You will find the basic facts about Plastic-Frictional Theories (Part. 4) - no details -. Detail is a matter of my current Ph.D. research and I will not show that here. If you wanna know more just email me or feel free to ask in the Volcano Discussion Forum. This general overview should help you to understand the modeling results and their interpretations that will be presented in this Granular Volcano Group Web Site. I purposely erased all the bibliographical references and detailed equations to keep the text simple and easy to read.

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Part. 4 - Constitutive Equations for Frictional Granular Flows

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We have seen on the preceding sections:
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)
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

On this page, you will find the last part of this Frictional-Plastic course:
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

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Done and updated by Sébastien Dartevelle, The WebMaster, June 29th, 2002.

V. Constitutive Equations for Frictional Granular Flows

V.1. Beyond the extended von Mises Yield locus

Again, I’d like to reemphasize that we need a flow rule to make the connection between the rate-of-strain tensor and the stress tensor when the material is just at yield … This can be done using the Plastic Potential Flow we have just seen this in paragraph III (Part. 2 - Plastic Potential Theory Page). Also, we now know from paragraph IV (Part. 3 - Critical State Theory Page) that if we adopt a yield locus that has the properties of Figures 13 in the /S-N plane or the properties of Figures 14 in the stress plane, we can successfully describe without no physical inaccuracies and inconsistencies the properties of concentrated frictional granular medium at yield, such as dilatancy, consolidation and constant density at critical state. The problem is that the Mohr-Coulomb doesn’t give us a function that looks like those yield loci as seen in Fig. 13 and Fig. 14. The same would apply for the extended conical von Mises yield function in 3D in the principal stress space. If we want to use the Plastic Potential theory and predict dilatancy as well as consolidation, we must modify those laws, otherwise we are screwed (which is not good).

As you may guess, changing the von Mises and Mohr-Coulomb law is not an easy task but it has been done in the past. To the best of my knowledge, there is only one function that can be used and have all the properties we have previously seen (dilatancy, consolidation, critical state in paragraph IV):

      Eq. 1

where II_dT is the second invariant of the deviator of the stress tensor, is the average of the three principal stresses, is the angle of internal friction and P is a positive function that measures the compressibility of the granular material and monotonically increases with the bulk density of the granular material. P will be a normal frictional isotropic/hydrostatic stress (i.e., Pressure). This yield locus function is shown on Fig.15 for three different bulk densities in the principal stress space. As a reminder from the previous sections, II_dT and are calculated by:

      Eq. 1_2

and,

      Eq. 1_3

From Eq. 1 and knowing II_dT (Eq. 1_3), we can find the radius of this yield function (for any fixed value of P, see Fig.15), which is:

      Eq. 2

At the two apex of this function, the radius is zero (on the hydrostatic axis for , and the radius is maximum whenever , which defines a critical state. A close inspection of Fig. 15 shows that this new yield function has a convexity. Hence the normal to the surface has a negative projection on the hydrostatic axis for (failure process), a positive projection for (consolidation process), and an orthonormal projection at (critical state with no change of the bulk density).

All this can also be shown if we apply the Plastic Potential Theory to this new yield function. Indeed,

      Eq. 3

So, we see that the divergence of the velocity will be positive or negative depending on the sign of . Since for a dilatancy process, we have , the divergence will be positive, which is an expansion as expected. While, if we have a consolidation process, , the divergence is negative, which is indeed a compression. And if , at a critical state, then we have neither expansion, nor compression as expected. So, we are happy very much indeed.

Of course, it is clear that the use of the Plastic Potential theory assumed that the Mohr-Coulomb/von Mises approach is not right for such a theory. Therefore, we had to modify it … You may like that or not, it is as it is.

Now, what I want is to use the preceding results for using them in a computer model, that is for solving the momentum equation. So, I have to rearrange Eq.3 in order to properly and easily use it. So, what I am gonna do, it’s a little bit algebra but don’t worry, I think it is a lot of fun … you’ll see.

You may see that Eq. 3b may be generalized as:

      Eq. 4

And using Eq. 3c, we may rewrite Eq. 4 as:

      Eq. 5

where is the deviatoric part of the rate-of-strain tensor (i.e., pure shear rate-of-strain) and is the deviatoric part of the stress tensor (i.e., pure shear stress). This is an extremely important result as this flow rule (Eq.5c) gives the relationships between shear rate-of-strain and shear stress. Such a flow rule, which is a direct result from the Plastic Potential Theory, is often named the Levy-von Mises flow rule. This Levy-von Mises flow rule necessarily implies that the granular flow is slightly compressible. If the flow is incompressible, then we would have , hence:

      Eq. 6

Notice the difference with Eq. 5c. Sometimes Eq. 6 is named the Levy flow rule, which can only be applied for incompressible medium (this is not good for granular flow). Notice that both flow rules imply the co-axiality between the stress directions and rate-of-strain directions since the shear rate-of-strain is zero on plane where the shear stress is zero (Please, if you do not know what co-axiality means you must read the previous section of this course: Part.2 - Plastic Potential Theory). The normality conditions is of course true since both flow rules are derived from the associated Plastic Potential theory (again see Part.2: Plastic Potential Theory Page).

