langford and Cohen [28] have conducted multipass wiredrawing on lowalloyed steel wires in order to prestrain the material by ε = 0.22 in each pass. By the use of more than 30 passes, the authors were able to reach a total maximum prestrain of ε ≥ 7. By the conduction of tensile and upsetting tests on the wires, they were able to generate the corresponding flow curve. Pöhlandt [29] has exploited the steadystate properties of forward rod extrusion for the production of prestrained specimens for the subsequent conduction of upsetting tests for the first time. The author observed large deviations between the results of upsetting tests on undeformed material and its corresponding extrapolation as well as on the specimens prestrained by forward extrusion. The author suggested, that due to forward rod extrusion, the formed shaft possesses an inhomogeneous strain distribution over the shaft radius which is caused by shearing in the vicinity of the shaft surface, leading to a larger overall flow stress. Doege et al. [26] pointed out that the unusual shape of the resulting flow curves is caused by the Bauschinger effect, due to the load reversal between forward rod extrusion and upsetting. This led the authors to the conclusion that a flow stress evaluation is not possible by the proposed method. Krause [23] utilized rolling to produce sheets with known prestrains in order to find the flow curves by subsequent tensile tests on the prestrained specimens leading to flow curves up to high strains. However, large deviations were observed in a comparison with results of conventional testing methods. Possible reasons for these deviations will be discussed in the following.
In addition to damage, a plastic load reversal leads to a direction dependence of the flow stress due to the Bauschinger effect. As a consequence, both damage and the Bauschinger effect must be considered in the evaluation of flow curves for large strains, especially for intermittent experimental procedures.
Forward rod extrusion is a unique process to produce cylindrical parts with large known prestrains. The amount of plastic strain along the central axis of the extrudate is directly prescribed by the reduction of the crosssection. To the knowledge of the authors of this paper, no publication exists, which deals with the conduction of tensile tests on material prestrained by forward rod extrusion to characterize the plastic behaviour of metals under large plastic strains. So far, the highest strains were reached by Langford and Cohen [28] utilizing wiredrawing. In wiredrawing, the maximum strain per pass is highly limited by the occurrence of necking. This, however, does not apply in forward rod extrusion, as the material is pushed through the die under high compressive stresses. In addition to this, forward rod extrusion allows for the extraction of standardized specimens for the conduction of subsequent tests, which means the influence of inhomogeneous strain distribution can be incorporated more accurately.

Homogeneous strain distribution over specimen length

Known strain distribution over specimen radius

Monotonic stress history for material points moving through the forming zone

Negligible residual stresses within specimens.
The requirements have been verified for standardized tensile and upsetting test specimens (diameter d = 8 mm).
In a second step, the proposed method was utilized on 16MnCrS5 casehardening steel in order to evaluate the corresponding flow curve as well as to assess the influence of hydrostatic pressure during forward extrusion as well as a load reversal on the resulting flow stress.
Characteristics of forward rod extrusion
Fundamental parameters
Strain distribution over extrudate length
A necessary requirement for the production of prestrained specimens by forward extrusion is a homogenous strain distribution over the specimen length. An inhomogeneous strain distribution would lead to a inhomogeneous stress distribution in the tensile test specimens during loading, which may cause premature necking. After necking the calculation of the corresponding flow stress from the tensile force is impossible.
The numerical results emphasize, that due to the steadystate properties of forward extrusion, the strains are homogeneously distributed over a sufficiently large region of the extrudate length. For a standardized tensile test specimen with a diameter d = 8 mm and a total length of l = 77 mm (DIN 50125 – B 8 × 40) the strains are homogenous over the whole specimen length, which means the first requirement is fulfilled.
Strain distribution over extrudate radius
From Fig. 7 and 8 the inhomogeneous distribution of the true strain over the extrudate radius becomes clear. The second requirement for the produced specimens is a known strain distribution over the specimen radius. The knowledge of the strain distribution allows for a definition of an effective strain, which can be used to shift flow curves of prestrained material by a known prestrain.
From the deviation of the areaweighted average strains and the actual strain distribution of prestrained specimens (Fig. 10) the question arises, whether a flow curve evaluation is possible with sufficient accuracy, although the local effective strain during tensile loading is unknown. To investigate this, simulations of tensile tests were conducted, considering specimens with the actual prestrain distribution generated by forward extrusion as well as specimens homogeneously prestrained by the areaweighted average strain \( \overline{\varepsilon} \) (Appendix B). As a result of the investigations, the strain inhomogeneity over the specimen radius is taken into account in the flow curve evaluation procedure by shifting the flow curves of the prestrained specimens by the corresponding amount of areaweighted average strain \( \overline{\varepsilon} \).
Load path
In general, the flow stress of a prestrained material depends on the stress history that previously led to its deformation. While ductile damage can result in a reduction of the flow stress of prestrained material, the Bauschinger effect introduces a direction dependence of the flow stress. In the case of tensile tests on rolled sheets Krause [23] observed lower flow stresses in comparison to conventional methods, i.e. tensile and upsetting tests. To account for this, the stress states during forward extrusion are investigated and compared to the stress states of tensile tests.
Hence by comparison of Eq. 13 and Eq. 17, forward extrusion yields the same deviatoric stress state along the rotational axis of the extrudate as the uniaxial tensile test. Both stress states differ only by the amount of radial stress σ_{r} in the hydrostatic stress (Eq. 12 and Eq. 16). Usually, σ_{r} is large in forward rod extrusion. Due to the large resulting hydrostatic pressure supressing damage, a high material formability is possible, which, for some materials, allows for large true strains of 1.6 and higher.
Thorough investigations have been conducted e.g. by Avitzur et al. [34] to predict the stresses occurring in the forming zone during forward rod extrusion by means of the upperbound method. However, the exact amount of hydrostatic stress cannot be predicted analytically with sufficient accuracy, making the use of numerical methods necessary.
Description of loading conditions by triaxiality and Lode parameter
Herein, σ_{1}, σ_{2} and σ_{3} again indicate the first and second and third principle stresses.
Maximum hydrostatic stress, maximum stress triaxiality and strainweighted stress triaxiality along the central axis of the extrudate, depending on the extrusion strain
ε _{ex}  0.1  0.3  0.5  0.7  1.0  1.2  1.5 

