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ߪ௘௤௘௡௚ ൌ ඨ 3 2 ටݎݐ൛T'௘௡௚ଶ ൟ ሺA.15 ሻ
Plastic stretching tensor (plastic “strain rate”).2D௉ ൌ Fሶ ௉F௉ିଵ൅ F ௉ି்Fሶ ௣் (A.16)
Equivalent plastic strain rate: εሶ ୣ୯୮ൌ ඨ 2 3 ටtrD୮ଶ (A.17 )
Special case of small strains - applicable to strain gage measurements൝ εଵଵ 0 0
0 εଶଶ 0
0 0 εଷଷൡ ൎ ൝ uଵଵ 0 0
0 u ଶଶ 0
0 0 uଷଷൡ
E ቐ εଵଵE 0 0
0 ԖଶଶE 0
0 0 εଷଷEቑ ൎ ൝ σଵଵ െ νσଶଶ 0 0
0 σଶଶ െ νσଵଵ 0
0 0 െνσଵଵ െ νσଶଶൡ
ቐ εଵଵ୮ 0 0
0 εଶଶE 0
0 0 εଷଷPቑ ൎ ቐ εଵଵ െ εଵଵE 0 0
0 ε ଶଶ െ εଶଶE 0
0 0 εଷଷ െ εଷଷEቑ
൞ Dଵଵ୮ 0 0
0 Dଶଶ୮ 0
0 0 D ଷଷ୮ൢ ൎ ൞ εሶଵଵ୮ 0 0
0 εሶ ଶଶ୮ 0
0 0 εሶ ଷଷ୮ൢ
Invariants appearing in evolution equationssଶ ൌ tr൛T'ଶ ൟ ൌ 2 3 ሺσଵଶ൅ σଶଶെ σଵσଶሻ ؠ 2 3 σୣ୯ଶ
sଷ ൌ tr൛T'ε୮ ൟ ൌ 1 27 ሾሺ2σଵ െ σଶሻଷ ൅ ሺ2σଶ െ σଵሻଷ ൅ ሺσଵ ൅ σଶሻଷሿ
µଵ ൌ tr൛ T'ε୮ ൟ ൌ ሼሺ2σଵ െ σଶሻlog pଵ ൅ ሺ2σଶ െ σଵ ሻlog pଶ െ ሺσଵ ൅ σଶሻlog pଷሽ ؠ
ؠ σଵ log pଵ ൅ σଶ log pଶ
µଶ ൌ tr൛T'ଶߝ୮ൟ ൌ 1 9 ሼሺ2σ ଵ െ σଶሻଶlog pଵ ൅ ሺ2σ ଶ െ σ ଵሻଶlog pଶ ൅ ሺσଵ ൅ σଶሻଶlog pଷሽ ؠ
ؠ σଵ ሺσଵ െ 2σଶ ሻlog pଵ ൅ σଶሺσ ଶ െ 2σଵ ሻlog pଶ
൫εሶ ୣ୯୮൯ଶ ൌ 4 3 ቈ൬ pሶ ଵ pଵ൰ଶ ൅ ൬ pሶଶ pଶ൰ଶ ൅ pሶଵ pଵ pሶ ଶ pଶ቉ ؠ 4 3 ൫εሶ୮ଵଵଶ ൅ εሶ ୮ଶଶଶ ൅ εሶ୮ଵଵεሶ ୮ଶଶ൯
൫ߝ௘௤௣൯ଶ ൌ 4 3 ሾሺ log pଵሻଶ ൅ ሺlog pଶሻଶ ൅ log pଵlog pଶሿ ؠ 4 3 ൫ε୮ଵଵଶ ൅ ε୮ଶଶଶ ൅ ε୮ଵଵε୮ଶଶ൯
It must be taken into account that ߝ ሶ
௘௤௉് ݀ ݀ߝ ݐ௘௤௉
unless for proportional plastic strain paths where ߝ௉ଶଶ ൌ ݉௉ߝ௉ଵଵwith mp = const: