Modeling of microstructures in a Cosserat continuum using relaxed

SISSA, Trieste, Italy, June 6–10, 2016
Modeling of microstructures in a Cosserat continuum using relaxed energies.
Klaus Hackl1 , Sergio Conti2 , Muhammad Sabeel Khan1
1
Lehrstuhl für Mechanik - Materialtheorie, Ruhr-Universität Bochum, Bochum Germany,
2
Institut für Angewandte Mathematik, Universität Bonn
A continuum model for granular materials exhibiting microstructures is presented using the
concept of energy relaxation. In the framework of Cosserat continuum theory the free energy of the material is enriched with an interaction energy potential taking into account the
counter rotations of the particles. The total energy thus becomes non-quasiconvex, giving rise
to the development of microstructures. Relaxation theory is then applied to compute its exact
quasiconvex envelope. It is worth mentioning that there are no further assumptions necessary
here. The computed relaxed energy yields all possible field fluctuations of displacements and
micro-rotations as minimizers. We show that the material behavior can be divided into three
different regimes. Two of the material phases are exhibiting microstructures in rotational and
translational motion of the particles, respectively, and the third one is corresponding to the
case where there is no internal structure of the deformation field. The properties of the proposed model are demonstrated by carrying out numerical computations. The obtained results
exhibit a number of unexpected features, for example the transition between distributed and
localized microstructures.
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