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,
or
where
n is a normal to the unit sphere
, and
is the domain on
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
the
and
may never coincide with the full unit sphere.
Let us consider the yield criterion of type (
1), (
2) for a more general function
where
and
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 where and 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
then
—the Schmidt cylinder; if
then
—cylinders intermediate between the Schmidt and von Mises cylinders; if
then
—the von Mises cylinder; if
then
—cylinders intermediate between the von Mises and Tresca cylinders; if
then
—the Tresca cylinder.
Examples: a)
b)
c)
where
are the principal stresses.
On the other hand, the integration of
over the unit sphere
gives
(formula (5.2)
1 follows from here when
).
However, as mentioned above, only the domains on the unit sphere where the local yield condition (4) is fulfilled, must be taken into account and therefore may never coincide with for
Then the resulting function will also depend on the third deviatoric invariant
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 -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.