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ܼு ൌ ඨ 2 3 ඥܾ ൅ ܿ െ 1ሺܿ െ 2݉ ൅ ܾ݉ଶሻ ଷ ଶ⁄
ܿଶ ൅ ܾଶ݉ଷ െ ݉ሺ2ܿ െ 1ሻ െ ݉ଶሺ2ܾ െ 1ሻ (30)
According to the above definition of equivalent stress and definition of equivalent
plastic strain, we haveߪത ൌ √2 √3 ሺܴଵ ൅ ܴ ଶ ൅ ܴ ଵܴଶሻିଵ ଶ⁄ ሼܴଵߪଶଶ൅ ܴଶߪଵଶ൅ ܴଵܴଶሺߪଵ െ ߪଶሻଶሽଵ ଶ⁄ ൌ
ൌ ቆ 3 2⁄ ܾ ൅ ܿ െ 1 ቇିଵ ଶ⁄ ሺܿߪଵଶ൅ ܾߪଶଶെ 2ߪଵߪଶ ሻଵ ଶ⁄ (31)
݀ߝ ҧ ൌ ඥ2 3⁄ ඥܴଵܴ ଶ ൬ ܴଵ ൅ ܴଶ ൅ ܴଵܴ ଶ 1 ൅ ܴଵ ൅ ܴଶ ൰ଵ ଶ⁄ ሼܴଵ݀ߝଵଶ൅ ܴ ଶ݀ߝଶଶ൅ ܴଵܴଶ݀ߝଷଶሽଵ ଶ⁄ ൌ
ൌ ඨ 2 3 ൬ ܾ ൅ ܿ െ 1 ܾܿ െ 1 ൰ଵ ଶ⁄ ሺܾ݀ ߝଵଶ൅ ܿ݀ߝଶଶ൅ 2݀ ߝଵ݀ߝଶሻଵ ଶ⁄ ൎ ݀ߝҧ
௣ (32)
such that evolution equations, so called Levi-Mises equations, read݀ߝଵ ܿߪଵ െ ߪଶ ൌ ݀ߝଶ ܾ ߪଶ െ ߪଵ ൌ െ ݀ߝଷ ሺܿ െ 1ሻߪଵ ൅ ሺܾ െ 1ሻߪଶ ൌ 3݀ߝ ҧ
2ሺܾ ൅ ܿ െ 1ሻ ߪത (33)
The solution procedure covers integration of Eq. (33), which is elementary for propor-
tional paths where m = const, which allows subsequently the equivalent limit strain.
At this place, however, limit strains are checked according to equality ZD = ZH. It was
assumed that ߪത ൌ ܿߝҧ௡, which allows from Eq. (27) the relation ߝ ҧ
௨ ൌ ܼ݊ for uniform (i.e.
limit) equivalent strain.
It should be mentioned that in the report /4/ for the sake of illustration the following
materials are considered: (1) orthotropic, with ܴଵ ൌ 1 0.38 ⁄ , ܴଶ ൌ 1 0.3092 ⁄ (rolled, by
aluminium killed steel); (2) transversely isotropic with ܴଵ ൌ ܴଶ ൌ 1 0.35 ⁄ ; (3) transver-
sely isotropic with ܴଵ ൌ ܴଶ ൌ 1 2;⁄ (4) isotropic with ܴଵ ൌ ܴଶ ൌ 1. Two Ramberg-
Osgood exponents, n = 0.431 and n = 0.148 are taken. Results are plotted on Fig.1.5 and
1.6. It was remarked that for a small value of n limit curves are almost coincident.
Note N-1
Consider an orthotropic material. For it an invariant yield function in the form /31/ 2݂ ؔ 2 3ߪതଶ
݄ ൌ 2 2ݎଵ ൅ 2 ݎଶ െ 1
4 ൅ ݎଵ ൅ ݎଶ ൜2ܬଶ ൅ 3 2ݎଵ ൅ 2ݎଶ െ 1 ൣሺ1 െ ݎଶሻܣఙభ ଶ ൅ ሺ1 െ ݎଵሻܣఙమ ଶ ൧ൠ ൌ 1
is chosen with, ܣఙభ ൌ ሺܽതଵ ٔ ܽതଵሻ:T', ܣఙమ ൌ ሺܽത ଶ ٔ ܽതଶ ሻ: ܂Ԣ, as stress invariants in
privileged directions, while r1 and r2 depend on initial yield stresses in material principal
directions Ref. /31/.
Here, strain increment invariants in privileged directions are ݀ܣఌభ ൌ ሺܽതଵ ٔ ܽതଵሻ: ݀ e௣,
݀ܣఌ ೛ ൌ ሺܽതଶ ٔ ܽതଶሻ: ݀e௣, so that Levi-Mises equations are given by ݀ߝ ଵ ܿߪଵ െ ߪଶ ൌ ݀ߝଶ ܾߪଶ െ ߪଵ ൌ െ ݀ߝଷ ሺܿ െ 1ሻ ߪଵ ൅ ሺܾ െ 1ሻߪଶ ൌ 3݀ߝ ҧ
ሺ2 ൅ ܾ ൅ ܿሻ ߪത (34)
By means of ݉ ൌ ߪଶ ߪଵ ⁄ we get the expression for critical subtangent as follows