Granular Matter

, Volume 12, Issue 3, pp 239–252 | Cite as

Stress, stress asymmetry and couple stress: from discrete particles to continuous fields

  • Isaac Goldhirsch


Explicit and closed expressions for the stress and couple-stress fields for discrete (classical) mechanical systems in terms of the constituents’ degrees of freedom and interactions are derived and compared to previous results. This is done by using an exact and general coarse graining formulation, which allows one to predetermine the resolution of the continuum fields. Since the full dynamics of the pertinent fields is considered, the results are not restricted to static states or quasi-static deformations; the latter comprise mere limiting cases, which are discussed as well. The fields automatically satisfy the equations of continuum mechanics. An explicit expression for the antisymmetric part of the stress field is presented; the question whether the latter vanishes, much like its nature when it does not, have been debated in the literature. Physical explanations of some of the obtained results are offered; in particular, an interpretation of the expression for the stress field provides an argument in favor of its uniqueness, yet another topic of debate in the literature. The formulation and results are valid for single realizations, and can of course be used in conjunction with ensemble averaging. Part of the paper is devoted to a biased discussion of the notion of coarse graining in general, in order to set the presented results in a certain perspective. Although the results can be applied to molecular (nanoscale included) and granular systems alike, the presentation and some simplifying assumptions (which can be easily relaxed) target granular systems. The results should be useful for the analysis of experimental and numerical findings as well as the development of constitutive relations.


Coarse graining Averaging Homogenization Stress Stress asymmetry Couple Stress Cosserat Continuum mechanics Discrete mechanical systems Granular systems Nanoscale systems 


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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.School of Mechanical Engineering, Faculty of EngineeringTel-Aviv UniversityTel-AvivIsrael

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