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Microstrain Analysis

2366 words - 10 pages

Micromechanics of materials with microstructure:
FORCE CHAINS

Contents

Introduction 2
Theory 3
Experimental Determination of Contact Forces 4
Derivation of equations in granular shear experiment 5
Goal and solution by this analysis 9
Micromechanical Strain Calculation or visualization using MATLB 9
Significance of the problem 11
Possible refinements 11
References 12


Introduction

Each of the grains is in itself a classical solid body the physics of which is extremely well understood, yet the conglomeration of many of them leads to novel collective behavior.

Inter-particle forces in granular media form an inhomogeneous distribution of filamentary ...view middle of the document...

In granular assemblies, the classical theories of matter are thus pushed to the limits of their validity. This is a fascinating situation which shines a new light both on long-established general theories and on specific properties of granular systems.

Theory

Under the action of external stresses, grains in dry granular materials form an inhomogeneous contact network, which carries most of the external load by way of force chains. The resultant network is different for shearing than for isotropic compression and is history-dependent owing to friction. Previous experiments have reported an exponential tail for the distribution of contact force magnitudes. This tail can be successfully predicted by many models with radically different mathematical structures and microscopic assumptions. Testing the validity of these models requires that the predicted force distributions be verified by measurements of full vectorial contact forces in the bulk of the sample.

It is also important to find other distinguishing signatures characterizing the nature of force chain networks under different boundary conditions—an important goal of the present work.

Experimental Determination of Contact Forces

From an experimental point of view, the macroscopic nature of particles forming granular matter presents a number of practical advantages for the study of contact properties. First, direct imaging of the grains is possible without having to resort to microscopy or scattering techniques. Second, by carefully choosing the studied system, the properties of the material of the grains can be exploited to directly visualize the contact forces. A full study of a three-dimensional packing remains however a daunting task and experiments have concentrated either on forces between grains and the boundaries, or on two-dimensional setups.

The first quantitative studies of contact forces were done in a very simple experimental apparatus. The forces between the grains and the walls were determined by placing a layer of carbon paper on the inner surface of the container. The magnitudes of the normal forces were deduced from the sizes of the marks left, which obey a linear relation with applied pressure. In later experiments, a number of improvements in the measurement techniques were introduced. Nevertheless, all these methods access only the boundary forces, the properties of which might significantly differ from the forces in the bulk.

An alternative approach is to use a two-dimensional system of particles made of a photo-elastic material, which displays stress-induced birefringence in response to applied forces. With the help of cross-polarizers, large stresses inside the grains can be directly observed. While such direct imaging gives a compelling qualitative picture of stresses inside a granular packing, precise quantitative analysis is difficult. Extracting individual contact forces from the polarization pattern is a highly non-trivial task, and full...

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