[Paper Review] Macroscopic Continuous Approach versus Discrete Approach, Fluctuations, criticality and SOC. A state of the question based on articles in Powders & Grains 2001

P. Évesque|arXiv (Cornell University)|Jan 1, 2001

Granular flow and fluidized beds17 references3 citations

TL;DR

This paper evaluates the validity of the macroscopic continuous approach in granular media by analyzing experimental and simulation data from Powders & Grains 2001, concluding that despite significant microscopic fluctuations and critical-like behavior, the continuous approach remains a robust approximation for typical laboratory-scale granular samples, with the Representative Elementary Volume (REV) typically spanning just a few grains, except in special cases where dynamical instabilities or size-dependent effects may extend the REV beyond this scale.

ABSTRACT

The macroscopic continuous approach of the mechanics of granular media assumes that the whole system of discrete variables (contact locations, contact forces,...) can be replaced by continuous field equations relating stress and strain on macroscopic scale. On the contrary, recent approaches contest this validity on the basis that microscopic studies show the existence of large fluctuations of forces, of chains of forces,... This paper tries and establishes the state of this question using recent works reported at Powders & Grains 2001 which have studied the microscopic variables, their fluctuations and their evolution. This paper shows that these results validate the macroscopic approach despite the existence of these fluctuations. It concludes that the representative elementary volume is of few grains most of the time, except in some peculiar cases. Pacs # : 5.40 ; 45.70 ; 62.20 ; 83.70.Fn

Motivation & Objective

• To assess the validity of the macroscopic continuous approach in granular mechanics against emerging evidence of large-scale fluctuations and criticality.
• To resolve the tension between classical continuum mechanics and recent findings of self-organized criticality (SOC) and force chain heterogeneity.
• To determine the size and conditions under which the Representative Elementary Volume (REV) becomes meaningful in granular systems.
• To reconcile the persistence of macroscopic behavior with microscopic complexity, such as force chains and intermittent avalanches.
• To clarify whether granular systems exhibit truly anomalous, non-averaging behavior or if classical averaging remains applicable under most experimental conditions.

Proposed method

• Analysis of experimental and simulation results presented at the Powders & Grains 2001 conference, focusing on microscopic variables like contact forces and force chains.
• Use of statistical analysis to evaluate fluctuations in force distributions and their scaling behavior across different system sizes.
• Comparison of discrete element method (DEM) simulations with continuum modeling to assess consistency in macroscopic responses.
• Application of concepts from statistical physics, including critical phenomena, power-law distributions, and renormalization group theory, to interpret observed behaviors.
• Evaluation of the BTW sandpile model as a prototype for self-organized criticality (SOC), and its relevance to real granular systems.
• Investigation of size-dependent transitions in mechanical response (e.g., from random stick-slip to quasi-periodic behavior) to infer REV size.

Experimental results

Research questions

• RQ1 To what extent do microscopic fluctuations in granular force chains invalidate the macroscopic continuous approach?
• RQ2 What is the effective size of the Representative Elementary Volume (REV) in granular materials under typical experimental conditions?
• RQ3 Under what conditions does the system transition from random, uncorrelated stick-slip behavior to quasi-periodic, correlated dynamics?
• RQ4 Does the observation of power-law distributed avalanches in simulations and experiments imply that granular matter is intrinsically critical, or can it still be described by continuum mechanics?
• RQ5 Can the self-organized criticality (SOC) paradigm be applied to real granular systems, or is it an artifact of specific model conditions?

Key findings

• Despite large fluctuations in contact forces and the presence of force chains, the macroscopic continuous approach remains a valid approximation for granular systems under typical laboratory conditions.
• The Representative Elementary Volume (REV) is generally on the order of just a few grains, indicating that macroscopic averaging is effective even at small scales.
• In some cases, such as stick-slip transitions, the REV size can extend to 200–500 grain diameters due to dynamical instabilities, not necessarily due to criticality.
• The transition from random to quasi-periodic stick-slip behavior in different-sized samples suggests a crossover influenced by system size and strain rate, not solely by critical scaling.
• The BTW sandpile model generates power-law distributed avalanches (P(N) ∝ N⁻α), but such behavior is not typically observed in real granular experiments except in very small systems.
• The existence of critical fluctuations and SOC-like behavior does not invalidate the continuum approach, as long as the system is not operating at special critical points or under extreme dynamical conditions.