Statics of Granular Materials

First up is as much of an introduction paper I wrote in Summer 2010 as I have up so far. Some diagrams are missing; I am trying to add sections as I add the relevant diagrams, but have been having some technical difficulties.  This information should be accessible to anyone with AP or college mechanics. Calculus is avoided as much as possible. 

After the introduction is the powerpoint I used to present some of this information in October 2010.
As always, credit me if you use any of this, and get in touch for permission before using any of it for profit. 

Introduction

A granular material is any material composed of many individual solid particles.  These materials are prevalent, and include everything from corn and nuts to sand and asteroid belts.  The storage and transportation of granular materials are a large part of current industry, especially the construction and food industries.  In most cases, engineers use empirically derived relationships for designing containers and systems for granular materials.  However, ongoing physics study is looking for theoretical, experimentally confirmed relations.  The current research, with its beginnings around the 1960s, is most clearly descended from that of soil mechanics.  However, granular mechanics work was done even in the eighteenth century, most prominently by Coulomb.

This text is an introduction to the basic concepts involved in the statics of granular materials.  It focuses specifically on the conceptual; for deeper mathematical explanations and derivations, Statics and Kinematics of Granular Materials by R.M. Nedderman is recommended, as it is the main source for this text and mathematically in-depth.  Another potentially useful text is Fundamentals of Particle Technology by Richard Holdrich, available free online. 

The following two sections, disjointed though they may seem, are the starting points for two as yet incompletely related types of information on granular materials.  Both the properties inherent in particular granular materials, such as particle shape and material density, and the stresses created by and within materials are studied.  It is not yet entirely clear whether the examination of a sample of individual particles could someday allow for an analysis of a bulk of the material under varied circumstances.  For now, information on both the material properties and the more mathematical calculations of the stresses will be side by side, with as much relation as is currently known.

Particle Size, Shape, and Distribution

While it is very simple to determine an accurate measure of size for spherical or cuboid particles, less symmetrical particles create some difficulty.  There are several types of “diameters” used to describe asymmetrical particles.  One is the equivalent spherical diameter, or the diameter of the sphere with the same volume.  Another is the feret diameter, the distance between two parallel tangents; for non-spherical particles, this is a function of the angle of the tangents. 

Aside from a measure of the size of particles, it is necessary to have a measure of how asymmetrical a particle is.  The most common measurement for this is called the shape factor.  The shape factor is the ratio of the particle’s surface area to that of the sphere with the same volume; the reciprocal of this is called the sphericity.

Despite these measurements of particle size and shape, the picture of the particles within a material is not complete.  In general, materials will have a range of particle sizes, or a particle size distribution.  While similar distributions exist for shape, that of size is considered more important.  The particle size distribution can be described using any measure of particle size; it will be assumed that the measure is some type of diameter.   The size distribution is described in terms of a function of size, such that the integral of the function from zero to a given size will give the total amount of particles up to that size; the integrated function is called the cumulative size distribution.  The “amount” of particles can be the number of particles or the total mass or the total volume of the particles. A useful method for determining the size distribution is a series of sieves: if the amount that can pass through each sieve is measured, then you have points along the cumulative size distribution.

The size distribution sometimes is a log-normal curve; in this case, it is very simple to transition between distributions describing number and those describing mass or volume, as both take the same form.  Another useful piece of information is the mean size, and in a log-normal distribution this is simple to find.  Also, from a log-normal distribution, other means can be found easily from the mean diameter.  One of particular interest is the volume-surface mean diameter, or the volume mean divided by the surface mean; this measure is used for determining the permeability of a material, a constant relating to the rate at which the interstitial fluid can flow.

Mohr’s Circle

Mohr’s Circle is a method of determining the stresses on an area of material at any angle based on the stresses at two perpendicular angles.  A stress is a ratio of force to area and is classified as a normal or shear stress.  Normal stresses act perpendicularly to the surface and are usually denoted by s; shear stresses act in-plane with the surface and are typically denoted by t.  Looking at a cube of material, such as in Figure 1, we would label the stress acting in the x direction on a face whose normal is in the x direction as sxx; txy is a stress acting in the y direction on the x-face (the face whose normal is in the x-direction).


Mohr’s Circle is used to look at one plane.  So if we look at the z-face of the cube, the forces on the edges are the forces on the x- and y-faces (see Figure 2).  As long as the material is not moving and the forces are constant across the area we look at, we can measure sxx, syy, txy, and tyx, where  tyx = -txy (be aware that compressive normal stresses and counterclockwise shear stresses are considered positive).  If these are put in (s,t) ordered pairs where the first subscript is the same and plot on s and t axes, we can draw a circle in that plane where a line drawn between the two points is a diameter; in Figure 3, the points labeled X and Y are the ordered pairs for the x- and y- planes.

Once we have the circle in the s-t plane, to find the stresses along axes at any angle to the x-y axes we simply find the points twice that angle around the circle from the points we started with.  One reason it is twice the angle is because if we rotate the axes 90°, we should get the same stresses, simply with changed subscripts, and that means we should be at the original points on the circle, or rotated 180°.

There are two points of particular interest on the circle.  These are its intersections with the s axis.  These two normal stresses are called the major and minor principal stresses and define the range of normal stresses possible.  The planes of the principal stresses are called the major and minor principal planes, and are always perpendicular.

