First of all, here are some comments on the microplane model MP1. On the microplane level, the yield condition is given by the function (5.11) 1  
fσ=σD2+σL2+σM2k2=0
(1)
It should be stressed that, for obtaining a global yield condition from the local one, some global criterion independent of the chosen coordinate system must be used eg,  
maxlimnfσ,n=0
(2)
or  
S+fσ,ndsn=c>0,
(3)
where n is a normal to the unit sphere S, and S+ is the domain on S where  
fσk2.
(4)
Criterion (2) defines a surface (in the σ-space) of the first onset of global plasticity, criterion (3)—a surface of more extended plasticity. It is clear that, for k>0, the S+S and S+ may never coincide with the full unit sphere.
Let us consider the yield criterion of type (1), (2) for a more general function f, 
fσ,n=Yσ,nk2Y=a1σD2+a2σL2+σM2,
(5)
where a1 and a2 are some non-negative constants.

Equations (2) and (5) define a family of cylinders in the 6D σ-space.

For the convenience of classification, let us introduce a parameter κ=σy±/τy2, where σy± and τy are the yield points in the uniaxial tension (compression) and shear, respectively.

An analysis of Eqs. (5) and (2) reveals that the following cases are possible 2:if a1a20, then κ=9/4—the Schmidt cylinder; if a2>a1>3/4a2, then 9/4<κ<3—cylinders intermediate between the Schmidt and von Mises cylinders; if a1=3/4a2>0, then κ=3—the von Mises cylinder; if 3/4a2>a1>0, then 3<κ<4—cylinders intermediate between the von Mises and Tresca cylinders; if a1=0,a2>0, then κ=4—the Tresca cylinder. Examples: a)  
a1=a2=1;
 
|2σασα+1σα+2|=κ;
 
α=1,2,3;-theSchmidtcylinder,
(6)
b)  
a1=0,a2=1;
 
|σασα+1|=2κ;α=1,2,3;-theTrescacylinder,
(7)
c)  
a1=34,a2=1;
 
σ1σ22+σ2σ32+σ3σ12=6κ2,
 
-thevonMisescylinder,
(8)
where σα,α=1,2,3, are the principal stresses.
On the other hand, the integration of Y over the unit sphere S gives  
14πSYσ,nds=1152a1+3a2J2σ
(9)
(formula (5.2) 1 follows from here when a1=a2=1).

However, as mentioned above, only the domains S+ on the unit sphere where the local yield condition (4) is fulfilled, must be taken into account and therefore S+ may never coincide with S for k>0.

Then the resulting function will also depend on the third deviatoric invariant J3σ, and the global yield criterion (3) gives  
14πS+fσ,ndsn=FJ2,J3.
(10)

So the local yield condition (5.1) together with the global criterion (2) defines the Schmidt cylinder 3 or, in the case of (5.1) and (3), the yield condition (10) and does not correspond to the J2-flow theory.

Finally, we should note that the so called “microplane model version MP2” (5.8) was first put forward by Malmeister in 1955 4 and was further elaborated by him and by his collaborators and followers in numerous Russian and English papers. We have also considered a number of other models based on general integral representations of arbitrary second-rank tensors. For more details and references see our book 2.

1.
Brocca
,
M
and
Bazˇant
Z
2000, Microplane constitutive model and metal plasticity, Appl. Mech. Rev. 53(10).
2.
Lagzdin¸sˇ A, Tamuzˇs V, Teters G, and Kregers A (1992), Orientational Averaging in Mechanics of Solids, Longman Scientific & Technical, London.
3.
Schmidt R (1932), U¨ber den Zusammenhang von Spannungen und Formaenderungen im Verfestigungsgebiet, Ingenieur-Archiv, Springer, Berlin, 3.
4.
Malmeister A (1955), Deformation of a medium capable of twinning, Problems of Dynamics and Dynamic Strength (in Russian), Vol. 3, Riga.