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A4-3. For proportional stress paths with constant ratio of two in-plane principal
stresses ݉ ؔ ߪଶ ߪଵ ൌ ܿ݋݊ݐݏ ⁄ ) we obtain݀ߝଷ ݀ߝଶ ൌ 1 ൅ ݉ݎ െ ሺݎ ൅ 1ሻ݉ ߙ ؠ (58)
equivalent stress - equivalent strain relationshipߪത ൌ ܿଵሺߝ଴ ൅ ߝ ҧሻ௡ (59)
Analysis
1. Cutting the specimen, depicted in Fig. 4, along the grooves across the minimal cross
section and writing equilibrium equations Marciniak and Kuczynski have obtainedߪଵ஺ݐ஺ ൌ ߪଵ஻ݐ ஻. For the grooved cross section stress path is in general non-proportional.
Thus, they introduced an additional scalar functionݑ ൌ √3 √2 1 √ܴ ൅ 1 ߪଵ஻ ߪ஻ (60)
whose variability is responsible for non-proportionality. Inserting it into the above
expression for ߪത and differentiating leads to݀ݑ ݑൌ ݀ߪത஺ ߪത஺ െ ݀ߪതଵ஻ ߪത஻ െ ݀ ߝଷ஺ െ ݀ ߝଷ஻ (61)
where strain increment components perpendicular to sheet plane are݀ߝଷ஺ ൌ ݀ݐ஺ ݐ஺ , ݀ߝଷ஻ ൌ ݀ݐ஻ ݐ஻
From the assumption A4-1 there follows that ݀ߝଷ஺ ൏ 0, ݀ߝଷ஻ ൏ 0. Then two
principal stresses in the grooved cross section are connected byߪଶ஻ ߪଵ஻ ൌ ܴ ܴ ൅ 1 ൅ √2ܴ ൅ 1 ܴ ൅ 1 √1 െ ݑଶ
ݑ (62)
Here, the authors tacitly assumed that stress state is nearer to equibiaxial case leading
to ߙ ൏ 0, ݀ߝଶ ൐ 0. Then Eqs. (52) allow݀ߝ ҧ
஻ ൌ ඨ 1 ൅ 2ܴ 1 ൅ ܴ ඨ 2 3 ሺ1 െ ݑ ଶሻିଵ ଶ⁄ ݀ߝଶ
݀ߝ ҧ
ଷ஻ ൌ െ ቆ √1 ൅ 2ܴ 1 ൅ ܴ ݑሺ1 െ ݑଶሻିଵ ଶ⁄ ൅ 1 1 ൅ ܴ ቇ ݀ߝଶ (63)
Now, inserting Eq. (63) into Eq. (61) and taking into account Eq. (59) in its differen-
tial form, Marciniak and Kuczynski obtained the following integral-differential equation݀ݑ ݑൌ ቊ 1ܣଵ ൅ ܤଵഄమ ൅ ܧଵ ൅ ሺ1 െ ݑଶሻିଵ ଶ⁄ ቆܥଵݑ െ 1
ܦଵ ൅ ܤଵ ׬ሺ1 െ ݑଶሻିଵ ଶ⁄ ݀ݑቇቋ ݀ߝଶ (64)
with constants ܣଵ ൌ √3 ൬1 ൅ ߙ ൅ 1 ൅ ܴ 2 ߙଶ൰ିଵ ଶ⁄ ߝ଴ 2݊ , ܤଵ ൌ 1 1 ൅ ܴ ൅ ߙ
ܥଵ ൌ √2ܴ ൅ 1 ܴ ൅ 1 , ܦଵ ൌ √3 2 ൬ 1 ൅ ܴ 1 ൅ 2ܴ ൰ଵ ଶ⁄ ߝ଴ ݊ ,ܧଵ ൌ 1 1 ൅ ܴ ൅ ߙ (65)
Its solution depends mostly on initial value of geometric imperfection