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Cylindrical cup drawing of a commercially pure titanium sheet: experiment and crystal plasticity finite-element simulation

  • Developments in modelling and simulation..Japan, South Korea and China
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Abstract

Because of the strong anisotropy in mechanical properties, press forming of commercially pure titanium (CP-Ti) sheets often creates significant defects, including earing formation during drawing. However, the predictive accuracy of CP-Ti sheet drawing processes by finite-element simulations is still not satisfactory because it is difficult to accurately represent the strong anisotropy with phenomenological constitutive models. In this study, a crystal plasticity model is employed to conduct finite-element simulations of a cold-rolled Grade 2 CP-Ti sheet cup drawing process, and its applicability to the process is examined in detail by comparing it with experimental results. Experimentally, the maximum cup height appears at an angle of approximately 50° from the rolling direction, and the heights at 0° and 90° are similar. The thickness strain distribution evolution is strongly dependent on the direction. Twinning activity during drawing is the largest at 90°, followed by 45°, and then 0°. The simulation qualitatively captures the overall tendencies well, but non-negligible discrepancies are also involved in the cup height at 90°, and the thickening at the cup edge at 0° and 90°. The mechanisms that yield the discrepancies between the experiment and the simulation are examined. Moreover, parametric studies are conducted to discuss the effects of twinning activity and friction on the drawability.

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Acknowledgements

The authors would like to acknowledge Mr. Sohei Uchida of the Osaka Research Institute of Industrial Science and Technology for assisting with the EBSD. This study was supported by the Japan Society for the Promotion of Science (JSPS) Grants-in-Aid for Scientific Research (KAKENHI) Grant number 20H02480 and the Amada Foundation Grant number AF-2019004-A3. We would like to thank Editage (www.editage.com) for English language editing.

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  1. Graduate School of Energy Science, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501, Japan

    Takayuki Hama, Kaho Hirano & Ryo Matsuura

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  1. Takayuki Hama
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Appendices

Appendix 1

Effect of numbers of initial orientations and finite elements on cup drawing simulation

Two additional cup drawing simulations were conducted. The number of orientations assigned to each integration point was set to 100 in the first case, while a finite element model with 1000 elements were used in the second case. Other simulation conditions remained unchanged. The pole figures obtained with the 100 orientations used in the first case are shown in Fig. 23a. The distributions are slightly different from those of the experimental results (Fig. 3a) especially in the \((10 \overline{1 } 0)\) pole figure and, moreover, the maximum intensity was slightly larger. Figure 23b shows the finite element model with 1000 elements used in the second case. The numbers of elements in the radial and circumferential directions were smaller than those of the original model (Fig. 5), while the number of elements through the thickness remained unchanged.

Figure 23c and d show the earing profiles and evolution of the thickness strain at the flange edge. The earing profiles were almost independent of the simulation condition. In contrast, the evolution of the thickness strains depended on the simulation condition. Especially, the evolution in the RD and TD differed largely. These results show that it is important to use appropriate simulation conditions to discuss strain evolution although the results of earing profile were not affected by the conditions.

Appendix 2

Determination of material parameters

Based on a previous study for a Grade 1 CP-Ti sheet [62], it was first presumed that the CRSS was the smallest for prismatic slip, followed in order by pyramidal < a > slip and pyramidal < a + c > slip. As in a previous study, the rank of the CRSS for basal slip was adjusted to reproduce experimental results. The CRSS for pyramidal < a + c > slip was initially assumed to be larger than twice as large as that for prismatic slip. Because twinning activities are smaller in a Grade 2 CP-Ti sheet than those in a Grade 1 CP-Ti sheet [11, 12], the rank of the CRSSs for twinning used in a previous study could not be utilized in this study. Our preliminary study for a Grade 2 CP-Ti sheet [95] reported that the CRSSs for twinning were set to be comparable or larger than those of pyramidal < a + c > slip to fit evolution of twin volume fraction. Therefore, the same assumption was used for initial guess also in this study.

Then, the hardening parameters were calibrated. Considering the role of each slip and twinning system on the work-hardening behavior, the parameters for prismatic slip and contraction twinning were primarily calibrated by referring to the stress–strain curve and the evolution of the r-value under RD tension. Similarly, the parameters for pyramidal < a > and basal slip were calibrated using the results of TD tension, whereas those of extension twinning were calibrated using the results of RD compression. These steps were repeated several times to obtain good fits with the experimental results under RD, 45D, and TD loadings with a single set of parameters. The reader is referred to a previous study [62] for the basic calibration procedures.

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Hama, T., Hirano, K. & Matsuura, R. Cylindrical cup drawing of a commercially pure titanium sheet: experiment and crystal plasticity finite-element simulation. Int J Mater Form 15, 8 (2022). https://doi.org/10.1007/s12289-022-01655-x

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