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A Review of PlasticFrictional Theory Part. 2 Plastic Potential Theory 
You will find the basic facts about PlasticFrictional Theories (Part. 2)  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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The yield condition laws I have reviewed in the preceding section says nothing about the nature of the motion which is initiated at yield. Alone, the MohrCoulomb/von Mises relations are helpless for describing a granular media in a continued motion. Constitutive relations are needed for describing the kinematics of motion at yield following the MohrCoulomb/von Mises theory. In addition, I must also account for the famous kinematics feature described for the first time by Osborne Reynolds: dilatancy. For a given normal stress, the material will sheared only if a critical shear is attained, once sheared, the granular medium may initially expand. Afterwards, it will settle to a fairly constant volume. This counterintuitive behavior is of a key importance in soil mechanics.
The plastic potential theory will provide us a way to predict the velocity distribution within the granular medium at yield. This theory makes the connection between stress and deformation (or velocity gradient) in using three concepts: a Yield Function (Y), a Plastic Potential Function (G), and a Flow Rule.
The yield function (Y) is what we have seen in the previous section (Part. 1). For instance, from the MohrCoulomb and the von Mises yield functions, we have:
where II_{dT} is the second invariant of the deviator of the stress tensor T. The other symbols have been seen in the 1st Part of this course. II_{dT} can be expressed as:
It is clear from all I said in the previous section (Part.1 ), at yield, we must have Y=0 for a plastic material (as indicated by Eq.1). Therefore, at yield, we can see that Eq.1a is the equation of a straight line (which is, in the mathematical terminology, a level curve for Y=0, see Fig.8) in the principal stress plane (), and Eq.1b is the equation of a cone (which is, in the mathematical terminology, a level surface for Y=0, see Fig.9) in the principal stress space (). Both functions have an central axis of symmetry which has the property of being the axis of equal principal stresses, i.e., for MohrCoulomb 2D case and for the von Mises 3D case. Hence, this axis of symmetry represents the hydrostatic/isotropic Pressure. Both functions have their vertex on this hydrostatic axis at the origin, when .

[Figure 8: Domain of nodeformation (rigid) and domain of plastic deformation for the 2D MohrCoulomb case. The central axis on which the principal stresses are equal represents the hydrostatic Pressure] 
[Figure 9: Representation of the 3D 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. Plastic deformation occurs at yield on the surface of this cone] 
The Plastic Potential Function (G) is defined so that the strain rate in any arbitrary directions (D_{i j}) is proportional to the derivative of G with respect to the corresponding stress (T_{i j}):
Eq. 3
where q is a positive scalar sometimes named "plastic multiplier". This scalar is not a property of the material but rather a property of the flow conditions. Eq.3 can also be written in terms of the principal directions of D and T (using their respective eigenvalues) as:
Eq. 4
Now, we are almost there, since we now specifically play with the rate of deformation (D), hence the velocity gradient. So, the remaining thing to do is to find out what is G exactly? This is given by the flow rule. One of the most widely used flow rule is the associated flow rule, which states that the Plastic Potential Function (G) is equivalent to the Yield function (Y), G=Y, hence, Eq. 4 can be rewritten as:
It is worth saying that there are other flow rules which do not assume G=Y, those are named nonassociated (I won’t comment this any further). It is clear that in Eq. 5, the RightHandSide (RHS) is nothing but the gradient of the Yield Function with respect to the stresses, which is important to notice for defining the properties of the plastic potential flow. But before doing so, let’s indeed derived Eq. 5 using the Yield Function defined by Eq. 1b (extended von Mises) and the definition of the second invariant of the deviator of the stress tensor (II_{dT}) in Eq. 2c, we have in 3D:
This equation can be safely generalized as:
where D, I, and are the rateofstrain tensor, the unit tensor and the deviator of the stress tensor respectively. The deviator of the stress tensor is:
Eq. 8
where is the spherical part of the stress tensor which is the average in the 3direction of space of the normal stresses (i.e., one third of the sum of the diagonal element of the stress tensor).
From those equations, we can now state the two key properties of the Plastic Potential Theory:
1 the coaxiality or alignment condition: the principal axe of rate of deformation are aligned with those of stresses. This to be in agreement with the intuitive idea that the material should respond to unequal stresses by contracting in the direction of greater stress and expand in the direction of lesser stress. In other words, it is equivalent to say that there is no shear strain on planes on which there is no shear stress. This is exactly what shows Eq. 6 and Eq. 7 since whenever the shear stress (T_{i j}) is zero on a given plane, so is the shear strain (D_{i j}). I will come back to this property latter in this course.
2 the normality condition: which is shown by the original equation of the Plastic Potential Theory (Eq. 5). Indeed, if we consider the particular level surface Y=0 (at yield) in Eq. 5, we know from the properties of gradient that D_{i} must be a vector perpendicular to the Yield (level) Surface, in the principal stress space.
Now, I’m pretty so you’re gonna ask: Sooooooooooooo what? Why the hell in the world do we want to know all that? Well, those last equations actually give us a connection between the velocity gradient within the granular flow and the state of stress within it. And basically, that’s what we were looking for, because it can be easily implemented into momentum equations in a computer model as we must know such relations.
Now, it would be interesting to calculate the divergence of the velocity field, which is nothing but minus the volumetric rate of deformation of the granular body (i.e., first invariant of D):
Therefore, with both MohrCoulomb and von Mises cases, the granular body, at yield, suffers dilatation (divergence of the velocity field is positive, i.e., that is extension). On the one hand, that is good because it is in agreement with Reynolds' Principle of Dilatancy and therefore that is what we expect from those laws. However, on the other hand, it is bad as Eq. 9 leads to a continued expansion of the granular medium without any limits, which is physically impossible.
But there is another trouble. Indeed, dilatancy is one phenomenon that can be observed in granular material at yield but there are other processes that may happen as well depending on the physical properties of the granular material and on its previous deformation history. If the granular medium is wellcompacted, it will expand when sheared (that is dilatancy), whereas loosely packed, a granular medium at large normal stress may contract (that is consolidation or contractancy). Moreover, the expansion or the contraction takes place only until a critical bulk density is reached, afterwards under continuous stress, deformation will continue without any change in the bulk density. And that is not shown at all in the Eq. 9. In addition, it can be shown that the rate of energy dissipation is always zero if we use a simple MohrCoulomb/von Mises law, which is an unacceptable results for frictional processes.
So, we got a problem down here … In conclusion, the MohrCoulomb/von Mises theory cannot be applied straightforwardly for describing flowing frictional granular material without committing unphysical instabilities and inconsistencies. The simple Mohrcoulomb/von Mises law must be somehow adapted to the plastic potential flow theory for describing flowing granular medium accurately. But before getting into the constitutive equations and the possible corrections, it is necessary to review the basic behaviors of granular material at yield. This is what I’m gonna do in this following section:
~~~~~~~~~ Part 3: IV. Critical State Theory ~~~~~~~~~
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