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A Review of Plastic-Frictional Theory
Part. 1
Introduction, Mohr-Coulomb, and von Mises Stresses

You will find the basic facts about Plastic-Frictional Theories (Part. 1) - no details -. If you wanna know more just email me or feel free to ask in the Discussion Forum. I purposely erased all the bibliographical references and detailed equations to keep the text simple and easy to read.

If you need an official reference for the content of this website, please, use:
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.

You will find on this page:
I. Introduction
II. Stress space, Slip Planes, Mohr-Coulomb and von Mises stresses
      II.1. Mohr-Coulomb case: a 2D representation of stress (particular case)
      II.2. von Mises case: a 3D representation of stress (general case)

While, on the following pages, you will find the continuation of this frictional course:
III. Plastic Potential Theory
IV. Dilatation, consolidation, yield locus, and critical state
      IV.1. Dilatancy
      IV.2. Consolidation
      IV.3. Yield locus and critical state
V. Constitutive Equations for Frictional Granular Flows
      V.1. Beyond the extended von Mises Yield locus
      V.2. Within the extended von Mises Yield locus

Please, don’t forget to sign our Granular Volcano Guest Book.

Done and updated by Sébastien Dartevelle, The WebMaster, Saturday November 22 2003.

I. Introduction

 

Concentrated granular flows are economically important phenomena, e.g., in pharmaceutical industry, corn flow in a silo, coal and flour in a bin, granular material flowing under the action of gravity in a hopper are a few common examples. Unfortunately, those granular flows are poorly understood. It is common for the industries to deal with enormous difficulties in the withdrawal process of the granular material from a container (bin, silo, hopper). This cause high financial losses, and it is not rare to see the complete collapse of silos as seen on Picture 1:

 


[Picture1: Collapse of a granular (corn) silo due to a poor design. Such a dramatic phenomenon are very common with silos]
[Most of the time, the silo is poorly designed due to a lack of understanding of concentrated granular flow behaviors and properties]
[In particular, not taking into account compressibility effects and using inadequate modeling techniques (like depth-average technique) are the main responsible causes of such disaster]

 

Most of the silos and hoppers are designed after computer modeling of granular flows. Unfortunately, those modeling are based on simple depth-averaged technique, sometimes called in engineering design, slice analysis. Those modeling techniques are inadequate since they only depend on one space variable. Flowing granular materials -even highly concentrated and frictional- display complex instability in time and space, and are fundamentally unsteady and slightly compressible. Those depth-average techniques -also commonly used in volcanology and sedimentology- are totally unable to approach such important complexities and common properties of granular flows.

 

At very high concentrations and low rate-of-strain, collisions cannot be seen as instantaneous anymore, since grains suffer longer and permanent contacts in rubbing, rolling on each other. Therefore, a kinetic stress model based on the Boltzmann equation is irrelevant and a frictional stress model must be taken into account. This can be done using plasticity and similar theories in which the material behavior is assumed to be independent of the velocity gradient or the rate-of-strain. Needless to say, this is atypical for a viscous Newtonian flow where stress specifically depends on rate-of-strain and, again, this shows that a Newtonian rheology cannot be chosen for granular flow whatever the concentration. Under a normal stress, a well-compacted granular material will shear only when the shear stress attains a critical magnitude. This is described by a Mohr-Coulomb law based on the laws of sliding friction. However, the Mohr-Coulomb law says nothing about how the granular material deforms and flows, it rather describes the onset of yielding. The Plastic Potential theory will provide the required constitutive equations for describing the deformation of a granular material under frictional motion. According to this theory, the concept of critical state plays a key role. However, a critical state described by a simple Mohr-Coulomb theory leads to physical inaccuracies (such a infinite dilatancy of the granular material). Actually, the Mohr-Coulomb law should only be seen as an asymptotical solution after the granular material has endured long and permanent plastic-frictional deformation. The Plastic Potential theory allied with the critical state approach can successfully described the phenomenon of dilatancy, consolidation, the independence between the rate-of-strain-tensor and the stress tensor. It will also become clear in this manuscript that the Mohr-Coulomb law does not say anything and should not play a central role for describing flowing and deforming granular materials at yield.

 

In establishing the constitutive equations for frictional granular media, I review in this page the basic theoretical concepts behind frictional and plastic theories, and I also review the important properties of granular flow, such as dilatancy and consolidation phenomena. Then the constitutive equations are developed. Those equations show a clear independence between the stress tensor and the rate-of-strain tensor of the flowing material as prescribed by the theory.

