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 
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,
where n is a normal to the unit sphere S, and S+ is the domain on S where
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,
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)
where σα,α=1,2,3, are the principal stresses.
On the other hand, the integration of Y over the unit sphere S gives
(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

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.

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