Abstract
The isostaticity theory for stress transmission in macroscopic planar particulate assemblies is extended here to non-rigid particles. It is shown that, provided that the mean coordination number in d dimensions is d + 1, macroscopic systems can be mapped onto equivalent assemblies of perfectly rigid particles that support the same stress field. The error in the stress field that the compliance introduces for finite systems is shown to decay with size as a power law. This leads to the conclusion that the isostatic state is not limited to infinitely rigid particles both in two and in three dimensions, and paves the way to an application of isostaticity theory to more general systems.
I am grateful to Prof. Robin Ball for critical comments.
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A typical grain size is defined here for convenience only; there is no loss of generality and the discussion is equally valid for packings of particles with any distribution of grain sizes.
The extension of the analysis presented here to packings of particles of different materials is straightforward.
It should be noted that for finite systems it is always possible to find a finite small force F y that satisfies this criterion. However, the magnitude of this force decreases with system size.
Without loss of generality, the packing is presumed to experience no body forces.
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Blumenfeld, R. (2005). Stress Transmission and Isostatic States of Non-Rigid Particulate Systems. In: Calderer, MC.T., Terentjev, E.M. (eds) Modeling of Soft Matter. The IMA Volumes in Mathematics and its Applications, vol 141. Springer, New York, NY. https://doi.org/10.1007/0-387-32153-5_10
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