Section 6: Material modelling in solid mechanics - GAMM 2012

6 Section 6: Material modelling in solid mechanics
kinematics at the continuum scale. As a result the total energy becomes non-convex, thus giving
rise to the development of microstructure. To guarantee the existence of minimizers an exact
quasi-convex envelope of the corresponding energy functional is derived. As a result microstructure
occurs in both the displacement and micro-rotation field. The corresponding relaxed energy
is then used for finding the minimizers of the two field minimization problem corresponding to a
Cosserat continuum. Finite element formulation and numerical simulations are presented. Analytical
and numerical results are discussed.
About the Microstructural Effects of Polycrystalline Materials and their Macroscopic
Representation at Finite Deformation
Eva Lehmann, Stefan Loehnert, Peter Wriggers (Universität Hannover)
In the sheet bulk metal forming field, the strict geometrical requirements of the workpieces result
in a need of a precise prediction of the material behaviour. The simulation of such forming processes
requires a valid material model, performing well for a huge variety of different geometrical
characteristics and finite deformation.
Because of the crystalline nature of metals, anisotropies have to be taken into account. Macroscopically
observable plastic deformation is traced back to dislocations within considered slip
systems in the crystals causing plastic anisotropy on the microscopic and the macroscopic level.
A finite crystal plasticity model is used to model polycrystalline materials in representative
volume elements (RVEs) of the microstructure. A multiplicative decomposition of the deformation
gradient into elastic and plastic parts is performed, as well as a volumetric-deviatoric split of the
elastic contribution. In order to circumvent singularities stemming from the linear dependency
of the slip system vectors, a viscoplastic power-law is introduced providing the evolution of the
plastic slips and slip resistances.
The model is validated with experimental microstructural data under deformation. The validation
on the macroscopic scale is performed through the reproduction of the experimentally
calculated initial yield surface. Additionally, homogenised stress-strain curves from the microstructure
build the outcome for a suitable effective material model. Through optimisation techniques,
effective material parameters can be determined and compared to results from real forming processes.
A rheological model for arbitrary symmetric distortion of the yield surface
A.V. Shutov, J. Ihlemann (TU Chemnitz)
A new rheological model is presented, which provides insight into phenomenological modelling of
combined nonlinear kinematic and distortional hardening. The model is constructed by coupling
idealized two-dimensional rheological elements like Hooke-body, Newton-body, and modified St.
Venant element [1,2]. The symmetric distortion of the yield surface and its orientation depending
on the recent loading path are captured by the rheological model in a vivid way. We emphasize the
flexibility of the proposed approach since it can be used to capture any smooth convex saturated
form of the yield surface observed experimentally, if the yield surface is symmetric with respect to
the recent loading direction. In particular, an arbitrary sharpening of the saturated yield locus in
the loading direction combined with a flattening on the opposite side can be taken into account
[2]. Moreover, the yield locus evolves smoothly and its convexity is ensured at each hardening
stage.
The rheological model serves as a guideline for construction of new constitutive relations. The
kinematic assumptions, the ansatz for the free energy and for the yield function are motivated by
the rheological model. Additionally to kinematic and distortional hardening, a nonlinear isotropic
hardening is introduced as well. Normality flow rule is considered, and a rigorous proof of ther-