Accuracy improvement of FLD prediction for anisotropic sheet metals using BBC 2008 advanced yield criterion

Abstract

In this paper, a theoretical framework was developed to incorporate BBC 2008 advanced yield criterion into the classical Marciniak–Kuczynski theory to compute limit strains of anisotropic metallic sheets. The classical Hill’s 48 yield function was also utilized to compare with the results obtained by BBC 2008. The anisotropy parameters contained in BBC 2008 yield function were identified by minimizing an error function using Levenberg–Marquardt method. The data from uniaxial and plane strain tensile testings were employed to establish the error function. All the experimental tests were performed on AA 3003-H19 aluminum sheets. The yield criteria were assessed from the standpoint of predicting material properties and the yield locus by comparing theoretical findings and experimental data. The influence of the use and non-use of the plane strain yield stresses in the identification procedure on the FLD calculation was also investigated. To verify the theoretical predictions of the FLD, a series of Nakajima tests were accomplished. Comparing the experimental results and theoretical predictions revealed that the best agreement was found when BBC 2008-16p yield function was utilized to express the yield locus. Failure to use the plane strain yield stresses in the calibration procedure of the BBC 2008-16p criterion also causes overpredicting the limit strains for positive strain ratios.

Appendix A

A more detailed description of each term of Eq. (8) is presented as follows:

$$\begin{aligned} \frac{{\partial \overline{\sigma }}}{{\partial L^{\left( i \right)} }} & = \frac{{\left( {w - 1} \right)}}{{\overline{\sigma } ^{{\left( {2k - 1} \right)}} }}\left\{ {w^{i - 1} \left\{ {\left[ {L^{\left( i \right)} + M^{\left( i \right)} } \right]^{2k - 1} + \left[ {L^{\left( i \right)} - M^{\left( i \right)} } \right]^{2k - 1} } \right\}} \right\} \\ \frac{{\partial \overline{\sigma }}}{{\partial M^{\left( i \right)} }} & = \frac{{\left( {w - 1} \right)}}{{\overline{\sigma } ^{{\left( {2k - 1} \right)}} }}\left\{ {w^{i - 1} \left\{ {\left[ {L^{\left( i \right)} + M^{\left( i \right)} } \right]^{2k - 1} - \left[ {L^{\left( i \right)} - M^{\left( i \right)} } \right]^{2k - 1} } \right\}} \right. \\ & \quad \left. { + w^{s - i} \left\{ {\left[ {M^{\left( i \right)} + N^{\left( i \right)} } \right]^{2k - 1} + \left[ {M^{\left( i \right)} - N^{\left( i \right)} } \right]^{2k - 1} } \right\}} \right\} \\ \frac{{\partial \overline{\sigma }}}{{\partial N^{\left( i \right)} }} & = \frac{{\left( {w - 1} \right)}}{{\overline{\sigma } ^{{\left( {2k - 1} \right)}} }}\left\{ {w^{s - i} \left\{ {\left[ {M^{\left( i \right)} + N^{\left( i \right)} } \right]^{2k - 1} - \left[ {M^{\left( i \right)} - N^{\left( i \right)} } \right]^{2k - 1} } \right\}} \right\} \\ \frac{{\partial L^{\left( i \right)} }}{{\partial \sigma_{11} }} & = l_{1}^{\left( i \right)} \\ \frac{{\partial L^{\left( i \right)} }}{{\partial \sigma_{12} }} & = \frac{{\partial L^{\left( i \right)} }}{{\partial \sigma_{21} }} = 0 \\ \frac{{\partial L^{\left( i \right)} }}{{\partial \sigma_{22} }} & = l_{2}^{\left( i \right)} \\ \frac{{\partial M^{\left( i \right)} }}{{\partial \sigma_{11} }} & = \frac{{m_{1}^{\left( i \right)} \left[ {m_{1}^{\left( i \right)} \sigma_{11} - m_{2}^{\left( i \right)} \sigma_{22} } \right]}}{{M^{\left( i \right)} }} \\ \frac{{\partial M^{\left( i \right)} }}{{\partial \sigma_{12} }} & = \frac{{\partial M^{\left( i \right)} }}{{\partial \sigma_{21} }} = \frac{{\left( {m_{3}^{\left( i \right)} } \right)^{2} \left( {\sigma_{12} + \sigma_{21} } \right)}}{{M^{\left( i \right)} }} \\ \frac{{\partial M^{\left( i \right)} }}{{\partial \sigma_{22} }} & = - \frac{{m_{2}^{\left( i \right)} \left[ {m_{1}^{\left( i \right)} \sigma_{11} - m_{2}^{\left( i \right)} \sigma_{22} } \right]}}{{M^{\left( i \right)} }} \\ \frac{{\partial N^{\left( i \right)} }}{{\partial \sigma_{11} }} & = \frac{{n_{1}^{\left( i \right)} \left[ {n_{1}^{\left( i \right)} \sigma_{11} - n_{2}^{\left( i \right)} \sigma_{22} } \right]}}{{N^{\left( i \right)} }} \\ \frac{{\partial N^{\left( i \right)} }}{{\partial \sigma_{12} }} & = \frac{{\partial N^{\left( i \right)} }}{{\partial \sigma_{21} }} = \frac{{\left( {n_{3}^{\left( i \right)} } \right)^{2} \left( {\sigma_{12} + \sigma_{21} } \right)}}{{N^{\left( i \right)} }} \\ \frac{{\partial N^{\left( i \right)} }}{{\partial \sigma_{22} }} & = - \frac{{n_{2}^{\left( i \right)} \left[ {n_{1}^{\left( i \right)} \sigma_{11} - n_{2}^{\left( i \right)} \sigma_{22} } \right]}}{{N^{\left( i \right)} }}. \\ \end{aligned}$$

(33)