#### Fig. 4 - Numerical and experimental thickness distributions along the symmetry path

### Model Set Up

The commercial FE Explicit code LS-DYNA was used for numerical simulations. The blank geometry and the die shape utilized for simulations are shown in Figure 3.

The blank was divided into two parts: the internal one, which is subjected to oil pressure and the external one (in contact with the blankholder) on which the closing force (or BlankHolder Force, BHF) is applied. Due to the symmetry, only half of the blank was modeled in order to reduce computational costs; in addition also a mass scaling technique was adopted.

The blankholder and the die were modeled as rigid parts (*MAT_20 in LS-DYNA) while the blank as deformable; different yield criteria were used to model the material behavior; in particular, the following anisotropic yield criteria, in plane stress condition, were taken into account: Hill 1948 (*MAT_122), Barlat 1989 (*MAT_36) and Hill 1990 (*MAT_243). Both Lankford’s parameters (R00, R45 and R90) and plastic flow curves along the investigated orientations (0°, 45° and 90°) were used for determining the parameters of the adopted yielding models. Also the isotropic yield criterion (*MAT_18) was used for comparison purposes. In this work the following process parameters were adopted (for both experimental and numerical tests): Temperature (T): 110°C; BHF: from 63 (BHFmin) up to 89 kN (BHFmax) by a linear profile; maximum pressure (pmax): 48 bar by a linear profile. Such process parameters were implemented through the LS-DYNA cards *LOAD_RIGID_BODY and *LOAD_SHELL_SET for BHF and pmax respectively (the adoption of the working temperature was simulated using the material behavior specific of that temperature).

### Step Analysis and Post-Processing

The COF value was evaluated by minimizing difference between thickness data along the longitudinal middle path (axis of symmetry) of the component. In particular, an optimal value was determined for every yield criterion investigated in the present work by comparing thickness results from numerical simulations with correspondent experimental data obtained using the DIC system Aramis. The graph in Figure 4 summarizes numerical results (in terms of thickness profiles along the symmetry path) obtained using the investigated yield criteria: Barlat’89 and Hill’90 (R = f (ε)) appears to be the ones which allow the best fitting of experimental data. In particular, the Hill’90 criterion is able to fit better the left part of the experimental curve (characterized by smaller strain levels) while the Barlat’89 criterion allows to fit better the right part of the curve (the one concerning the deepest part of the component). Also an additional parameter was investigated for checking the robustness of numerical models: the Flatness (it is as the ratio between the length, LC, of the symmetry path in contact with the die and the length, LD, of the bottom part of the die). The Figure 5 shows the flatness values calculated using models adopting different yield criteria: using as reference the experimental value of 0.2141, the Barlat’89 allowed the best approximation.

In order to predict critical areas characterized by an elevated risk of ruptures or wrinkling, material FLCs (which represent the limit values of major and minor strains) were implemented in the numerical models. The experimental FLCs shown in Figure 2 were used in LS-PrePost as reference for the principal strain values calculated in the FE analyses for all the sheet elements. The Figure 6 shows results obtained using anisotropic Barlat89 model with COF equal to 0.068.

It is possible to note that none sheet element exceed FLCs curve, therefore risks of rupture are not highlighted. The quality of formed component is quite good, recording a severe thinning in correspondence of deepest part of the component. The map of the quality areas is corroborated by that of the thinning.