IFMASS10

߱ሶ ൌ 3ܵ

ଵ

߱ሺ1 െ ߱ሻ

ത

ெ

ቀࣛ

ଵ଴

ത ൅ ࣛ

ଶ଴

ݎݐ

Tሶ

ሶ ቁ

(80)

Their integration for given stress history

ଵ

ሺ

ሻ,

ଶ

ሺ

ሻ,

ଷ

ሺ

ሻ,

gives limit strains

ଵ

ሺ

ሻ,

ଶ

ሺ

ሻ,

where temporal onset of instability is found from

߱ሺ

ሻ ൌ ߱

Numerical

results of such an integration and the corresponding diagrams are given in /4/.

3. SOME COMMENTS ON EXPERIMENTAL EVIDENCE

The principal question is how to identify experimental problems of interest from the

comprehensive literature on the localization and ductile failure. An effort is made to give

a concise approach, since there are not clear boundaries between different applications.

3.1. Forming limit diagrams (FLD)

Due to their importance for applications in automotive industry there is a general

agreement on main items, although sometimes misleading and wrongly interpreted.

Azrin and Backofen concluded /8/:

–

the upper bound for limit strain is obtained from the condition

ଵ஺

ଵ஻

⁄ ൌ 0

(M-K

recommendation), whereas the lower bound corresponds to initiation of non-proporti-

onality, i.e.

ଶ஻

ଵ஻

⁄ ് ܿ݋݊

ݐݏ

(equivalent to

് 0

);

–

for AISI 301 commence of the upper bound comes abruptly (upper and lower bound

almost coincide), dislike to ferritic steels where such a transition is smooth;

–

the dependence of limit strain on strain ratio

ଵ஻

݀ሺ

ଶ஺

ଵ஺

⁄ ሻ

⁄

is positive for ferritic

steels, slightly negative for AISI 301 and almost constant for brass;

–

making an artificially produced through (i.e. a straight scratch) they concluded that

initial imperfection must be larger than in M-K theory.

Ghosh and Backofen report /10/:

–

pronounced influence of strain history, as given by prestraining in diverse

directions

ଶ஺

ଵ஺

⁄

and then uniaxially pulled by

ଵ஺

് 0,

on limit strains;

–

with respect to “orange peel“ on surface their main outcome was that local neck did not

evolve from early strain-induced roughening.

Tadros and Mellor in their paper /14/:

–

show that localized neck is accompanied by a large increase of surface roughness for

rimming steel, aluminium and soft 70/30 brass;

–

conclude that experimentally found limit strains are lower than theoretically predicted

instability strains (by M-K analysis);

–

found initial imperfection

in M-K theory fitting experimental data and theoretical

curve;

–

state that Hill's equations for plastic orthotropic materials with yield function defined by

means of modified equivalent stress are reasonable for the killed steel but fail for

aluminium and 70/30 brass;

–

assume that deformation is uniform up to the general instability point.

The last conjecture is still open since it depends mainly on physical model and initial

geometrical imperfection.

Aleksandrović /7/ has also investigated non-proportional stress paths taking special ac-

count to abrupt change of principal axes between phases and trying to apply some quasi-

theoretical best-fit connecting experimental points of different phases, following /9/.