Now, we know that the Pitman-Schaeffer-Gray-Stiles yield function states that (see Eq. 1):

      Eq. 7

It is worth noting that the Euclidian norm of the deviator of the stress tensor is:

      Eq. 8

And we can also rewrite the Levy-von Mises flow rule as (see Eq. 5c):

      Eq. 9

Therefore, taking Eq.7, and using Eq.9 along with Eq. 3c, we have:

      Eq. 10

Cool, so what? Well, good question, why the hell we have done all this? Typically in a computer model of granular flow we must solve a momentum equation in which one term will account for the momentum contribution from all the stresses within the flow. The total stress can be written as:

      Eq. 11

where Eq.11a is the total stress tensor, Eq.11b is the spherical part of the stress tensor, Eq.11c is the deviatoric part of the stress tensor, Eq.11d is the rate-of-strain tensor (note the minus sign which is explained by the fact that compression is positive following our sign convention), is the deviatoric part of the rate-of-strain tensor, I is the unit tensor, are the frictional bulk and shear viscosities. So, we must also find how to calculate the "frictional viscosities". Here how. We know from Eq. 3c, Eq. 11b and Eq. 10:

      Eq. 12

Cool, and now the shear frictional viscosity … a piece of cake! From the Levy-von Mises flow rule (Eq. 5c), we know:

      Eq. 13

Voila! We’re done folks! And the nice thing is that a close inspection of Eq.12 and Eq.13 show that the frictional stress is independent of the rate-of-strain tensor (D) as required by the frictional theory. Indeed, if the components of D are multiplied by a factor, the components of the stress tensor remain unchanged (because at the denominator we have II_dD and ). And that is exactly what we want since frictional flow must be rate-of-strain independent.

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V.2. Within the extended von Mises Yield locus

As we have seen in the previous paragraph, we cannot use the Mohr-Coulomb/von Mises yield function with the plastic potential theory as it leads to physical inconsistencies. Therefore, we had to modify the von Mises theory in order to make them work within the plastic potential. This is kinda done often, and it has my favor as I think it makes sense in terms of physics. On the other hand, in some conditions, we may consider to keep the von Mises yield function. For instance, if we are only interested into failure processes at very high concentration, the extended von Mises/Mohr-Coulomb yield function may suffice for our needs.

Let’s see some properties of the extended conical von Mises conical yield function associated with the Plastic Potential Theory (we have already seen some):

      Eq. 14

As a reminder, we see that the divergence is always positive predicting a continued dilatation (Eq.14c). Now, in using Eq.14c, we can generalize Eq.14b as:

      Eq. 15

where II_dD and II_dT are the second invariant of the deviator of the rate-of-strain and the stress tensors respectively. The last equation is again the famous Levy-von Mises flow rule, which is a direct consequence of the compressibility of the material in applying the Plastic Potential Theory. We know at yield we must have a conical function given by:

      Eq. 16

And as in the previous paragraph (V.1.), we note that the Euclidian norm of the deviator of the stress tensor is (for the extended von Mises case):

      Eq. 17

Now, thanks to the Levy-von Mises flow rule, we have:

      Eq. 18

Now, in the previous paragraph (V.1.), we have seen that it is possible to find an expression for the bulk density since we knew that , where P would be a frictional isotropic Pressure, which is a function of the bulk density of the granular material. You may see that now there is an impossibility to find an expression for the bulk viscosity as we deal with an indeterminate solution for the bulk viscosity (try to use Eq.18 with Eq. 14c and you will see it cannot be done). Hence, the bulk density cannot be known and we will assume it is equal to zero, hence .

Now, as we have done in the previous paragraph, we want to use those results for solving the momentum equations in a computer model for instance. Typically, we must know the total stress at any time anywhere. The total stress can be written as:

      Eq. 19

Since, the bulk viscosity is assumed to be zero, the viscous stress tensor is traceless and is given by Eq.19c. Now we just have to find an expression for the shear viscosity in using Eq.19c with Eq. 18:

      Eq. 20

A close inspection of Eq.19 and Eq.20 shows that the frictional stress is independent of the rate-of-strain tensor (D) as required. Indeed, if the components of D are multiplied by a factor, the components of the stress tensor remain unchanged.

Ooooaaaaaah! It’s over, we’re done! I don’t know if you realize that it took me three months of hard labor for finding all the books, papers, articles and writing down all this (besides drawing all the figures) … hope you found this frictional course useful. Please, let me know what you think: email me … Now since it is the summer time here in Michigan (June 29th, 2002), I’m gonna take a few days off … (I'm kidding)

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