Effect of a strain rate dependent material modeling of a steel on the prediction accuracy of a numerical deep drawing process

Part quality in production chains of sheet metal components is affected by batch and process fluctuations, which often lead to production rejects and thereby decrease the material efficiency [1]. In the context of increasing pressure on the manufacturing industry to save resources and CO₂ emissions, compensation of these effects becomes of major importance. Therefore, a reliable prediction of the resulting part properties depending on such fluctuations is needed. The use of analytical models is a promising approach for this purpose [1], since they require less computing time than numerical simulations [2] and thus offer possibilities for inline process adaption [3]. A huge database is usually required to create these models. The corresponding number of experiments is therefore hardly manageable within the framework of real tests, which is why numerical simulations are often used [4]. In order for the analytical models generated from this to provide a reliable prognosis, the accuracy of the simulation as well as the modelability of all input parameters to be varied are of great importance.

One process parameter that has great potential as a control variable in terms of economic efficiency is the stamping velocity [5]. By taking strain rate dependent material behavior into account, it could be included as a design variable in the process design. A prerequisite for this is strain rate dependent modeling of the material behavior. A frequently used model for this purpose is the Johnson-Cook [6] approach, the Cowper-Symonds [7] or Zerilli-Armstrong [8] models are also widely applied. In the latter, the strain rate dependent portion of the yield stress is added to the strain rate independent portion, in the other two, the terms are multiplied. For most steel materials, an increase in the strain rate leads to an increase in the yield stress as the dislocation formation occurs at a higher rate than the softening by dynamic recovery [9]. However, the strain rate dependence is usually neglected in numerical simulations at room temperature because it is not pronounced in this temperature range for most metals [10].

Nonetheless, the forming velocities used as standard in material characterization are often far below those occurring in real forming tests [11]. In these cases, the strain rate sensitivity of a material can be important even at room temperature [11]. In particular, this concerns the prediction of crash behavior. Therefore, characterization tests are often carried out at high strain rates in the range of typically up to 200 s⁻¹ [12] and the strain rate dependence is taken into account in the material model. Additionally, some steel materials already show a significant strain rate dependency m at room temperature. It is calculated according to [12] as follows:

Hereby, \(\varepsilon\) is the true plastic strain, \(\dot{\varepsilon }\) the strain rate, \(T\) the temperature and \(\sigma\) the flow stress. For example, the microalloyed steel HX340LAD, according to Larour [12], has a strong strain rate sensitivity compared with other steels in the low strain rate range 10^–3–20 s⁻¹ with an m of about 0.2 at room temperature.

Against this background, the question arises whether a consideration of the strain rate sensitivity of the flow behavior leads to an improvement of the prediction quality of numerical simulations not only in high-speed applications, but already at lower stamping velocities. Additionally, since strain rates vary locally and over process time, strain rate dependent modeling could also be relevant for processes with fixed stamping velocity.

Several approaches to this topic can already be found in the literature, whereby the strain rate dependency is often considered in the context of higher temperatures. Winklhofer et al. [13] modeled the flow behavior of an aluminum alloy of the 5xxx series in the temperature range between 25 °C and 250 °C using the Cowper-Symonds approach. They validated the model via deep drawing experiments of a round deep drawn cup at a punch speed of 2 mm/s. Afterwards, the model was used for the numerical investigation at a punch velocity of 20 mm/s. However, the main subject of the investigation here was also not the strain rate influence, but the temperature influence [13].

Prakash and Kumar [14] investigated the deep drawing of a round cup made of AA5083-O for strain rates dependent on temperatures up to 300 °C and punch velocity. To characterize the material behavior, they performed uniaxial tensile tests in the strain rate regime between 10^–2 s⁻¹ and 0.333 s⁻¹. They then used the resulting material models to numerically investigate the deep drawing process at punch speeds between 20 mm/min to 720 mm/min in terms of thinning and process force.

Tulke et al. [15] also chose the geometry of a deep drawn cup for their numerical investigations with strain rate dependent material model for the material DC06. Here, the results from tensile tests with strain rates of up to 180 1/s from the work of [16] were used and validated by means of tensile tests at a plastic strain rate of 50 1/s. In the experiments, the strain rates occurred at the punch speeds of up to 180 1/s. In the real test, maximum strain rates of 100 1/s occurred at punch speeds of 9 m/s [15].

