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Non-Associated Flow Rule Constitutive Modeling Considering Anisotropic Hardening for the Forming Analysis of Orthotropic Sheet Metal

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Abstract

Background

The evolution of anisotropy has an important influence on the forming of parts under large deformation. However, most of the current yield criteria do not consider the evolution.

Objective

An anisotropic constitutive model based on non-associated flow rule (non-AFR) was established for orthotropic sheet metal. The classical quadratic Hill48 model was used to describe the yield anisotropy and plastic deformation anisotropy, respectively.

Methods

According to the principle of equivalent plastic work, the existence and significance of anisotropy evolution with plastic deformation were revealed. In order to improve the prediction accuracy of the model, a continuous capture scheme considering anisotropic hardening was proposed.

Results

The evolution of directional yield stress, directional r-value and yield locus was well captured by the developed model. To further verify the model, square box deep drawing tests of different strokes of the punch were carried out. Compared with the experimental results, the developed model could predict the material flow behavior in flange area and thickness thinning behavior, which actually reflected the evolution behavior of directional flow stress and directional r-value of sheet metal respectively.

Conclusion

The developed model improves the prediction accuracy of anisotropic sheet metal forming, and can provide an effective reference scheme for large deformation problems in industrial production.

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Data Availability

The authors confirm that the data supporting the findings of this study are available within the article.

Abbreviations

\(\phi\) :

Yield criteria

σ :

Cauchy stress tensor

\(\bar \sigma ({\bar \varepsilon_p})\) :

Equivalent stress-strain relationship of uniaxial tension

\({\bar \varepsilon_p}\) :

Equivalent plastic strain

\({f_y}\) :

Equivalent stress of yield function

\({f_p}\) :

Equivalent stress of potential function

\(\boldsymbol d{\boldsymbol\varepsilon}_p\)  :

Plastic strain increment

\(d\lambda\) :

Plastic multiplier increment

m :

First order gradient of the yield function

n :

First order gradient of the potential function

dw :

Equivalent plastic work increment

d σ :

Stress tensor increment

\({\boldsymbol C}_{e}\)  :

Elastic stiffness matrix

\(\boldsymbol d{\boldsymbol\varepsilon}_e\)  :

Elastic strain increment

\(\boldsymbol d\boldsymbol\varepsilon\)  :

Strain tensor increment

F σ,  G σ,  H σ, L σ,  M σ,  N σ :

Anisotropic parameters of yield function

F r,  G r,  H r,  L r,  M r,  N r :

Anisotropic parameters of potential function

X:Y :

Tensor double contraction operation of X and Y

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Funding

This project was funded and supported by National Natural Science Foundation of China (51975509 and 52205353).

Author information

Authors and Affiliations

  1. Key Laboratory of Advanced Forging & Stam** Technology and Science (Yanshan University), Ministry of Education of China, Qinhuangdao City, People’s Republic of China, 066004

    Y. Zhang, Y. Duan, P. Fu & J. Zhao

  2. College of Mechanical Engineering, Yanshan University, Qinhuangdao City, People’s Republic of China, 066004

    Y. Zhang, Y. Duan, P. Fu & J. Zhao

  3. Hebei Key Laboratory of Material Near-Net Forming Technology, (Hebei University of Science and Technology), Ministry of Education of China, Shijiazhuang City, People’s Republic of China, 050018

    Z. Mu

  4. College of Material, Hebei University of Science and Technology, Shijiazhuang City, People’s Republic of China, 050018

    Z. Mu

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  1. Y. Zhang
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  4. P. Fu
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Correspondence to Y. Duan or Z. Mu.

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Zhang, Y., Duan, Y., Mu, Z. et al. Non-Associated Flow Rule Constitutive Modeling Considering Anisotropic Hardening for the Forming Analysis of Orthotropic Sheet Metal. Exp Mech 64, 305–323 (2024). https://doi.org/10.1007/s11340-024-01032-6

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