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From Granular Matter to Generalized Continuum

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Mathematical Models of Granular Matter

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1937))

Following a cursory review and synthesis of multipolar continua, the rudiments of graph theory, and granular mechanics, a graph-theoretic, energy-based homogenization is proposed for the systematic derivation of multipolar stress and kinematics in granular media. This provides a weakly non-local hierarchy of multipolar field equations for quasi-static mechanics based on polynomial representations of the kinematics of the type employed in past works. As an improvement on those works, a method is proposed for avoiding “overfitting” of fluctuations based on the so-called “Generalized Additive Method” of statistics. Among other results, it is shown that the standard formula for Cauchy stress in granular media may break down owing to multipolar effects, and that granular rotations in the typical granular medium should not lead to Cosserat effects, as the lowest-order departure from the simple-continuum model.

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Author information

Authors and Affiliations

  1. Department of Mechanical and Aerospace Engineering, University of California at San Diego - La Jolla, La Jolla, CA, 92093-0411, USA

    J. D. Goddard

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  1. J. D. Goddard
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Editors and Affiliations

  1. Department of Mathematics, University of Pisa, largo Bruno Pontecorvo 5, 56127, Pisa, Italy

    Gianfranco Capriz

  2. Accademia Nazionale dei Lincei, via della Lungara 10, 00165, Roma, Italy

    Gianfranco Capriz

  3. DICeA, University of Florence, via Santa Marta 3, 50139, Firenze, Italy

    Paolo Maria Mariano

  4. Dipartimento di Meccanica e Materiali, Università “Mediterranea” di Reggio Calabria, via Graziella, località Feo di Vito, 89060, Reggio Calabria, Italy

    Pasquale Giovine

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Goddard, J.D. (2008). From Granular Matter to Generalized Continuum. In: Capriz, G., Mariano, P.M., Giovine, P. (eds) Mathematical Models of Granular Matter. Lecture Notes in Mathematics, vol 1937. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78277-3_1

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