However, studies investigating actual forming processes on the strain rate influence at room temperature and comparatively low forming speeds, as they usually occur in cold forming processes, are rarely found. In addition, it is of interest how the strain rate dependence affects the forming process and its result in the case of a more complex shaped geometry. For this reason, within the scope of this work the strain rate dependent material behavior of a steel material is investigated in characterization tests and transferred to a material card. A numerical model of a deep drawing process using this material card is set up and validated based on experiments. Moreover, variant simulations are performed in order to investigate to what extent other process parameters affect the strain rates developing in the component as well as process force and strains.

Figure 3 shows the flow curves determined from the uniaxial tensile tests under different strain rates.

With increasing strain rate, a significant increase of the yield stress level can be observed. The initial yield stress is rising from YS = 410.6 MPa ± 2.4 MPa to YS = 425.8 MPa ± 2.4 MPa in the investigated strain rate range. The values determined for the uniaxial anisotropy coefficient are shown in Fig. 4. Strain rate is found to have no significant effect on the uniaxial anisotropy, but the r-value is highly depending on the angle to RD.

The material card *MAT133_BARLAT_YLD2000 is used to represent the material properties in the numerical simulations, since it is capable of taking into account both anisotropic material behavior and a dependence of the yield stress on the strain rate. Accordingly, the yield stress is composed of a deformation dependent and a strain rate dependent component.

In order to determine which yield curve approximation approach is most suitable for the material used, the flow curves from the experiments at an angle of 0° to the rolling direction are first being approximated separately using different approaches. Excel solver was used to determine the respective coefficients under the premise of root mean square error (RMSE). It was found that the Hockett-Sherby [23] approach reflects the real strain hardening behavior particularly well. The strain rate dependent portion of the yield stress is represented in the material card by the Cowper-Symonds [7] approach. This results in Eq. (2) for the true stress:

The coefficients \({c}_{1}\), \({c}_{2}\), \({c}_{3}\), \({c}_{4}\), \(C\) and \(p\) are now determined using the solver function while simultaneously taking into account the experimental data of the flow curves of all three strain rates in such a way that the RMSE is minimized overall. The experimental data are used up to a strain of 0.17 for all three curves in order to prevent different weighting of the individual strain rates. The resulting curves are shown in Fig. 5. The flow curve mathematically determined with this approach for a theoretical strain rate of 0%/s up to a strain of φ = 3.0 is stored in the material card via *DEFINE_LOAD_CURVE.

The constants \(C\) and \(p\) are included as parameters, whereby \(C\) is multiplied with a correction factor due to the artificially increased stamping velocity in the simulation. The yield function is modeled according to the approach of Barlat et al. [24].

Additionally, a material card without strain rate dependency of the flow stress is generated. For this purpose, the Cowper-Symonds parameters \(C\) and \(p\) in the material card are set to zero. The flow curve from the tensile tests, which were carried out at the usual strain rate \(\dot{\varepsilon }\) =0.4%/s according to the standard, is used in this material card with an approximation according to Hockett-Sherby.

For validation of the numerical model, the computational result is being compared with the data generated from the experiments in terms of force–displacement curves and part geometry.

First, the calculated sheet thickness distribution is compared with the actual distribution in the part. For this investigation, the untrimmed component geometry is used, since this also allows for the thinning and thickening behavior in the area of the drawing beads to be investigated. Figure 6 shows that the thinning behavior is generally well predicted by the simulation.

The areas of most severe thinning are well identified. The increasing of the overall thinning from experimental series V1 to V4 also is reflected by the simulation. In addition, the local thinning in the concave radius increases from V2 to V4. This is also predicted by the simulation. For V1, it should be noted that sheet thickness is somewhat overestimated in the simulation in general, but especially in the part wall. There is no clear difference between the simulation with and the one without strain rate dependency. It can be concluded that the consideration of the strain rate dependency in the investigated parameter space has no noticeable influence on the calculated sheet thickness distribution in the component.

Figure 7 shows that the prediction of the component geometry with strain rate independent material modeling agrees slightly better with experiment than that with strain rate dependent modeling. This refers in particular to the transition radius between the flange and the part wall.

With strain rate dependent modeling, the largest deviation occurs at V2. However, the accuracy of the simulation for the strain rate dependent material model is rated as sufficient.

An improved modeling of the springback behavior could be achieved by performing tension–compression tests and incorporating the corresponding material behavior into the material model, since the alternating load when passing through a drawing bead has been shown to influence the strength [25].

