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indicates that the two Yld2000-2d predicted loci are in good agreement with the experimental points, with a slightly better approximation whenM= 5.85. However, the yield loci predicted with vonMises and Hill’s 48 yield conditions are far from the experimental data. This figure illustrates that the choice of a yield condition is important and many studies, such as this one, have shown that vonMises and Hill’s 48 are rarely suitable for metallic materials.

Fig. 6 indicates that the crack initiates at about 45° from the RD after strain localization occurs. This is consistent with the minimum thickness strain measured for this orientation (Fig. 5) and predicted only with Yld2000-2d for M= 5.85. For M= 6 the minimum thickness strain, and presumably the crack, would occur in the TD.

Figure 5: Experimental radial thickness strain profile at 45° from RD for HE specimen and predictions with different yield conditions

Figure 3: Yield loci for different models: von Mises, Hill‘s 48 and Yld2000-2d for M = 5.85 and M = 6.

Figure 6: HE specimen showing neck and crack initiation at 45° from RD (Reproduced from [4] with the permission of AIP Publishing.) This example clearly illustrates that the predicted strain field in the hole expansion test is extremely sensitive to the consti- tutive equations of plasticity. In these conditions, to perform accurate simulations is a challenge that can be achieved only at the expense of advanced mechanical characterization techniques [5] and constitutive models. References: [1] Yoon, JW, Barlat, F, Dick, RE, Karabin, ME, 2006. Int. J. Plasticity 22, 174-193. [2] Lee, JY, Lee, M-G, Barlat, F, Chung, KH, Kim, DJ, 2016. Int. J. Solids Structures 87, 254–266. [3] Nakano, H, Hakoyama, T, Kuwabara, T, 2017. ESAFORM, AIP Conference Proceedings, Vol 1896, 020014. [4] Hakoyama, T, Nakano, H, Kuwabara, T, 2017. ESAFORM, AIP Conference Proceedings, Vol 1896, 020024. [5] Kuwabara, T, 2014. In ComprehensiveMaterials Processing; Van Tyne, CJ, Ed.; Elsevier Ltd., Vol. 1, pp 95-111. CAEWissen by courtesy of Frédéric Barlat Pohang University of Science and Technology, Republic of Korea. Toshihiko Kuwabara Tokyo University of Agriculture and Technology, Japan

Figure 4: Experimental radial thickness strain profile in TD for HE specimen and predictions with different yield conditions Fig. 4 shows the radial strain distribution from the hole edge towards the specimen outside, experimentally determined and simulated with the four constitutive descriptions as discussed above. It is clear that the results obtained with Hill’s 48 and vonMises are not in good agreement with the experiments. Even the trends of the strain profile, which exhibits a minimum at about 8 mm from the hole edge, cannot be captured with vonMises and Hill’s 48. In contrast, the curves predicted with Yld2000-2d are much closer to the experimental points. In addition, a minimum is predicted at distances of 8 and 7 mm for M= 5.85 and 6, respectively, in agreement with the experiment. Finally, for each location, there is a significant strain difference depending on the value of the exponent M, with 5.85 leading to the best agreement with the experiments. The same trends are observed for the thickness strain profiles measured along the RD and at 45° (Fig. 5), although the two profiles predicted with Yld2000-2d are closer to each other.

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