 

It is assumed that the material is slightly compressible, dry, cohesionless, and perfectly rigid-plastic. Such properties are relevant for modeling the granular flows commonly seen in volcanology-geophysics. As done throughout this website, and as done in Soil Mechanics, I consider that compressive stress, compressive strain, and their rates are taken to be positive or, if you prefer, tensile stress, elongative strain, and their rates are negative (:-).

 

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II. Stress space, Slip Planes, Mohr-Coulomb and von Mises stresses

 

In a 3D with a Cartesian coordinate system, we can write the total stress tensor for a frictional granular flow as:

 

 

      Eq. 1

 

 

where Txx, Tyy, Tzz represents the normal stress acting on the YZ, XZ and XY planes respectively, Txy, Txz are the shear stresses acting in the Y-direction and Z-direction respectively on the plane YZ (whose normal is X), Tyx, Tyz are the shear stress in the X- and Z-directions respectively acting on the plane XZ (whose normal is Y), and Tzx, Tzy are the shear stress in the X- and Y-direction acting on the plane XY (whose normal is Z). Since a stress tensor must be symmetric, we have Txy = Tyx, and Txz = Tzx and Tyz = Tzy (see Fig. 1).

 


[Figure 1: Positive directions of stress in the different planes]
[The sign convention is that if we assume that the X-, Y-, Z-velocity gradients are positive in the X-, Y- and Z-directions,
then the stress is positive in the opposite directions. In other words, compressive stress is always positive]

 

Let’s note that the eigenvalues of T are with and their associated eigenvectors (principal directions) are n1, n2, and n3 respectively. It is always possible to decompose the stress tensor into two parts, spherical and deviatoric:

 

 

      Eq. 2

 

 

where I is the unit tensor and IT is the first invariant or the "trace" of the stress tensor (i.e., the sum of the diagonal element). IT is said to be invariant because it is independent of the chosen coordinate system (e.g., the sum of the diagonal element in the Cartesian or Principal direction coordinate system is the same) as indicated by Eq.2b. Obviously, the sum of the diagonal elements of the deviatoric stress tensor (Eq. 2c) is equal to zero, hence the deviatoric is said to be "traceless".

 

It is now convenient to write the relationship between the stress tensor and the rate-of-deformation (or rate-of-strain) tensor. Let’s write this way:

 

 

      Eq. 3

 

 

where are the bulk and shear viscosities, P is an isotropic/hydrostatic Pressure (normal stress), is the divergence of the velocity, are the gradient and its transpose of the velocity, is the deviatoric part of the rate-of-strain tensor, and ID is the first invariant or the trace of the rate-of-strain tensor (i.e., sum of its diagonal elements). Clearly, Eq.3 associates shear stress with shear rate-of-strain (Eq.3b) and normal stress with normal rate-of-strain (Eq.3a). Many of you may be puzzled by the fact that Eq.3 looks like a rheology very Newtonian indeed. But as we will see later (paragraph V) the shear and bulk viscosities are not constant and actually are complicate functions of the velocity gradient, hence Eq.3 is not at all a Newtonian rheology. Also, if you are surprised by the minus sign in Eq.3c and Eq.3d, please, recall that my sign convention imposes that compressive stress and compressive strain are positive.

 

Having said those very general relationships, we may now move on (please, keep them in mind as we will need them a lot in the followings).

 

If you already struggle here, you must then visit my web page All I wanna know ’bout viscous stress! where I explain all this from the very beginning. Also, you may ask any question you want in the Granular Discussion Forum.

 

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II.1. Mohr-Coulomb case: a 2D representation of stress (particular case)

 

Let’s assume that , and that the principal direction n1 forms an angle measured counterclockwise with the X-axis (see Fig 2A and Fig. 2B). A close inspection of Fig. 2 shows we may carry out a 2-dimensional analysis of stress. Since , the plane XZ only suffers normal stress (which is ), while the plane XY and YZ suffers shear (e.g., Tzx, Txy, so forth) and normal stress (Txx and Tzz).

 


[Fig. 2A]

[Fig. 2B]

[Convenient geometry that permits a 2-dimensional analysis of stress]
[In Fig. 2A, one coordinate axis (Y) is parallel to one of the principal coordinates (the eigenvector, n2)]
[Therefore, the principal directions n1 and n3 stand in the plane XZ]
[The great advantage of this geometry is that the stress can simply be analyzed in 2D in the plane XZ as shown in Fig. 2B]

 

We can now rewrite Eq. 1 as:

 

 

      Eq. 4

 

 

Let’s define the normal stress as the average between the principal stress components (in 2D, within the plane XZ):

 

 

      Eq. 5

 

 

while the shearing stress will be defined as:

 

 

      Eq. 6

 

 