In addition, the forming force is used to evaluate the prediction quality of the simulation model. Here, more distinct differences between the strain rate dependent and the strain rate independent simulation become apparent, which is illustrated by Fig. 8. The punch speed specified in the input deck of the simulation is the same for both simulation variants in order to exclude the influence of dynamic effects, e.g. in the contacts. In general, it is noticeable that the curves calculated with strain rate independent material model are consistently lower than those calculated with strain rate dependent modeling. This is in line with expectations, as higher strain rates are expected at the forming speeds of 10 mm/s and 40 mm/s than the value of 0.4%/s in the tensile test according to the standard.

It can be seen that the increase of the forming force in all simulations at the beginning is in very good agreement with the data from the experiments. For V1 with standard yield curve, at the beginning of the plateau there is a deviation of − 46 kN, which corresponds to about − 11.1%. With strain rate dependent modeling, the maximum deviation is − 6.7% at the beginning of the plateau. In the further course of the process, the force in the real process tends to decrease. Thus, there is a somewhat larger deviation of 10.6% at the end of the process. It was found that the blankholder force tends to drop in the real process, which indicates an issue in the press control. This drop can be seen as the cause of the decrease in the forming force in the real process, since a higher blankholder force raises the force level in the process (see Sect. 5.2). For the parameter combination V3, the beginning of the force plateau is predicted very well. Therefore, in the following, primarily the forming force at the beginning of the plateau is used as a measure for the quality of the prediction. Using the strain rate independent material card, the calculated force level is significantly lower than the experimentally determined. At the beginning of the plateau, the difference is − 11.0%. For the parameter combination V2, the strain rate independent simulation model already hits quite well with a deviation of − 6.8%. That the strain rate independent simulation fits better for V2 and V4 than for V1 and V3 was to be expected, since these two were carried out with a forming velocity of 10 mm/s instead of 40 mm/s. Hence, the difference to the standard strain rate is smaller than for V1 and V3. However, by taking the strain rate dependence into account in the material card, an improvement can be achieved here as well. For parameter combination V4, a very good prediction is already possible by means of the strain rate independent flow curve with a maximum deviation of − 6.8%. With strain rate dependent modeling, the curve radius becomes somewhat smaller in the transition to the plateau, so that the deviation there reduces to − 4.6%. Summarizing, compared to a modeling of the flow behavior with the purely strain dependent flow curve determined at standard speed, the forming force curve calculated with strain rate dependent flow curve agrees significantly better with the experimentally determined curve.

Overall, it can be stated that the simulation model with strain rate dependent material modeling predicts component geometry and process force well. The simulation model is thus considered validated in the parameter space investigated.

Nonetheless, it should be mentioned that the consideration of the strain rate dependency has a negative effect on the calculation time. When using the strain rate dependent material cards, the computation consistently takes about twice as long as with the strain rate independent material card. The calculations were performed on Intel Core i7 with parallel use of 12 cores.

Adjustment of the drawbead height and blankholder force are specifically used in the design of sheet metal forming processes to control the sheet metal draw-in and the stretching of the material. This suggests that these process parameters also have an influence on the local strain rate distribution. In order to analyze these effects, the simulation model with strain rate dependent material card is used for numerical investigations.

In this work, the sheet material HC340LA was characterized in uniaxial tensile tests and layer compression tests with strain rate variation. Flow behavior was integrated into the numerical simulation of a deep drawing process via an anisotropic material card. Experimental tests were carried out in order to validate the simulation model in terms of component geometry and forming force. With this material card, an improvement could be achieved with respect to the accuracy of the forming force compared to a strain rate independent flow curve determined at standard speed. However, under consideration of the strain rate dependency the computation time was significantly higher. Furthermore, the quality of the geometry prediction decreased slightly. To improve the latter, future work could take into account the material behavior under tension–compression loading in the material card. Numerical analysis of the effects of changing the drawbead height and blankholder force showed that taking strain rate dependency into account leads to nonlinearities in the force response or at least amplifies them. An improvement of prediction accuracy by catching these characteristics of the interactions correctly is desirable. In summary, strain rate dependent modeling has both advantages and disadvantages. If the variation of the stamping velocity is not explicitly to be investigated, it could be beneficial to implement a strain rate independent material model and to accept slight deviations in favor of a lower computation time. In particular, when generating a database for analytical process modeling comprising a huge number on simulations, computation time must be given high priority. On the other hand, if the forming force is the key object of investigation, strain rate dependent modeling would be advantageous. Thus, especially when intending to use force data for an enhanced process monitoring with adaptive process control in order to reduce production rejects, consideration of the strain rate dependency in the numerical simulation is recommended. For further investigations, it is suggested to examine the effect of strain rate dependent modeling on further geometries within the framework of comprehensive test and simulation series and to investigate the effects by means of an in-depth statistical analysis.