PSI - Issue 42

Gauri Mahalle et al. / Procedia Structural Integrity 42 (2022) 570–577 Mahalle et al./ Structural Integrity Procedia 00 (2019) 000 – 000

574

5

= ̇ ( )

(4)

] (− )

̇ = [ ( )

(5) Here, material constant α can be written α = β/ n 1 ; material constants can be evaluated from curves slopes ( ln έ ) vs ( σ ), ( ln έ ) vs ( ln σ ) respectively. At test temperature, the reciprocal of mean value of slopes of lines ln[sinh( ασ)] -ln ε̇ define the value of material constant n. Q can be calculated from the mean of slope of curves of ln [sinh(ασ)] vs (1/T) , which is plotted for both strain rate and strain rate. After considering the strain effect, the expression can be written as = 0 − 1 exp (− 2 ̅) (6) By combining above Eq. (3-6), the final constitutive equation is expressed as = 1 . 0 − 1 exp (− 2 ̅) ln { ( ) 1 + [( ) 2 + 1]} (7) The material constants 0 , 1 & 2 are calculated from experimental stress-strain data at various deformation temperatures and strain rates. An example equation of these constants that links strain rate and temperature to the Zener-Hollomon parameter (Z) is given as = × ln + (8) Where i = 0,1 & 2 For SPT, the above constitutive equation is modified by the von Mises criterion described by Karami and Mahmudi 2012 to define the shear deformation behavior. The ̇ will be replaced by ̇ and with τ , as per the criterion for a pure shear state of kinematical harden materials. The modified constitutive equations are: = ̇ ( ) (9) ̇ = [ ( ) ] (− ) = . − (− ̅) { ( ) + [( ) + ]} Fig. 3 gives the relation between the test temperature and shear strain rate used for the calculation of material constants.