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ߪ ൌ ܿଷሺߝ଴ଷ ൅ ߝ ௡ଷ ሻ ؠ 418ሺ0.4337 ൅ ߝ ଴.ସଵ଻ሻ (direction inclined by 45)
according to experiments and a best fit made in Ref./9/. Although the above functions are
not suitable for small strains, when ՜ ߝ 0, ሺ݀ߪ ݀ ߝ ⁄ ሻ ՜ ∞ they permit determination of
material constants r1 and r2 appearing in expressions for equivalent stress and equivalent
strain. Indeed, rewriting them in the following form reminding us to the “universal”
stress-strain curveߪതࣥଵ ൌ ܿଵ ሾߝ଴ଵ ൅ ሺ݀ߝ ࣥଵ ⁄ ሻ௡ଵ ሿ, ߪതࣥଶ ൌ ܿଶ ሾߝ଴ଶ ൅ ሺ݀ߝ ࣥଶ ⁄ ሻ௡ଶሿ
ߪതࣥଷ ൌ ܿଷሾ ߝ଴ଷ ൅ ሺ݀ߝ ࣥଷ ⁄ ሻ௡ଷ ሿ (68)
where 3ࣥଵଶൌ 2 ൅ ܾ ൅ ܿ ܿ , 3ࣥଶଶൌ 2 ൅ ܾ ൅ ܿ ܾ , 3ࣥଷଶൌ 2 ൅ ܾ ൅ ܿ5ܾ ൅ 5ܿ െ 12
we may find in this case c = 1.38; b = 1.3092; n = (1/3) (0.448 + 0.428 + 0.417) = 0.431.
Deflections from universality given by (ni-nj)/nj are 6.92% and 4.46%, whereas mea-
sured by ቀܿ௜ࣥ௜ ିଵି௡೔ െ ܿ௝ ࣥ௝ ିଵି௡ೕቁ ቀܿ௝ ࣥ௝ ିଵି௡ೕቁ ൗ amount even to 12.05% and 58.29%.
This indeed gives rise dramatically to the question how correct is it to use a single curveߪത ൌ ߪതሺߝ ҧሻfor advanced strains.
Similar conclusion for cruciform specimen has been drawn before in Albertini et al.
/26/ by simultaneous measurements of (ߝଵߪ ,ଵ) and ( ߝଶߪ ,ଶ) diagrams.
4. A comprehensive graphical representation of results of M-K analysis for orthotropic
materials with diverse type of line imperfections has been given in /4/. In this report
results of M-K analysis are also compared with results of Hill's localized necking.
Concluding this subsection it is possible to say:
– Range of applicability of Hill-Korhonen criterion is wider than that of M-K criterion,
since for the later McClintock's assumption of rigid shoulders along the groove is
essential ingredient (equivalent to ߝଶ஺ ൌ ߝଶ஻ ). Linearity of FLD by means of ߝଵ௨ ൅
ߝ ଶ௨ ൑ ݊, Eq. (53), for all materials is doubtful and should be examined by experiments.
– Ductility ࣧ found from Hill-Korhonen's criterion is insensitive to Ludwick exponent n,
whereas ࣧ found from M-K analysis is highly sensitive to value of n.
2.5. Gurson's approach to ductile rupture by void nucleation and growth
In this subsection Gurson’s paper /18/ is analysed with some additional modifications
concerning strain hardening of matrix materials, being neglected in /18/.
Assumptions
A5-1. Let the average (“macroscopic”, which means measurable) stress and strain be
denoted by T and whereas matrix stress and its strain be TM and ߝெ . Then, for either
cylindrical or spherical voids and whole matrix plastified (i.e. Plastically deformed) he
derived the following yield function݂ ൌ ሺߪത ߪതெ ⁄ ሻ ଶ ൅ 2߱ܿ݋ݏ݄ ൬ ࣲ 2 ݎݐT
ߪതெ ൰ െ 1 െ ߱ଶ ൌ 0 (69)
where ࣲ ൌ √3 for cylindrical and ࣲ ൌ 1 for spherical voids, while ߱ is volume fraction
of voids, total volume of voids divided by body volume. Main assumptions in Eq. (69)
are: (a) random voids are idealized by a single void; (b) matrix is rigid perfectly plastic
and obeys Mises flow rule. Then integration leading to macroscopic stress when
axisymmetric deformation takes place brings to Eq. (69).