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A Review of PlasticFrictional Theory Part. 4 Constitutive Equations for Frictional Granular Flows 
You will find the basic facts about PlasticFrictional Theories (Part. 4)  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.
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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 rateofstrain 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 /SN 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 MohrCoulomb 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).

[Figure 13: Yield locus curve for a given bulk density, which is the combination of a failure locus curve and a consolidation locus curve] [At the intersection, we define a point called the critical state. The straight line passing through all the critical sates is named the Termination locus] [In Fig. 13C, I show two possible situations: dilatation path and consolidation path. Clearly, as the material deformed at yield, it will change from one yield locus to another till it reaches and finds its critical state] [See text for further explanation] 
[Figure 14: Nearly the same as Fig. 13 but seen from the principal stress perspective. See Part 4.  Critical state Theory page for detailed explanation] 
As you may guess, changing the von Mises and MohrCoulomb 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):
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:
and,

[Figure 15: Modified von Mises yield function accounting for compressibility effects, failure and consolidation processes] [Compare with the extended von Mises conical yield function as shown on Fig.9 of paragraph II.2.] [One apex of this function lies at the origin as for the extended von Mises yield function, but in this case there is a second apex on the hydrostatic central axis] [Since the function P increases with the bulk density of the granular material, any yield surface containing smaller yield surfaces represents a higher density than the ones it contains] 
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,
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 MohrCoulomb/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:
where is the deviatoric part of the rateofstrain tensor (i.e., pure shear rateofstrain) 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 rateofstrain and shear stress. Such a flow rule, which is a direct result from the Plastic Potential Theory, is often named the Levyvon Mises flow rule. This Levyvon Mises flow rule necessarily implies that the granular flow is slightly compressible. If the flow is incompressible, then we would have .
Notice that the Levyvon Mises flow rule implies the coaxiality between the stress directions and rateofstrain directions since the shear rateofstrain is zero on plane where the shear stress is zero (Please, if you do not know what coaxiality means you must read the previous section of this course: Part.2  Plastic Potential Theory). The normality conditions is of course true since this flow rule is derived from the associated Plastic Potential theory (again see Part.2: Plastic Potential Theory Page).
Now, we know that the PitmanSchaefferGrayStiles 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 Levyvon Mises flow rule as (see Eq. 5c):
Eq. 9
Therefore, taking Eq.7, and using Eq.9 along with Eq. 3c, we have:
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:
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 rateofstrain 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 rateofstrain 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 Levyvon 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 rateofstrain 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 rateofstrain independent.
V.2. Within the extended von Mises Yield locus
As we have seen in the previous paragraph, we cannot use the MohrCoulomb/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/MohrCoulomb 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):
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 rateofstrain and the stress tensors respectively. The last equation is again the famous Levyvon 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 Levyvon Mises flow rule, we have:
Now, in the previous paragraph (V.1.), we have seen that it is possible to find an expression for the bulk viscosity since we knew that , where P would be a frictional isotropic Pressure, which is a function of the macroscopic 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 viscosity 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 rateofstrain 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 …
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