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Modeling of Microstructures in a Cosserat Continuum Using Relaxed Energies

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Trends in Applications of Mathematics to Mechanics

Part of the book series: Springer INdAM Series ((SINDAMS,volume 27))

Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. Lehrstuhl für Mechanik-Materialtheorie, Ruhr-Universität, Bochum, Germany

    Muhammad Sabeel Khan & Klaus Hackl

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  1. Muhammad Sabeel Khan
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  2. Klaus Hackl
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Correspondence to Klaus Hackl .

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Editors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, Pavia, Pavia, Italy

    Elisabetta Rocca

  2. Fakultät für Mathematik, Universität Wien, Wien, Austria

    Ulisse Stefanelli

  3. CNRS, Paris, Paris, France

    Lev Truskinovsky

  4. Dipartimento di Matematica, Università degli Studi di Trento, Povo di Trento, Trento, Italy

    Augusto Visintin

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Khan, M.S., Hackl, K. (2018). Modeling of Microstructures in a Cosserat Continuum Using Relaxed Energies. In: Rocca, E., Stefanelli, U., Truskinovsky, L., Visintin, A. (eds) Trends in Applications of Mathematics to Mechanics. Springer INdAM Series, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-319-75940-1_6

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