Modeling of Tension Instabilities in Sheet Metals

Patrick Zerbe

Patrick Zerbe did his honours project at the Fachgebiet Computational Mechanics (TUM)
and was supervised by Prof. Dr.-Ing. habil. Fabian Duddeck.

The onset of local necking in thin metal sheets during deep drawing is modeled according to criteria proposed by Marciniak [1,2]: maximum tension for tension-compression states and an imperfection analysis for tension-tension states. The imperfection analysis is enhanced to additionally account for tension-compression states. The limit strains are presented in forming limit curves (FLC) that are typical for deep drawing applications. The criteria are formulated and implemented into a framework to output FLCs based on elasto-plastic material models.

Elasto-Plastic Framework

Elasto-plastic return mapping algorithms are implemented for the plane stress case with isotropic hardening. The von Mises and Barlat89 yield surface are incorporated. Arbitrary isotropic strain-rate dependent hardening functions can be considered, e.g. a power law for plastic strain and strain rate.

Maximum Tension

A neck starts to develop as soon as maximum tension is reached. Tension is defined as product of principal stress and current sheet thickness. The state of maximum tension defines the limit strains.

Maximum Tension Algorithm Tension Curves Resulting FLC
Fig.1: Maximum Tension Algorithm
Fig.2: Tension Curves (for several strain ratios)
Fig.3: Resulting FLC

Imperfection Analysis

A slight thickness reduction in the sheet is defined by an imperfection parameter: f0=(tB/tA)0. With increasing load a neck normal to the maximum load develops until all strain localizes there. This leads to a plane strain state in the neck and failure of the sheet. For tension-compression states the algorithm is modified considering a rotated neck in a way that the resulting limit strains are minimized.

Imperfection Analysis Algorithm Strain Paths Resulting FLCs
Fig.4: Imperfection Analysis Algorithm
Fig.5: Strain Paths
Fig.6: Resulting FLCs

Conclusions

The maximum tension algorithm predicts qualitatively the tension-compression region of the FLC. The imperfection analysis algorithm allows prediction of the FLC over the full range if the neck alignment is considered. By introducing the imperfection parameter a quantitative adjustment is possible. Therefore, it seems to be well suited for analysis of tension instabilities in thin sheets with elasto-plastic material models.

 Forming Limit Curves

Fig.7: Exemplary FLC Results

References

[1] Marciniak, Z., Duncan, J.-L., Hu, S.-J.; “Mechanics of Sheet Metal Forming”; Butterworth-Heinemann; 2002
[2] Marciniak, Z., Kuczynski, K.; “Limit Strains in the Process of Stretch-Forming Sheet Metal”; Int. J. Mech. Sci.; Vol. 9; 1967