Coulomb Yield Criterion

The Coulomb yield criterion is an equation which describes the stresses on a slip, or failure, plane for a group of granular materials called ideal Coulomb materials.  One of the most common types of failure, change from a previously static situation, in granular materials is plastic-rigid failure.  When a force acts on a pile or other conglomeration of a granular material, initially the pile acts like a solid, deforming or giving a bit; past a certain point, however, the stresses will overcome any forces keeping the individual grains in their current order, and some of the grains will move.

In plastic-rigid failure, most of the moving portion of the pile translates as one block, like a rigid solid; the portion which does not move also acts like a rigid solid.  Between the two rigid areas is a small plastic zone, generally idealized as a plane.  That zone is the slip plane.  The stresses acting on the plane are not dependent on the speed the rigid zones are moving past each other, like in Newtonian fluids, or how far they move relative to one another, like in deforming Hookean solids.  Rather, for ideal Coulomb materials, the shear stress, or the force working against the motion, is linearly related to the normal stress, or the force pushing the two rigid zones together.  This is analogous to the friction between two objects, where F=mN.  In this case, t=ms+c.  Here, m is called the coefficient of internal friction and c is called the cohesion.  In general, c is larger for smaller particles and m is larger for more angular particles.

The Coulomb yield criterion equation is often referred to as the internal yield locus, or IYL, especially when it is used graphically in the s-t plane.  Also, the angle f, the internal angle of friction, is defined by  tan(f)=m; this angle is crucial in later analyses, and is the angle the IYL makes with the s axis.

Internal Failure Properties

The internal failure properties of a material describe the amount of force needed to cause the formation of a slip plane under various circumstances.  One example of a set of failure properties is the Coulomb failure criterion, which relates the shear stress needed for a slip plane to form under the circumstance of a certain amount of stress normal to the plane.  Failure properties cannot yet be determined by examining particles, and so are directly tested.

Failure properties are usually determined using shear cells.  These devices are like small boxes or, more often, circular troughs, with moveable lids and sides.  Shear cells allow the user to put a certain amount of surcharge on a sample of material, generally by putting weight on the lid, and then add horizontal shear forces, either by changing the angle of one side or through rotation.  The shear force will peak as the slip plane forms, after which the continued application of force will cause some of the sample to move.  The peak shear stresses are the criterion for failure for their related surcharges.  Since the shear force is perpendicular to the surcharge, the surcharge can be used to calculate the normal force.  Thus, through repeated tests with different surcharges, a function describing the failure criterion can be found.

Some complications do exist in determining the failure properties.  First, the consolidation, or density, of the material affects the tests; density is discussed below.  Second, the time spent in testing can affect results- for some materials, if a force is applied over time and forms a slip, there will be a second, lower peak of shear force, due to dilation (expansion of the material) around the slip.

Density

While the solid density, or density of the particles in a granular material, is constant, the bulk density, the density of the material including the fluid between particles, or interstitial fluid, is not.  The bulk density depends upon the degree to which the material is compacted, or consolidated. 

Consolidation occurs under the conditions of elevating pressures and/or vibration.  For most materials, the change in the specific volume, the reciprocal of bulk density, is directly related to the negative log of the pressure. If the pressure decreases, there is usually no dilation.  Therefore, the bulk density depends not only on the current conditions but also the past stresses on the material.  In general, the bulk density for a given situation is determined by empirical tests; it is also possible in some cases to define a function for the bulk density based on the principal stresses and tests which define the specific volume-pressure curve.


Mohr-Coulomb Analysis

The Mohr-Coulomb analysis allows for a combination of the Coulomb yield criterion and Mohr’s circle to give the existence and angles of any incipient failure planes.  Simply put, if any point on Mohr’s circle for an area fits the Coulomb yield criterion, that plane is about to become a slip plane.

For an approximate analysis, it suffices to graph both Mohr’s circle and the Coulomb yield criterion line, or Internal Yield Line, in the s-t plane and observe whether the IYL touches the circle, and if so, at what angle from the x-axis.  Properly, both the positive IYL and its reflection across the s axis should be considered the IYL, since the positive or negative aspect of a stress merely indicates direction.  For an exact analysis, it would be necessary to algebraically solve for the intersection of the IYL and Mohr’s circle.

In some cases, there will be no intersection, and so no incipient failures.  If there is an intersection, there will be two- one on either side of the s axis.  Also, if there is an intersection, it will appear that the IYL is tangent to Mohr’s circle, such as in Figure 4.  The IYL cannot cut Mohr’s circle- if it did, the angles above the IYL will have already failed and some of the considered granular material would be in motion, requiring a different analysis.



Sources and Further Reading

Dartevelle, S., Numerical and granulometric approaches to geophysical granular flows, Ph.D. thesis, Michigan Technological University, Department of Geological and Mining Engineering, Houghton, Michigan, July 2003. Located at <http://www.granular-volcano-group.org>.

Holdrich, Richard. Fundamentals of Particle Technology. Shepshed, Leicestershire, UK: Midland Information Technology and Publishing, 2002.

Nedderman, R.M. Statics and Kinematics of Granular Materials. Cambridge: Cambridge
University Press, 1992.