H U U D H V d d M Gb d d 1 2 1 (10) Combining Eqs. (1), (9) and (10), the following equation can be derived 2 2 1 2 2 1 2 1 V D V H V V M GbK MK d d (11) or 2 2 2 1 V T V H V V MK d d II (12) where 1 2 2 1 M GbK II D T is the stage II hardening rate, i.e., the slope of the stress-strain curve in stage II It is recognized that TII is the limit value of the strain hardening rate for V o 0. On the other hand, when V approaches its saturation value Vs, the strain hardening coefficient H T V H ( / ) d d o 0. So, from Eq. (11) 2 1 K K GbM S V D (13) and Eq. (12) can also be expresed in the form ¸ ¸ ¹ · ¨ ¨ © § S II V V T T 1 (14) The constants K1 and K2 for a polycristalline material can be calculated if the value of the stage II hardening rate, TI,, and the value of the saturation stress Vs are estimated. In references [12,13] it is established that Vs is independent of strain rate and varies from 0.6 to 3 x 10-2 G for different materials, the higher values being those for materials with a lower Stacking Fault Energy (SFE). According to Refs. [11,12], the work hardening rate TII which is almost independent of temperature or strain rate, ranges from 1/15 to 1/30 of the shear modulus G. In Ref. [14], the following expressions are documented: 1000 4.5 2 .0 G M d d i M M II i ¸ ¹ · ¨ © § V V H V T (15) i y i i s M M M M 213. 213. 0 V V V (16) In absence of other expressions, the constants K1 and K2 calculated from a coarse-grained single-phase material basis, can be used in polycristalline materials M Gb K i II D T 2 1 2 = b D 0.009 (17) i S II M K V T2 2 = M GM 213 0.009 0 V (18)
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