Quasi-rigidity: some uniqueness issues

TL;DR

This paper investigates the concept of quasi-rigidity in granular assemblies, demonstrating that it provides a complete and unique predictive framework for disk-shaped grains under slow loading, resolving potential indeterminacies.

Contribution

It proves that quasi-rigidity theory is complete and unique for disk-shaped grains by addressing and resolving potential indeterminacies related to zero-frequency modes and contact status.

Findings

Zero-frequency modes do not cause indeterminacy when force equilibrium is considered.

Only one contact sliding choice is permitted under Coulomb friction.

Quasi-rigidity provides a complete predictive framework for disk-shaped granular assemblies.

Abstract

Quasi-rigidity means that one builds a theory for assemblies of grains under a slowly changing external load by using the deformation of those grains as a small parameter. Is quasi-rigidity a complete theory for these granular assemblies? Does it provide unique predictions of the assembly's behavior, or must some other process be invoked to decide between several possibilities? We provide evidence that quasi-rigidity is a complete theory by showing that two possible sources of indeterminacy do not exist for the case of disk shaped grains. One possible source of indeterminacy arises from zero-frequency modes present in the packing. This problem can be solved by considering the conditions required to obtain force equilibrium. A second possible source of indeterminacy is the necessity to choose the status (sliding or non-sliding) at each contact. We show that only one choice is permitted, if contacts slide only when required by Coulomb friction.