PSI - Issue 14

Ritam Chatterjee et al. / Procedia Structural Integrity 14 (2019) 251–258 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

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by Peczak (1995) (based on temperature) and by Peczak and Luton (1994) (based on strain rate). Two important assumptions are:  Initial dislocation density is same for all primary grains. DRX occurs on achieving a critical value of dislocation density. For secondary grains formed due to DRX, it increases from zero to a saturating value with deformation.  New nuclei are formed only on grain boundaries. This is based on grain boundary bulging mechanism which causes strain induced boundary migration (SIBM). The nucleation rate is defined as: ̇ = ∗ ̇ m ∗ ( − ) (1) where Q is the activation energy, T is the temperature, R is the universal gas constant, C is a constant, ̇ is the true strain rate and exponent m ~ 1. The activation energy has been taken to be 61.8KJ/mol (Stringer (1960)) and the constant ‘C’ is taken as 10000 via trial and error to obtain a high nucleation rate ~ 100s -1 per grain. 2.2. Dislocation Density Hardening based model Beyerlein and Tome (2008) developed a dislocation density hardening based algorithm for multiple slip and twinning modes in hcp material. The shear increment in each slip mode ‘ m ’ due to deformation can be written as the sum of shear increments in all the slip systems ‘ s ’ contained in the mode ‘ m ’: = ∑ ̇ ∈ (2) where, ̇ = ̇ | : ( ̇, ) | ( : ) (3) τ c refers to the slip resistance to dislocation movement caused due to local microstructural obstacles. The ‘ n ’ value in the above expression is taken to be quite high ~20 so that slip or twinning only occurs when the resolved shear stress approaches the slip resistance value. The slip resistance of a slip system is the sum of contributions due to initial slip resistance, forest dislocations, debris and twin boundaries: = + + + (4) The initial slip resistance depends on the temperature, strain rate, solute content and slip mode. The last term is the Hall-Petch term that is considered only when twinning effects are taken into consideration. To calculate the debris contribution to slip resistance, the debris generation must be evaluated first using the following equation: , = ∑ (5) where q is the coefficient that represents how debris is generated from point defects via thermally activated mechanism, f is the fraction of removed dislocations (due to recovery or as defects) that are converted to debris. = = √ (6)