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
$fσ=σD2+σL2+σM2−k2=0$
(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,
$maxlimn fσ,n=0$
(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
$fσ⩾k2.$
(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 $f,$
$fσ,n=Yσ,n−k2Y=a1σD2+a2σL2+σM2,$
(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 $a1⩾a2⩾0,$ then $κ=9/4$—the Schmidt cylinder; if $a2>a1>3/4 a2,$ then $9/4<κ<3$—cylinders intermediate between the Schmidt and von Mises cylinders; if $a1=3/4 a2>0,$ then $κ=3$—the von Mises cylinder; if $3/4 a2>a1>0,$ 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)
$a1=a2=1;$

$|2σα−σα+1−σα+2|=κ;$

$α=1,2,3;-theSchmidtcylinder,$
(6)
b)
$a1=0,a2=1;$

$|σα−σα+1|=2κ;α=1,2,3;-theTrescacylinder,$
(7)
c)
$a1=34,a2=1;$

$σ1−σ22+σ2−σ32+σ3−σ12=6κ2,$

$-thevonMisescylinder,$
(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
$14π ∫SYσ,nds=1152a1+3a2J2σ$
(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
$14π ∫S+fσ,ndsn=FJ2,J3.$
(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.