And it is clear from Eq. 4, 5, and 6, we must have:

 

 

      Eq. 7

 

 

Consider an element of surface whose unit normal vector n is parallel to the XZ plane so that the projection of n on that plane has components , where is the angle between the n and the X-direction (see Fig. 3):

 


[Fig. 3A]

[Fig. 3B]

[Nearly the same situation as in Fig. 2A and 2B]
[In Fig. 3B, we focus on a particular plane (in green) within the infinitesimal volume, where n represents a normal unit vector to that plane]
[In Fig. 3B, the same situation but projected over the plane XZ in order to 2-dimensionalize the stress analysis]

 

Then the normal stress (N) and shear stress (S) acting on that surface element in terms of the normal and shearing principal stresses will be (Fig. 3B):

 

 

      Eq. 8

 

 

Eq.8 defines a circle on the N-S plane centered at the point (,0) and of radius equal to (see Fig. 4). The interpretation of the angle can be easily understood from Fig. 3B. The onset of yielding can be described with a Mohr-Coulomb model which asserts that a material will yield by shearing on a surface element with normal n if S attains a critical value given by (see Fig. 4):

 

 

      Eq. 9

 

 

where k is a known material property (describing the cohesive state of grains, and is therefore a cohesive shear), and is the angle of repose (or the angle of internal friction of the material). Most granular materials are cohesionless, k=0. The angle of friction can be easily understood from Fig. 5.

 


[Figure 4: Mohr-Coulomb relationships in the N-S plane]

[Figure 5: Angle of internal friction]

 

This angle of repose is low when grains are smooth, coarse or rounded, and, it is high for sticky, sharp, or very fine particles. Typically, it is between 15° and 45°. Experiments suggest that this coefficient of friction drops when motions begins, i.e., the kinetic friction coefficient is less than the static coefficient. However, no data exist for granular material, and the universal assumption is that the kinetic and static coefficient of friction are more and less equal.

 

The Mohr-Coulomb law is a linear law between the Shear (S) and the Normal (N) Stress. This line is a yielding condition for shearing (see Figure 4 and 6 where Eq. 9 is drawn in the N-S stress plane). Below this yield line, the material response will be rigid and does not suffer any strain. If the shear stress is increased for a given normal stress such that the stress state of the material is exactly on the yield line, then plastic strain or yielding will results. It is impossible to have a state of stress above this yield Mohr-Coulomb line.

 

Before continuing any further, it is important to recall that the angles on the physical plane (e.g., Fig. 2 and 3) are doubled on the N-S plane in a Mohr-Coulomb circle (e.g., Fig. 4 and 6).

 

When the yield stress is reached then particles will slide over each other. Therefore, at each point in the material, this can formally written as:

 

 

      Eq. 10

 

 

All those relations can be shown and explained on a Mohr’s circle as represented on Fig. 6:

 


[Figure 6: Mohr-Coulomb circles in the S-N stress plane]

 

It is clear that the left hand side of Eq. 10 is maximum (only at yield when equality holds) whenever (recall angle is positive counterclockwise) we have:

 

 

      Eq. 11

 

 

Therefore using Eq. 8, we can calculate the yielding value of the normal stress (N) and the shear stress (S) in term of :

 

 

      Eq. 12

 

 

And Eq. 10 may be written as:

 

 

      Eq. 13

 

 

which is another way of writing Mohr-Coulomb’s yield condition: Eq.13 shows the admissible states of stress. Slip may occur if and only if equality holds in Eq.13. The linear relation between and is of a fundamental importance, this line is called the yield line in the plane. Eq.13 is a yield condition in the plane. Recall that it is a linear function because of the Mohr-Coulomb relation given by Eq. 10. The orientations of the surface on which slip occurs are given by Eq. 11 and are shown on Fig. 7. From Eq. 13, Eq.5, and Eq. 6, at yield, we must have:

 

 

      Eq. 14

 

 

For a given angle of friction, Eq.14 gives a linear relation between the major () and the minor principal stresses () that follows from Mohr-Coulomb law (Eq. 9) (see Figure 8).

 


[Figure 7: Orientation of the Slip-planes relative to the major and minor principal stresses.
Phi is the angle of internal friction of the granular material. Those geometrical relations are given by Eq. 11]

[Figure 8: Domain of no-deformation (rigid) and domain of plastic deformation (represented by two lines having slopes given by Eq. 14.
Theoretically, the inside domain can represent the elastic deformation, however, for most of granular materials, elastic strain is negligible, and this inner domain can be assumed as perfectly rigid.
The central axis on which the principal stresses are equal represents the hydrostatic Pressure.
Compare this figure with Fig. 9 below, which is a generalization in 3D]

 

So, let’s summarize, at yield (when equality holds) and in 2D, It can be shown that the Mohr-Coulomb condition (in Eq. 14) can be written as:

 

 

      Eq. 15

 

 

which is four different ways of expressing the exact same Mohr-Coulomb yielding Law.

