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54
Figure 3: Principal stretching direction for a localized strip in rigid plastic thin sheet
Then proportionality of stress path requires ߪሶ
ଶ ߪଶ ൌ⁄ߪሶ
ଵ ߪଵ i.e ߪሶ
ଶ ߪଶ ൌ⁄ ⁄ െ ߝሶ
ଷ. Thus,
localized necking commences with simultaneous maximum of two in-plane principal
engineering stresses.
A3-3. Hill postulated yield function for plane stress in the form.݂ሺ ߪଵߪ ,ଶ ሻ െ ܨሺܹሻ ൑ 0 (40)
where equality stands for plasticity and inequality for elasticity. The so called consistency
condition states that ݂ ሶ െ ܨሶ ൌ 0 should hold as well. According to his main assumption,
localized instability commences whenܨᇱ ሺܹሻ ൑ ݂ ሶ ܹሶ ൌ ߲݂ ߲ߪଵ ൅ ߲݂ ߲ߪଶ (41)
The last equality follows from the assumption A3-2.
Analysis
1. Consider an orthotropic material (Note N-1) having equivalent stress and equivalent
strain (according to Korhonen):ߪത ൌ ൬ 3 2ܾ ൅ 2ܿ െ 2 ൰ଵ ଶ⁄ ሺܿߪଵଶ൅ ܾߪଶଶെ 2ߪଵߪଶሻଵ ଶ⁄
݀ߝ ҧ ൌ 1 √3 ൬ 2ܾ ൅ 2ܿ െ 2ܾܿ െ 1 ൰ଵ ଶ⁄ ሺܾ݀ߝଵଶ൅ ܿ݀ߝଶଶ൅ 2݀ߝଵ݀ߝଶሻଵ ଶ⁄
݂ሺߪଵߪ ,ଶ ሻ ൌ ݂ ҧ ሺߪതሻ ฺ ߲݂ ߲ߪଵ ൅ ߲݂ ߲ߪଶ ൌ ݂݀ ҧ ݀ߪത ܿ െ 1 ൅ ݉ሺܾ െ 1ሻሺܿ െ 2݉ ൅ ܾ݉ଶሻଵ ଶ⁄ ൬ 3 2ܾ ൅ 2ܿ െ 2 ൰ଵ ଶ⁄ (42)
Such that (with ݉ ߪ ؠଶ ߪଵ ⁄ ). ݂ ሶ െ ܨሶ ൌ ݂݀ ҧ ݀ߪത ߪതሶ െ ܨᇱ ሺܹሻܹሶ ൌ ݂݀ ҧ ݀ߪത െܨ ᇱሺܹሻߪതߝ ҧ ሶ ൌ 0 (43)
Thus ܨᇱ ሺܹሻ ൌ ݂݀ ҧ ݀ߪത 1ߪത ݀ߪത ݀
ߝ ҧ ൌ 1 ܼ ݂݀ ҧ ݀ߪത (44)
where Z is the subtangent found from universal curve ߪത ൌ ߪതሺߝ ҧሻ. If the universal flow is of
Ramberg-Osgood type (A1-3), then