Keywords: inverse identification, optimization, finite element, deep drawing.

This paper shows a method, which enables determination of material parameters in advanced constitutive models in elasto-plastic strain range. The approach is based on inverse method in which experimental methods, FE (finite element) simulations and the optimization based on Levenberg-Marquardt technique[1] are employed. Classical experimental tests (e.g. uniaxial tension test, bulge test or compression tests) can be carried out only in limited strain range and therefore it is believed that the data obtained from inverse method, which includes the whole stain range of applied experiment, can be considered for better approximation of reality.

The inverse identification procedure is as follows:

Experimental tests are carried out and the data are recorded, e.g. the force- displacement curve. The tests are also simulated using FE-software starting with some chosen material parameters. The material parameters and the friction coefficient are treated as unknowns in the optimization procedure. The objective function is build as a difference between the calculated and the measured punch force formulated in a least square sense. The parameters are then changed using optimization techniques until the objective function reaches a minimum. The values of the parameters at the minimum point are the searched parameters. In case of anisotropy a more complex objective function, which includes also geometric quantities, is implemented.

Two identification problems based on deep drawing experiment are presented. In the first analysis the material is considered to be isotropic. Material parameters are identified using objective function expressed by:

where is experimentally determined punch force and is calculated punch force. The drawing process is divided on NR_STAGES intervals (stages). In each stage the punch force is collected from FE software output in order to build up the objective function. Hollomon's power law [2] was used to describe the plastic behaviour in this case. The isotropic identification is based on real experiment made with aluminium 6016-T4. Good agreement between measured and calculated punch force curves was achieved. Results from inverse based simulations are compared with simulations based on experimental data achieved from uniaxial tension test carried out in our laboratory and the friction determined by rotation test taken from literature[3]. In case of anisotropy the following objective function has been implemented:

where is experimentally determined punch force and is calculated punch force, is experimentally determined flange diameter taken in direction , , or degrees to the rolling direction, is calculated flange diameter in direction , , or , degrees to the rolling direction. The index is used to distinguish the three applied rolling directions. The index is used to distinguish three applied rolling directions. The drawing process is divided on NR_STAGES intervals (stages). In each stage the punch force and flange diameters are collected to build up the objective function. Material behaviour is described by linear hardening model with Hill's yield criterion[2]. Identification in this case was examined on artificial data taken from FE simulation. The FE simulation with known material parameters was used for generation of "experimental" data in this case. Very good convergence with artificial data was achieved. We are currently working on coupling this inverse problem with real experiment.

1

Arora, J. S., "Introduction to optimum design", McGraw-Hill Book Company, Singapore, 1989.

2

Hosford W. F., Caddel M.C., "Metal forming", Prentice-Hall, 1993.

3

Benchmark data from "Proceedings of NUMISHEET'99 Volume 2", Eds.: Gekin, J.C., Picart. P., University of Franche-Comte and ENSMM Besancon, France, 1999.

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