# DEVELOPMENT OF A MATERIAL MODEL FOR METAL SHEETS AT FINITE DEFORMATION

Leitung: | P. Wriggers, S. Löhnert |

Team: | E. Lehmann, S. Zeller |

Jahr: | 2009 |

Ist abgeschlossen: | ja |

Within their granular microstructure, metals show crystalline nature: the atoms are arranged on a regular space lattice. The macroscopically observable plastic deformation can be traced back to shearing or sliding of atoms on defined slip planes in the crystal. Depending on the present structure, predictions can be made about the slip behaviour. At the same time, crystal grains can be observed in the microstructure; regions, where atomic dislocation movements in the same directions take place and at whose borders atoms can pile up*.*

A suitable anisotropic material model is considered for the crystals on the microlevel, whereas the elastic and plastic material parameters are received in collaboration of the Institute Of Materials Science. In order to develop an anisotropic elastoplastic material model, the slip systems are considered in the plastic part of the deformation gradient and plastic gliding is modelled within these systems. The continuum mechanical model is gained through consistent derivation of the stress response and the elastic and plastic material tangent. A volumetric-deviatoric split is performed, whereas the slip systems are more easily actived under deviatoric deformation.

For the material model of the polycrystalline structures, a crystal plasticity model for finite deformation is considered, taking into account 24 slip systems of body-centered cubic crystals. For the elastic part of the deformation, isotropy is assumed, so that the Schmid stress can be expressed in terms of a scalar relation of the slip sytem vectors. In order to circumvent singularity stemming from the linear dependency of the latter ones, a viscoplastic law for the evolution of the slip rates is introduced, initially activating every slip system.

To carry out the entire computational simulation, it is necessary to model and discretise the polycrystalline microstructure. The geometry of polycrystalline materials will be simulated by 3D Voronoi tesselations, where a single crystal of the material is represented by a Voronoi cell. Meshing will be done with respect to the borders of every crystal, so that crystals may have different material properties and orientations. The rotation of the slip systems in each grains is performed through three randomly given EULER angles.

The problems concerning the geometry generation and the interface development lead to questions about the manipulation of three-dimensional subdivisions. In principle, a 3D Voronoi tesselation can be obtained by its dual graph, namely the Delaunay tesselation, by making use of space duality concepts. For the two-dimensional case, space duality, Delaunay- and Voronoi decompositions can be implemented by using the quad-edge data structure, so that a three dimensional generalisation of the quad-edge data structure and its topological operations have to be found.

For the representative volume elements, randomly distributed geometries of different sizes on the microscale are assumed, based on the received microstructural data.

Through homogenisation, effective stress-strain relations are gained for different boundary conditions being the outcome for an effective material model.

After an assumption for an effective material model has been made, the material parameters have to be determined through optimisation techniques. The results obtained by simulations will be validated with experimental data obtained from material specimens and formed structures.