Modeling of Microstructures in a Cosserat Continuum Using Relaxed Energies

  • Muhammad Sabeel Khan
  • Klaus HacklEmail author
Part of the Springer INdAM Series book series (SINDAMS, volume 27)


Granular materials tend to exhibit distinct patterns under deformation consisting of layers of counter-rotating particles. In this article, we are going to model this phenomenon on a continuum level by employing the calculus of variations, specifically 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 displacement and micro-rotation field fluctuations as minimizers. Based on a two-field variational principle the constitutive response of the material is derived. Results from numerical computations demonstrating the properties of relaxed potential are shown.


Cosserat Continuum Theory Quasiconvex Envelope Interaction Potential Energy Material Regime Modular Interaction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The first author gratefully acknowledge the funding by Higher Education Commission (HEC) of Pakistan and highly appreciate the support by Deutscher Akademischer Austausch Dienst (DAAD) for this research work.


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Lehrstuhl für Mechanik-MaterialtheorieRuhr-UniversitätBochumGermany

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