σ _{ h, max} [MPa]  289  236  131  −30  −250  −395  −541 
η _{max}  0.48  0.33  0.18  −0.04  −0.34  −0.53  −0.72 
\( \overline{\eta} \)  0.46  0.1  −0.05  −0.27  −0.61  −0.85  −1.16 
In the case of η_{max} an extrusion strain of ε_{ex} = 0.3 leads to a maximum triaxiality of η_{max} = 1/3, which corresponds to the stress state of uniaxial tensile test. For \( \overline{\eta} \) the corresponding value lies between extrusion strains of ε_{ex} = 0.1 and ε_{ex} = 0.3. Up to extrusion strains of ε_{ex} = 0.7, the triaxiality values are above or close to zero. For both triaxiality measures a further increase of the extrusion strain leads to a progressive shift towards negative triaxiality values, which makes the evolution of damage highly unlikely for larger extrusion strains.
The maximum stress triaxiality values along the centre lines from Table 1 are indicated in the diagram at a radius of r = 0 mm. With increasing radius the maximum triaxiality tends to decrease toward the specimen surface. This holds for all extrusion ratios up to ε_{ex}= 1.0. For higher extrusion strains, the curves flatten. Though the maximum stress triaxiality is not constant over the specimen radius, no intersections are present for the lines corresponding to varying extrusion strains. Hence, the triaxiality values calculated in Table 1 are sufficient to compare the stress history of extrudates by the use of just one characteristic value such as η_{max}(r = 0).
Residual stresses
Cold forging can lead to high forming induced residual stresses, caused by inhomogeneous elasticplastic deformations. To isolate the influence of forming induced residual stresses from the resulting flow stress, a numerical analysis has been performed to evaluate the residual stresses in the test specimens after extraction by machining.
While the core of the part is subjected to compressive stresses of 50% of the initial flow stress of the annealed material σ_{f, 0} = 340 MPa, the stress increases toward the shaft surface, resulting in a positive stresses of 25% of σ_{f, 0} in this region. After extraction of a cylindrical specimen (diameter d = 8 mm) by turning, the selfequilibrating nature of residual stresses leads to a second drop, as the region containing large positive stress is removed. The residual stresses after turning were evaluated analytically by shifting the remaining part of the curve to zero, in order to reach a stress balance. The remaining residual stresses are within the small range of ±50 MPa (±14.7% of σ_{f, 0}). Since in this procedure it is assumed that the turning operation does not cause additional residual stresses, the results are an approximation of the actual stress distribution.
Experimental procedure
It was shown, that a production of cylindrical specimens from forward extruded rods with a known prestrain is possible. In the following, the experimental procedure to produce and test the prestrained specimens to achieve flow curves up to high strains is presented and the results are discussed.
Cold extrusion
Investigated extrusion strains
ε _{ex}  0.1  0.2  0.3  0.5  0.7  1.0  1.2  1.5 

d _{1} [mm]  28.6  27.2  25.9  23.4  21.2  18.3  16.5  14.2 
Chemical composition of 16MnCrS5 steel
Mat. No.  DIN / SAE  C  Si  Mn  S  Cr 

