The computational code was developed in a commercial version of ABAQUS software which was fed with data from the DP600 steel. The results of the present work were compared to the results obtained with the experimental data presented in the study [12], where the formability of the DP600 steel with the variation of the blank holder load (BHL) were previously evaluated. To prove the model validity the configurations of the blank, die, BHL and the punch were defined in order to get closer to the parameters used by [12]. The Nakazima tool geometry used in the practical tests and in the computational model are shown in Fig. 2.

The blank holder, die and punch were modelled as solid shell elements since their deformation are not evaluated during the analysis. The sheet was modeled as deformable shell due to purpose of analyze its deformation during the process. The advantage of the shell format is reduced iteration time. The shell element type for the sheet was S4R integration type. The model was divided in three parts. The 1st part was the contact between blank holder and die, the 2nd part is the blank holder force application and the 3rd part was the vertical motion from the punch pushing the sheet downwards through the die. For this model it was used the penalty contact method.

The failure criteria used in the work are approximated for the plane stress state and this approach is also interesting for the sheet modeling in the shell format. The shell format will not consider the analysis of stress and deformation in the direction of the sheet thickness [13]. Fig. 3 summarizes the formats and types defined for the simulations.

The finite element simulation was done in three steps: 1st step - contact between the blank holder and the blank; 2nd step - application of the BHL load and the 3rd step - vertical displacement of the punch. The default value of the time period in ABAQUS application was 1, i.e., the time varies from 0.0 to 1.0 throughout the simulation step. The time increments in each analysis are simply fractions of the total period of each simulation step [13]. In this study the punch displacement step time is 10x longer than the time of the 1st and 2nd steps. For the computational simulations the same BLF levels used by [12], in his practical tests, were defined for the present study. In this case, three BLF loads were used: 58tf, 80tf and 130tf.

It is important to clarify the conditions under which the failure criteria are reached in the ABAQUS models defined in this work. Abaqus software automatically delete the elements at the crack location. It was selected an increment of 0.1mm from punch displacement to extract the exact punch displacement when the crack initiates.

To define the fracture envelopes of Eq. 2, 3, 4, 5 and 6 it was necessary to leave all terms on the right side of the equation as a function of \(\text{ɳ}\). This allows to vary the value of \(\text{ɳ}\) with an increment value and observe the corresponding value of \({\stackrel{-}{\epsilon }}_{f}\) for each value of \(\text{ɳ}\). In this way it is possible to obtain the trace of the fracture envelope curve. According to [2] for the plane strain state the Lode angle can be related to triaxiality through the Eq. 10:

\(-\frac{27}{2} \text{ɳ}\left({\text{ɳ}}^{2}-\frac{1}{3}\right)=\text{sen}\left(\frac{\stackrel{-}{{\theta }}{\pi }}{2}\right)\) (10)

The Lode angle and the Lode parameter are related by the Eq. 11.

\(tg\left(\stackrel{-}{\theta }\right)=\frac{\sqrt{3}(1+\mu )}{3-\mu }\) (11)

Using Eq. 10 and 11 to leave Eq. 2, 3, 4, 5 and 6 in function only of triaxiality we obtain the equations below:

Modified Mohr Coulomb:

\({{\stackrel{-}{\epsilon }}_{f}=\left\{\frac{A.{f}_{3}}{{C}_{2}}\left[\sqrt{\frac{1+{C}_{1}^{2}}{3}}.{f}_{1}+{C}_{1}\left(\text{ɳ}+\frac{{f}_{2}}{3}\right)\right]\right\}}^{-\frac{1}{n}}\) (12)

Maximum Shear Stress:

\({\stackrel{-}{\epsilon }}_{f}={\left\{\frac{A.{f}_{1}}{{C}_{2}\sqrt{3}}\right\}}^{-\frac{1}{n}}\) (13)

Johson-Cook:

\({\stackrel{-}{\epsilon }}_{f}={C}_{1}+{C}_{2}{e}^{(-{C}_{3}\text{ɳ})}\) (14)

Lou e Huh:

\({\stackrel{-}{\epsilon }}_{f}={C}_{1}{\left(\frac{2\sqrt{3}}{9}{f}_{1}\right)}^{-{C}_{2}}{\left(\frac{1+3\text{ɳ}}{2}\right)}^{{-C}_{3}}\) (15)

Oh et al.:

\({\stackrel{-}{\epsilon }}_{f}={C}_{1}{\left(\text{ɳ}+\frac{{f}_{1}}{\sqrt{3}}+\frac{{f}_{2}}{3}\right)}^{-1}\) (16)

The values of \({f}_{1}, {f}_{2}\)and \({f}_{3}\) are called simplifying functions and it can be obtained according to the Eq. 17, 18 and 19 [14].

\({\text{f}}_{1}=\text{c}\text{o}\text{s}\left\{\frac{1}{3}{\text{s}\text{e}\text{n}}^{-1}\left[-\frac{27}{2}\text{ɳ}\left({\text{ɳ}}^{2}-\frac{1}{3}\right)\right]\right\}\) (17)

\({\text{f}}_{2}=\text{s}\text{e}\text{n}\left\{\frac{1}{3}{\text{s}\text{e}\text{n}}^{-1}\left[-\frac{27}{2}\text{ɳ}\left({\text{ɳ}}^{2}-\frac{1}{3}\right)\right]\right\}\) (18)

\({\text{f}}_{3}={\text{C}}_{3}+\frac{\sqrt{3}}{\left(2-\sqrt{3}\right)}\left(1-{\text{C}}_{3}\right)\left(\frac{1}{{\text{f}}_{1}}-1\right)\) (19)

Using the calibration constants obtained by [1] for the DP600 steel, the five fracture envelopes in Fig. 4 were obtained. The curve data were useful to feed the ABAQUS software as a failure criteria. Usually tests that provides a known stress test are chosen to calculate the constants. Pure tensile test, pure compression or pure shear tests are used for that, Fig. 1.

The shape of the curve from Fig. 4 is explained by the behavior from the ductile damage equation for each criteria. In this case Eq. 14 for Johson-Cook and Eq. 16 for Oh et al. Johson-Cook has an exponential equation while Oh et al. has a linear equation. It was not selected to use criteria with similar curve shapes since Oh et al. per example presented minor errors when compared to other criteria as illustrated by Fig. 7.

The values of \({\epsilon }_{1}\) and \({\epsilon }_{2}\) (major and minor true strains) were also obtained by [12] via the Nakazima test of DP600. These values were used to implement the FLC criteria in the computational model. The fundamental mechanical properties and true stress-by-true strain curve for DP600 steel obtained by [15] as illustrated in Fig. 5 were also used.