[Paper Review] Deformation Measures for Granular Materials

Matthew R. Kuhn|arXiv (Cornell University)|Dec 3, 2024

Geotechnical and Geomechanical Engineering25 citations

TL;DR

The paper develops a micromechanical, void-cell based representation of deformation in 2D granular assemblies, extending Bagi and Satake frameworks, and demonstrates it via a 2D DEM simulation with observations of highly nonuniform deformation and micro-bands.

ABSTRACT

The paper presents a micromechanical representation of deformation in 2D granular materials. The representation is a generalization of K. Bagi's work and is based upon the void-cell approach of M. Satake. The general representation applies to a material region partitioned into polygonal subregions. This representation possesses a certain consistency that allows for a unique assignment of the contribution that each contact displacement makes to the average deformation of an assembly. The paper addresses construction of the particle graph and appropriate data structures for use with the Discrete Element Method. The approach is applied in a numerical simulation of a two-dimensional assembly of disks. The author presents results of the distributions of deformation and particle-group rotation, with a resolution of about a single particle diameter. Deformation was very nonuniform, even at low strains. Micro-bands, thin linear zones of intense rotation, were also observed.

Motivation & Objective

• Motivate a micromechanical view of deformation in granular materials.
• Generalize existing void-space partitioning methods to polygonal subregions.
• Establish a consistent method to assign the contribution of each contact displacement to the assembly's average deformation.
• Develop particle-graph and data structures suitable for Discrete Element Method (DEM) implementations.
• Demonstrate the approach with numerical simulations of a 2D disk assembly and analyze deformation and rotation distributions.

Proposed method

• Represent a 2D material region as a partition into polygonal subregions (void cells).
• Express the average velocity gradient in terms of vertex and edge motions within each subregion.
• Define a consistent Q^m matrix for relating edge-relative velocities to local deformation, with a recursive construction (Eq. 11).
• Generalize Bagi’s triangular case to m-edge polygons for local deformation computation (Eq. 7).
• Describe planar graph construction (vertices at particle centers, edges at contacts, faces as voids) and data structures (SLL, DLL, DCEL) for DEM integration.
• Outline an algorithm to construct void-cell faces and maintain topological consistency (Eq. 6–11).

Figure 1: 2D topological representations

Experimental results

Research questions

• RQ1 How can deformation in a polygonally partitioned granular region be related to motions of polygon vertices and edges?
• RQ2 Can a consistent, unique assignment be made for the contribution of contact displacements to the average deformation of a granular assembly?
• RQ3 What data structures and algorithms are suitable for constructing and updating the particle-graph/void-cell graph in DEM simulations?
• RQ4 What does 2D numerical simulation reveal about the spatial distribution of deformation and rotation in granular assemblies?
• RQ5 Do micro-scale deformation features such as nonuniform fields and micro-bands emerge under typical loading?

Key findings

• Deformation in 2D granular assemblies is highly nonuniform even at low strains.
• Micro-bands, thin linear zones of intense rotation, are observed in the simulated assembly.
• A consistent, polygon-based deformation measure is obtainable via a Q^m matrix, enabling unique contributions from contact displacements to local deformation.
• The approach yields a detailed, particle-scale view of deformation and rotation distributions within single particle diameter resolution.
• A planar graph framework with DEM-ready data structures is proposed for efficient topological tracking of contacts and voids.

Figure 2: Triangular face: edge, velocity, and relative velocity vectors