1.7139  16MnCrS5 / 5515  0.14–0.19  ≤ 0.4  1.0–1.3  0.02–0.04  0.8–1.1 
From the extruded rods, tensile and upsetting test specimens were extracted by means of turning. Details on the specimen extraction and the testing procedures will be discussed in the following.
Tensile tests
All tensile tests were conducted according to DIN EN ISO 68921. The velocity was controlled in order to ensure a constant strain rate of 0.0067 s^{−1}. The specimen elongation was measured directly on the test specimens by means of a tactile macroextensometer with a gauge length of 40 mm.
Upsetting tests
The upsetting tests were conducted according to DIN 50106, with a constant strain rate of 0.0067 s^{−1}. In order to reduce the friction between the test specimens and the dies, the contacting surfaces were sprayed with Teflon spray after each upsetting test. The change in height of the specimen was measured indirectly by the crosshead travel. To account for the unavoidable elastic deflection of the testing machine a stiffness correction curve was utilized. The recorded values were validated against the actual deformed specimen heights after upsetting, whereby only small deviations of ±2% were observed.
Results and discussion
The procedure of flow curve evaluation for large strains by means of tensile tests on material prestrained by forward rod extrusion will be discussed in the following. In addition, the variation of the extrusion strain ε_{ex} allowed for an investigation of the influence of hydrostatic pressure on the resulting flow curves. In order to evaluate the forming induced Bauschinger effect a comparison of tensile and compressive flow curves of the prestrained material was conducted.
The dark red curve corresponds to the result of the tensile test on annealed material. The specimens necked at true strains of approximately ε = 0.12. The dark blue curve, corresponding to the result of the upsetting test on annealed material, was evaluated up to a strain of ε = 0.7. After that, barrelling of the specimen due to friction became too pronounced for a correct calculation of the true stress, noticeable by a discontinuity in the flow curve. The true stressstrain curves of the tensile and upsetting tests on material prestrained by forward rod are plotted as bright red and blue lines, respectively. The beginning of each prestrained curve was shifted by their corresponding areaweighted prestrain \( \overline{\varepsilon} \), as explained in the previous section (Fig. 11).
So far, no special emphasis was given to the change in strain rate between extrusion and subsequent tensile and upsetting tests. However, Fig. 22 suggests that the change in strain rate does not have a significant influence on the workhardening behaviour of the material, as the tensile and compressive flow curves of extruded material are consistently lower than the flow curves of annealed material. This is underlined by results from Doege et al. [26], where no significant strain rate sensitivity was observed for strain rates below 8 1/s. Following from these observations, the strain rate sensitivity of the flow curves was neglected in the subsequent investigations.
Flow curve evaluation
Herein, F is the experimental upsetting force and D and h are the current diameter and height of the deformed specimen, respectively. The Coulombfriction coefficient is symbolized by μ. According to Eq. 22 the influence of friction on the apparent flow stress becomes more pronounced with a decrease in the height to diameter ratio, since the contact surface increases in a quadratic manner with decreasing specimen height.
C, ε_{0} and n are model parameters corresponding to a scaling in stress, a shift in strain and the hardening exponent, respectively. Due to the larger number of data points in the upsetting test regime, the support point was given more weight to be considered by the optimization algorithm. Generally, the choice of the flow curve model is arbitrary, as long as the model is capable to capture the experimental data with sufficient accuracy. In Fig. 23b, the resulting flow curve extrapolation is shown as dashed green line.
Damage
The remaining deviation between the compressive flow curve of annealed material and the tensile flow curves of prestrained material suggests that the variation of hydrostatic stress superposition during extrusion affects the strain hardening behaviour. In the second section, it was shown that the amount of hydrostatic stress along the central axis of a forward rod extruded shaft depends on the extrusion strain. The influence of hydrostatic and deviatoric stresses on the evolution of ductile damage is the topic of several ongoing research projects. Generally, it is assumed that for a given constant Lode parameter a high triaxiality may lead to an evolution of ductile damage, causing a material weakening, while negative triaxiality values tend to cause little or no damage accumulation.
In correspondence to this, Figure 24 shows that the tensile flow curve of annealed material and the tensile flow curves prestrained up to ε_{ex} = 0.7 saturate toward a shared flow stress level (black dashed line on the left), which is significantly lower than the “damagefree” compressive flow curve of annealed material (green line). Prestrains above ε_{ex} = 0.7 lead to a second flow stress level (black dashed line on the right), which saturates toward the extrapolation of the damagefree compressive flow curve (green dashed line), emphasizing that little or no damage was accumulated in the forward extruded shafts with these extrusion strains.
