Modeling Anisotropic Ductile Fracture Behavior of Sheet Metals Considering Non-directionality of Equi-Biaxial Tensile Fracture

Abstract

An accurate characterization of fracture behavior under pure shear, uniaxial tension, plane strain tension, and equi-biaxial tension plays a vital role in high-fidelity numerical simulation of the sheet metal forming. From a plastic deformation viewpoint, the non-directionality of the equi-biaxial tensile fracture is the inherent characteristic of sheet metal. However, little attention has been paid to modeling the non-directionality of the equi-biaxial fracture strain. This paper develops a new anisotropic ductile fracture model by including an empirical weight function into the DF2014-based fracture criterion to consider the non-directionality of the equi-biaxial tensile fracture. Then, the proposed model is utilized to depict the anisotropic ductile fracture behavior of DP980 steel, AA6082-T6 aluminum alloy, and Ti-6Al-4 V titanium alloy to verify its fracture predictability under various stress states. The prediction results are compared with the DF2014 and DF2016-based criteria. The results show that the proposed model correctly captures the non-directionality of equi-biaxial tensile fracture strain and depicts the anisotropic ductile behavior of these metals with high accuracy under proportional loading conditions. Meanwhile, the proposed model provided a similar fracture prediction accuracy to the DF2016-based criterion for different metal sheets, which indicated that the fracture predictability of this model had been successfully enhanced. In addition, finite element analysis for the square cup drawing test of AA6016-AC200 alloy is conducted in ABAQUS/Explicit to validate its performance under non-proportional loading conditions. The simulation results of the punch force–stroke curve and fracture shape in good agreement with the experimental measurements. The comparison study demonstrates that the proposed anisotropic ductile fracture model provides quite accurate predictability in depicting the anisotropic ductile fracture behavior of different metallic materials. Accordingly, the proposed model is recommended to be applied in FE simulation to improve the reliability and accuracy of numerical design and optimization of metal sheets product and forming process.

Appendix A: Calibrated Parameters of Lou and Yoon's (2021) Model for Different metal Sheets

See Tables 12, 13, and 14.

Table 12 Fracture parameters of Lou and Yoon's model in Eq 1 for DP980 steel

Table 13 Fracture parameters of Lou and Yoon's model in Eq 1 for AA6082-T6 aluminum alloy

Table 14 Fracture parameters of Lou and Yoon's model in Eq 1 for Ti-6Al-4 V titanium alloy

Lou and Yoon (Ref 53) suggested a user-friendly anisotropic ductile fracture defined below:

$$\begin{array}{*{20}c} {\left( {\frac{2}{{\sqrt {L^{2} + 3} }}} \right)^{{C_{1} \left( \theta \right)}} \left( {\frac{{f\left( {\eta ,L,C_{0} } \right)}}{{f\left( {2/3,1,C_{0} } \right)}}} \right)^{{C_{2} \left( \theta \right)}} \overline{\varepsilon }_{f}^{p} = C_{3} \left( \theta \right)} \\ \end{array}$$

(A1)

with

$$\begin{array}{*{20}c} {f\left( {\eta ,L,C_{0} } \right) = \eta + C_{4} \left( \theta \right)\frac{{\left( {3 - L} \right)}}{{3\sqrt {L^{2} + 3} }} + C_{0} } \\ \end{array}$$

(A2)

where C_i(θ) (i = 1, 2 , 3, 4) is the interpolated fracture parameters along the arbitrary loading direction, for instance, the interpolated fracture parameters are expressed as:

$$\begin{gathered} C_{i} \left( \theta \right) = A^{i} \cos^{4} \theta + B^{i} \sin^{2} \theta \cos^{2} \theta + D^{i} \sin^{4} \theta ; \hfill \\ A^{i} = C_{i}^{0} ; B^{i} = 4C_{i}^{45} - C_{i}^{0} - C_{i}^{90} ; D^{i} = C_{i}^{90} \left( {i = 1, 2, 3, 4} \right) \hfill \\ \end{gathered}$$

(A3)

where $C_{i}^{0}$, $C_{i}^{45}$, and $C_{i}^{90}$ (i = 1, …, 4) are calibrated fracture parameters along RD, DD, and TD, respectively.

In order to calibrate the fracture parameters in the above Lou and Yoon’s model, it is generally required three steps as follows:

• (1) Fracture experiments are performed to determine the fracture strains along different loading directions under various stress states of in-plane shear, uniaxial tension, plane strain tension, and balanced biaxial tension.

• (2) Based on measured fracture strains in Step 1, the fracture parameters in the above Lou and Yoon’s model are identified along specific loading directions, e.g., $C_{i}^{0}$ for RD, $C_{i}^{45}$ for DD, and $C_{i}^{90}$ for TD by minimizing the error equation defined in Eq A4.

  $$\begin{array}{*{20}c} {{\text{err}} = \mathop \sum \limits_{i = 1}^{N} \left( {\frac{{\overline{\varepsilon }_{f}^{{{\text{pred}}.}} }}{{\overline{\varepsilon }_{f}^{{{\text{exp}}.}} }} - 1} \right)^{2} \times 100\% } \\ \end{array}$$

  (A4)

  where the superscript exp. and pred. denote whether the data are obtained from the experiments or predictions.

• (3) Substituting the identified fracture parameters along specific loading directions in Step 2 into Eq A3, after simple linear operations, the interpolation coefficients, viz., A_i, B_i, and D_i, are finally identified.

Furthermore, to take into account the non-directionality of the balanced biaxial tension fracture strain, C₃(θ) in Eq A1 is set as identical to equi-biaxial fracture strain, viz., $C_{3} \left( \theta \right) = \overline{\varepsilon }_{{\text{f}}}^{{\text{b}}}$.

Accordingly, based on the experimental data provided in Table 1 for DP980 steel, Table 6 for AA6082-T6 alloy, and Table 8 for Ti-6Al-4 V alloy, the corresponding identified fracture parameters of the Lou and Yoon's model are briefly summarized below.