 

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II.2. von Mises case: a 3D representation of stress (general case)

 

The problem of the Mohr-Coulomb law is that we have to assume that . Therefore, the Mohr-Coulomb failure criterion is independent of the intermediate principal stress (). This a mathematical model (among others), and such Mohr-Coulomb assumptions may not be always possible or valid, specially for (some) granular materials. It is now clear that the velocity predictions based on the Mohr-Coulomb model may greatly differ from what it can be measured. Hence we must consider other approaches. An alternative approach would be to generalize in 3D without imposing any conditions on the intermediate principal stress.

 

Let’s define the normal stress as the average of the three principal stresses:

 

 

      Eq. 16

 

 

and the shear stress as:

 

 

      Eq. 17

 

 

Sometimes those stresses are called octahedral normal stress (Eq.16) and octahedral shear stress (Eq.17) respectively. In Eq.17 each principal stress difference is equal to the diameter of the appropriate Mohr’s circle and is therefore equal to twice the greatest shear stress in that set of planes ( see, for instance, Fig. 6). The octahedral shear stress is proportional to the root mean square of the three maximum possible shear stress as shown by Eq.17.

 

Taking the analogy with the Mohr-Coulomb law, we can postulate that there must be a proportionality relationship between octahedral shear stress and octahedral normal stress. We therefore have:

 

 

      Eq. 18

 

 

Which is therefore a generalized yield criterion in 3D (again I have written 3 times the same Law in Eq.18). Eq.18 gives the state of the stress of the granular medium just at yield. This yield criterion is named the extended von Mises yield criterion (or conical yield criterion) as opposed to the Mohr-Coulomb yield criterion as seen in Eq. 15. Whenever the intermediate principal stress is equal to the average between the minor and major principal stress (), we find back the Mohr-Coulomb yield criterion. It should be noted that the angle of internal friction may not have the same value depending on whether the material deforms in 2D only (plane strain), or within a volume (compression or extension in a triaxial test for instance): care with this frictional angle must be taken when working with the von Mises yield criterion.

 

In 2D, the Mohr-Coulomb can be represented by two straight line in the principal stress plane as shown on Fig. 8. In 3D, in the principal stress space, the extended von Mises yield has the geometrical shape of a cone with its apex at the origin as shown on Fig. 9. Both yields have an axis of symmetry, which represent the isotropic/hydrostatic Pressure for in the Mohr-Coulomb case.

 


[Figure 9: Representation of the von Mises Yield Surface in the principal stress space.
The central axis of symmetry on which all the principal stresses are equal is the hydrostatic/isotropic Pressure.
Theoretically, the inside domain of the cone represents the elastic deformation, however, for most of granular materials, elastic strain is negligible, and this inner domain can be assumed as perfectly rigid (no-deformation).
Plastic deformation occurs at yield on the surface of this cone. Compare this Figure with the 2D Mohr-Coulomb case as shown on Fig.8]

 

Uh Oh! Wow, if you have followed me till here then I am amazed because I can tell we have done a lot. As you may have noticed this page was mainly a "frictional warm up". Basically, we have learned the state of the stress within a granular material when it is just at yield. But we dont know how the material will react or move at yield … Basically, we don’t know anything ’bout the state of strain within the material. And we don’t know either the relationship between rate-of-strain and stress. Well, that is exactly what I’m gonna do in the second part of this course where I will define such a relationship thanks to the Plastic Potential theory. Please, follow this link if you wanna know more:

 

~~~~~~~~~ Part 2: III. Plastic Potential Theory ~~~~~~~~~

 

If you have enjoyed this Plastic-Frictional (Part 1) webpage, pleaaase, before you leave, sign my Guestbook. It's all I ask!

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Other Granular Volcano Group Webpages:

- What is a Granular Medium?

A complete review of Plastic-Frictional theories:

   - Part 1. Introduction, Mohr-Coulomb, and von Mises Stresses

   - Part 2. Plastic Potential Theory

   - Part 3. Critical State Theory

   - Part 4. Constitutive Equations for frictional granular flow

- All I wanna know about viscous stress!

- Granular Theory: an Overview

- Compute Your Own Atmospheric Profile

Numerical Results:

   - Go to the Numerical Results Page (Introduction and all Results)

   - Go to the Plinian Cloud simulations Page

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