Tekkaya et al. [37] have found, by means of SEM analysis as well as fatigue testing of specimens made from forward extruded rods that large deviations occur, when extrusion strains of ε_{ex} = 0.5 and ε_{ex} = 1.0 are compared. For the lower extrusion strain of ε_{ex} = 0.5 a more pronounced void nucleation was observed, leading to a significantly lower fatigue strength than the part with ε_{ex} = 1.0. The influences of workhardening and residual stress were ruled out by experimental and numerical analyses. The observations are in accordance with the present results regarding the occurrence of triaxialitydependent flow stress levels of prestrained specimens.
Load reversal
As shown in Fig. 22, the compressive flow curve of material prestrained by ε_{ex} = 0.1 saturates toward an elongation of the compressive flow curve of the annealed material. However, for higher prestrains, the curves saturate toward lower stress levels. During forward rod extrusion, the material was subjected to a tensile deviatoric stress state, whereas in the upsetting test, the deviatoric stress direction reverses. This load reversal not only leads to a decrease of the yield stress with increasing prestrain, known as Bauschinger effect, but also to a permanent softening effect, as observed by many authors e.g. Sun and Wagoner [38]. In addition to that, when prestrains exceed ε_{ex} = 0.2 a region of workhardening stagnation becomes apparent. It is stated e.g. by Yoshida und Uemori [39], that the general existence of workhardening stagnation depends on the investigated material. However, the present results indicate that even when a material does not show a clear region of workhardening stagnation up to a certain prestrain, the phenomenon may just be shifted toward larger prestrain regions. In addition to that, for prestrains above ε_{ex} = 0.7, the material seems to show not only one, but two consecutive regions of workhardening stagnation. The phenomenon becomes more pronounced for higher prestrains.
The evaluated Bauschinger coefficients depending on the corresponding prestrain reached by forward rod extrusion are shown in Fig. 25b.
For the annealed material, the yield stresses in both load directions are equal which leads to a Bauschinger coefficient of \( {\overline{\sigma}}_{\mathrm{f}} \) = 1. With increasing prestrain, both Bauschinger coefficients decrease to about \( {\overline{\sigma}}_{\mathrm{f}}= \) 0.6 to \( {\overline{\sigma}}_{\mathrm{f}} \)= 0.7, which means, that the compressive yield stress is about 30–40% lower than the tensile yield stress. Starting from the minimum investigated prestrain of ε_{pre} = 0.1 the Bauschinger coefficients are almost constant for the investigated prestrains. It is assumed, that the Bauschinger coefficient saturates in a prestrain region prior to ε_{pre} = 0.1, which is true for various materials in the literature. The same method of determining the Bauschinger coefficient can be applied to any associated effect, e.g. permanent softening and transient hardening.
Conclusions
A new experimental method has been proposed to evaluate flow curves of materials by tensile tests on specimens prestrained by forward rod extrusion. Due to the Bauschinger effect upsetting tests on prestrained specimens do not suffice for a flow curve evaluation. In addition to this, tensile flow curves with small prestrains cannot be utilized either, since the strength of these specimens seems to be affected by ductile damage, accumulated during the process of forward rod extrusion. With increasing extrusion strain, however, the stress state is increasingly superposed by hydrostatic pressure, yielding the accumulation of damage unlikely, which means that the resulting curves can be used as support points for a flow curve in the high strain region. The method has been conducted on the casehardening steel 16MnCrS5, generating a flow curve up to a strain of \( \overline{\varepsilon}= \) 1.7.
Since the amount of hydrostatic stress can be varied by changing the extrusion strain in forward rod extrusion, the experimental procedure allows for an investigation of the influence of hydrostatic pressure on the apparent flow stress and thus, a qualitative assessment of ductile damage. In addition, the difference between the tensile and compressive yield stress allows for an evaluation of the Bauschinger effect for prestrained material. Following from that, the experimental procedure can be used to further develop material models taking into account forminginduced damage and the Bauschinger effect for large prestrains.
If the strain rate sensitivity or the temperature is of interest the procedure can be adjusted accordingly by changing the extrusion speed or temperature, as well as the tensile test conditions.
Notes
Acknowledgements
The authors thank the German Research Foundation (DFG) for the financial support of project A02 in the Collaborative Research Centre CRC/Transregio 188 “Damage Controlled Forming Processes”.
Funding
This study was funded by the German Research Foundation (DFG), Project A02 in the Collaborative Research Centre CRC/Transregio 188 “Damage Controlled Forming Processes”.
Compliance with ethical standards
Conflict of interests
The authors declare that they have no conflict of interest.
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